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Quartile in R – Efficient Ways To Calculate

We’re going to show you how to calculate a quartile in R. This is particularly useful when you’re doing exploratory analysis and reporting, especially if you’re analyzing data which may not be normally distributed.

We’re going to use the r quantile function; this utility is part of base R (so you don’t need to import any libraries) and can be adapted to generate a variety of “rank based” statistics about your sample.

To calculate a quartile in R, set the percentile as parameter of the quantile function. You can use many of the other features of the quantile function which we described in our guide on how to calculate percentile in R.

In the example below, we’re going to use a single line of code to get the quartiles of a distribution using R.

You can also use the summary function to generate the same information.

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Create a quartile column for each value in an R data frame column.

Any numerical data can be divided into four parts by using three quartiles, first quartile at 25%, second quartile at 50% and third quartile at 75% hence there will be four quarters to represent first 25%, second 25%, third 25% and the last 25% in a set of data.

If we want to create a quartile (1 to 4) column for each value in an R data frame column then we can use the quantile function and cut function as shown in the below Examples.

Following snippet creates a sample data frame −

The following dataframe is created

To create a quartile column for column x in df1 on the above created data frame, add the following code to the above snippet −

If you execute all the above given snippets as a single program, it generates the following Output −

To create a quartile column for column y in df2 on the above created data frame, add the following code to the above snippet −

Nizamuddin Siddiqui

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quantile Function in R (6 Examples)

This tutorial shows how to compute quantiles in the R programming language .

The article is mainly based on the quantile() R function. So let’s have a look at the basic R syntax and the definition of the quantile function first:

Basic R Syntax of quantile():

Definition of quantile():

The quantile function computes the sample quantiles of a numeric input vector.

In the following R tutorial, I’ll explain in six examples how to use the quantile function to compute metrics such as quartiles, quintiles, deciles, or percentiles.

Let’s dive in!

Example 1: Basic Application of quantile() in R

In the first example, I’ll illustrate how to use the quantile function in its simplest way. Let’s create an exemplifying numeric vector first:

Our example vector contains 1,000 elements between the range of 1 and 100.

Now, we can apply the quantile R function to this vector as follows:

As you can see based on the RStudio console output, the quantile function returns the cutpoints (i.e. 0%, 25%, 50%, 75%, and 100%) as well as the corresponding quantiles.

Note: By default, the quantile function is returning the quartile (i.e. five cutpoints). Later on, I’ll show you how to get other metrics as well.

However, let’s first have a look at a common problem when the quantile function is applied…

Example 2: Handling NA Values with the quantile Function

In this example, you’ll learn how to deal with missing data (i.e. NA values ) in the input vector. Let’s first insert an NA value to our example data:

Now, if we apply the quantile function to this vector, the quantile function returns an error message:

Fortunately, we can easily fix this error by specifying na.rm = TRUE within the quantile command:

Same output as in Example 1 – Perfect.

Example 3: Extract Quantile Values Only

AS you have seen based on the previous examples, the quantile function returns the cutpoints AND the corresponding values to the RStudio console. In some cases, however, we might prefer to keep only the quantile values.

In this case, we can simply apply the unname function to the output of the quantile function. Have a look at the following R code:

Based on this R code, we only get the quantile values.

Example 4: Quantile by Group

In this example I’ll show you how to calculate the quantiles of certain subgroups. For the example, I’m going to use the Iris data matrix . Let’s load the data to R:

nrow function in R - Iris Example Data Frame

Table 1: The Iris Data Frame.

The Iris data set contains several numeric variables and the grouping variable Species.

We can now produce a data matrix of quantiles of the first column grouped by the Species column with the following R syntax:

Note that it would also be possible to calculate quantiles by group based on functions of the tidyverse . This tutorial demonstrates how to calculate quantiles by group using the dplyr package.

Example 5: Quartiles, Quintiles, Deciles, Percentiles & Many More

As I told you before, the quantile function returns the quartile of the input vector by default. However, we can use the probs argument to get basically any quantile metric that we want.

With the following R codes, we can calculate the median…

…tertiles…

…quartiles (as it would also be computed by default)…

…quintiles…

…sextiles…

…septiles…

…octiles…

…deciles…

…duo-deciles…

…hexadeciles…

…ventiles…

…percentiles…

…or permilles:

Example 6: How to Visualize Quantiles

Quantiles are often used for data visualization , most of the time in so called Quantile-Quantile plots .

Quantile-Quantile plots can be created in R based on the qqplot function. Let’s do this in practice!

First, we need to create a second vector:

Now, we can print a qqplot of our two example vectors with the qqplot function as follows:

qqplot in r

Figure 1: Basic Quantile-Quantile Plot in R.

Video, Further Resources & Summary

Below, you can find a video on the Statistics Globe YouTube channel where I describe the steps of this tutorial in expanded detail:

Quantiles can be a very useful weapon in statistical research. A topic we haven’t talked about yet is the commonly used quantile regression . If you want to learn more about quantile regressions, you can have a look at the following YouTube video of Anders Munk-Nielsen:

Furthermore, you may have a look at the other R tutorials on Statistics Globe:

  • Quantile-Quantile Plot in R
  • Compute Interquartile Range (IQR) in R
  • The Empirical Cumulative Distribution Function (ecdf R Function)
  • The do.call R Function
  • R Functions List (+ Examples)
  • The R Programming Language

At this point, I hope you know how to deal with the quantile function in the R programming language. However, if you have any questions don’t hesitate to let me know in the comments section below.

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17 Comments . Leave new

' src=

do.call(“rbind”, tapply(iris$Sepal.Length, # Specify numeric column iris$Species, # Specify group variable quantile)) in this scrip how to set the row.names or how to know which row belongs to which category?

' src=

Thank you for your question. Could you elaborate your question in some more detail? I’m not sure if I understand the question correctly.

' src=

Hi, Thanks for the post! I was wondering how would you do if you want a first group which contained between 0-5% and then separate in tertiles (5-100%).

Thanks for the comment! If I understand your question correctly, then this is what is shown in Example 5 – ventiles. Does this solve your problem?

' src=

Hi Joachim, thank you for your post. It’s very useful, but i have some questions about type argument which you haven’t mention in your post.

# method for default quantile(x, probs = seq(0, 1, 0.25), na.rm = FALSE, names = TRUE, type = 7, …) type an integer between 1 and 9 selecting one of the nine quantile algorithms detailed below to be used.

i have just some informations, but i can’t understand it very well. Could you please explain to me if you know about the usage of this argument ? Thanks in advance.

Hey Solène,

First of all, thank you for the kind words! Glad you like the article!

Regarding your question: The type argument allows you to specify different algorithms for the computation of the quantiles. The different types and formulas are described in the help documentation of the quantile function.

You may open the help documentation using the code ?quantile, and then you will find a detailed description of the algorithms under the section “Type”.

Regards, Joachim

' src=

I am looking at the variable ‘thorax’ from the data ‘fruitflies’ in the faraway package. I ran the R function ‘quantile’ on ‘thorax. > data(fruitfly) > dim(fruitfly) [1] 124 3 > head(fruitfly) thorax longevity activity 1 0.68 37 many 2 0.68 49 many 3 0.72 46 many 4 0.72 63 many 5 0.76 39 many 6 0.76 46 many > Deciles Deciles 10% 20% 30% 40% 50% 60% 70% 80% 90% 0.72 0.76 0.80 0.82 0.84 0.84 0.88 0.88 0.92

Identify all the four quintiles:

I quintile = II quintile = III quintile = IV quintile =

Can you explain this and let me know the answer. I am a beginner. please help me with this.

Hey Nandini,

Quintiles can be calculated as shown in the following code:

You would have to replace x by the data you want to calculate the quintiles for.

' src=

Hi, This is super helpful. I have been trying to obtain 98th percentile from a data frame with grouping. This basically combines example 4 with the first example of 5. I tried this:

do.call(“rbind”, tapply(MT_2.5_2001$Arithmetic.Mean, MT_2.5_2001$County.Name, quantile( probs=0.98))

**No default data, so then tried this:

do.call(“rbind”, tapply(MT_2.5_2001$Arithmetic.Mean, MT_2.5_2001$County.Name, quantile(MT_2.5_2001$Arithmetic.Mean, probs=0.98)))

With this error code: Error in match.fun(FUN) : ‘quantile(MT_2.5_2001$Arithmetic.Mean, probs = 0.98)’ is not a function, character or symbol

error in match.fun(FUN) : ‘quantile(MT_2.5_2001$Arithmetic.Mean, probs = 0.98)’ is not a function, character or symbol

and tried using dplyr, which spits out one number, not grouped data

MT_2.5_2017.98% group_by(County.Name) %>% summarize(quant98 = ~quantile(Arithmetic.Mean, probs= 0.98)) MT_2.5_2017.98

Any suggestions?

Hi Bethany,

Thank you so much for the kind words, glad you find my tutorials helpful!

I apologize for the delayed reply. I was on a long vacation, so unfortunately I wasn’t able to get back to you earlier. Do you still need help with your syntax?

Hi Joachim, No problem. I either figured it out or found a work around. I am an R novice so doing a lot of learning by internet blogs like yours. Thank you!

It’s great to hear that you found a solution, and thanks a lot for the very kind words regarding my blog! 🙂

' src=

I have a continuous variable in my dataset with such a distribution:

summary(emissions$NMVOC_gram) Min. 1st Qu. Median Mean 3rd Qu. Max. 0 256 547 15802 1074 50818630 how can i categorize this variable to unequal levels of extremely high to extremely low, low, high, and medium in R or excel?

thank you for the help

I tried the cut function in r but the result was not what I expected, actually, I do not know how I should define the breaks, in my data the 3rd Qu. is lower than the Mean.

' src=

Sorry for the late response. Could you find a solution? Here is what I found by a small search. You can split your values into quantiles via the split_quantiles() function of the fabricatr package . Then you can rename your values. See below:

Regards, Cansu

thank you very much for your help,

' src=

how to get a particular quantile value like q1 or q3? help is much appreciated.

Hello Rokesh,

You can use the same code setting shown in the tutorial as follows:

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There are several quartiles of an observation variable. The first quartile , or lower quartile , is the value that cuts off the first 25% of the data when it is sorted in ascending order. The second quartile , or median , is the value that cuts off the first 50%. The third quartile , or upper quartile , is the value that cuts off the first 75%.

Find the quartiles of the eruption durations in the data set faithful .

We apply the quantile function to compute the quartiles of eruptions .

The first, second and third quartiles of the eruption duration are 2.1627, 4.0000 and 4.4543 minutes respectively.

Find the quartiles of the eruption waiting periods in faithful .

There are several algorithms for the computation of quartiles. Details can be found in the R documentation via help(quantile) .

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Quartile in statistics: detailed overview with solved examples.

Posted on January 29, 2022 by finnstats in R bloggers | 0 Comments

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If you want to read the original article, click here Quartile in Statistics: Detailed overview with solved examples .

Quartiles are widely used in statistics to divide the given set of values into four equal parts. These four terms of the quartile are used to find the first, second, and third quartile which are widely used in the five-number summary.

The main purpose of the quartile is to calculate the interquartile range of the given set of data. Using the quartile median of the set can also be calculated easily. The interquartile range is used to measure the variability around the median of the given set of data.

In this article, we will go through the definition and formulas of quartile with a lot of examples.

Artificial Intelligence and Data Science » finnstats

What is a quartile?

A term that divides the given set of numbers into four equal parts or quarters is known as a quartile . These four parts are the first quartile, second quartile or median, third quartile, and interquartile. The interquartile range is used to determine the difference between the third and the first quartiles.

To measure the central point of the given set of data second quartile is used which is 50% of the given data. The lower and upper parts or the first and third quartiles are used to get information set before and after the median respectively.

First of all, arrange the given set of data in ascending order then take the middlemost value that is the median. The lower half of the set is the first quartile and the upper half is the third quartile. The difference between the lower and upper half can be identified by using the interquartile range.

r assign quartile

Formulas of quartile

There are four basic formulas of the quartile used to find the first, second, third, and inter quartiles.

  • For the first quartile or Q 1 .

First quartile = Q 1 = ((n + 1) / 4) th term

  • For the second quartile or Q 2 .

Second quartile = Q 2 = ((n + 1) / 2) th term

  • For the third quartile or Q 3 .

Third quartile = Q 3 = (3(n + 1) / 4) th term

  • For interquartile.

Interquartile = Q 3 – Q 1 = (3(n + 1) / 4) th term – ((n + 1) / 4) th term

By using the above three formulas for the first, second, and third quartiles, we can write a general formula to calculate the quartile.

Ogive curve in R » Quick Guide » finnstats

Q k = k (n + 1) / 4) th term

Where k = 1, 2, 3

How to calculate quartile?

By using formulas, we can easily calculate quartile .

Evaluate all parts of the quartile of the given set of data, 2, 9, 7, 29, 34, 61, 25, 19, 16?

Step 1: Take the given set of numbers.

2, 9, 7, 29, 34, 61, 25, 19, 16

Step 2: Arrange the given set of numbers according to ascending order.

2, 7, 9, 16, 19, 25, 29, 34, 61

Step 3: Now count the given set of numbers and put it equal to n.

Step 4: Now take the general formula of the quartile to find the first, second, and third quartiles.

Step 5: Put k = 1, 2, 3 one by one to calculate the first, second, and third quartiles.

Q 1 = 1 (9 + 1) / 4) th term

Q 1 = 1 (10) / 4) th term

Q 1 = (10) / 4) th term

Q 1 = (5) / 2) th term

Q 1 = 2.5 th term

Q 2 = 2 (9 + 1) / 4) th term

Q 2 = 2 (10) / 4) th term

Q 2 = (10 / 2) th term

Q 2 = 5 th term

Q 3 = 3 (9 + 1) / 4) th term

Q 3 = 3 (10) / 4) th term

Q 3 = (30 / 4) th term

Q 3 = (15 / 2) th term

Q 3 = 7.5 th term

Step 6: Now take the calculated values from the arranged data set.

Q 1 = 2 nd term + 3 rd term / 2

Q 1 = 7 + 9/2

Q 3 = 7 th + 8 th / 2

Q 3 = 29 + 34 / 2

Step 7: Now take the general formula to calculate interquartile and put the values.

Interquartile = Q 3 – Q 1

Interquartile = 31.5 – 8

Interquartile = 23.5

Hence, the quartiles of the given set are Q 1 = 8. Q 2 = 19, Q 3 = 31.5, and interquartile = 23.5

Intro to Tensorflow-Machine Learning with TensorFlow » finnstats

Find the interquartile of the given set of data, 23, 19, 3, 12, 22, 18, 11?

23, 19, 3, 12, 22, 18, 11

3, 11, 12, 18, 19, 22, 23

Step 4: Now take the general formula of the interquartile.

Step 5: Now calculate the first and third quartile.

Q 1 = (n + 1) / 4) th term

Q 1 = (7 + 1) / 4) th term

Q 1 = (8) / 4) th term

Q 1 = 2 nd term

Q 3 = 3(n + 1) / 4) th term

Q 3 = 3(7 + 1) / 4) th term

Q 3 = 3(8) / 4) th term

Q 3 = (24 / 4) th term

Q 3 = 6 th term

Step 6: Put the result of the third and first quartile in the interquartile formula.

Interquartile = 6 th term – 2 nd term

Interquartile = 22 – 11

Interquartile = 11

Now you can grab all the basic concepts related to quartile just by following this article. All the problems of the quartile can easily be solved by using the above-mentioned formulas. Once you practice the above examples, you will be able to solve any problem related to this topic.

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Quantiles in R

Quantiles and percentiles in R

Considering a value \(p\) , being \(0 < p < 1\) the quantile of order \(p\) is the value that leaves a proportion of the data below ( \(p\) ) and the rest \((1-p)\) above that value. Notice that quantiles are the generalization of the median which is the quantile for \(p = 0.5\) . In R, you can make use of the quantile function to calculate any quantile for any numeric vector.

The quantile function calculates the sample quantiles of a numeric vector ( x ). By default, this function calculates the quartiles specified inside probs , but you can also input any other probabilities to compute any percentile.

Quartiles are quantiles of order 0.25, 0.5 and 0.75 and they divide the sample into four parts with the same frequency . Usually, quartiles are denoted by \(Q_1\) , \(Q_2\) and \(Q_3\) .

Recall that the quartile 0.5 is equal to the median :

Note that you can remove the name attributes from the output setting names = FALSE .

Remove missing values

If your numeric vector contains missing values you won’t be able to calculate the quantiles, so you will need to set na.rm = TRUE to remove the missing values before the calculation.

Quantile algorithms

The calculation of the quantiles are based on one of the nine algorithms discussed in Hyndman and Fan (1996) . By default, the seventh algorithm is used, but you can select other passing an integer between 1 and 9 to type . Read the previous reference for further information about each algorithm.

Visual representation

It is important to note that a box plot can be used to visualize quartiles, but the method used inside the boxplot function is not the same as the one used inside quartile , so the output may vary slightly.

Box plot quartiles in R

Deciles are quantiles of order 0.1, 0.2, …, 0.9 and divide the sample into 10 equal-frequency parts . In order to calculate them you can input a sequence from 0 to 1 by 0.1 to probs , as shown in the example below.

Percentiles

Percentiles are quantiles of the order 0.01, 0.02, … , 0.99 and divide the sample into 100 equal-frequency parts . If you want to calculate the percentiles of a numeric vector you will need to specify a sequence from 0 to 1 by 0.01 inside probs .

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quartile_: Quartiles Calculus

Description.

Calculates the 3 Quartiles of a vector of data

A vector sorted with the elements divided by 4 parts

Should be a vector

Dennis Monheimius, [email protected] Eduardo Benito, [email protected] Juan Jose Cuadrado, [email protected] Universidad de Alcala de Henares

r assign quartile

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Calculate Quartiles in R

Quartiles are values which divide a dataset into four equal parts, each of which contains 25% of the data. It is helpful to understand the spread and distribution of a dataset by using quartiles.

In general, there are three quartiles (Q1, Q2, and Q3) used. Q1 (first quartile), Q2 (second quartile), and Q3 (third quartile) are the values below which 25%, 50%, and 75% of the data fall.

calculate quartiles in R

In R, the quartiles can be calculated using the built-in quantile() function.

The general syntax of quantile() looks like this:

Where, x is a vector of the dataset.

The following three examples explains how to use the quantile() function from R to calculate quartiles from vector and data frame.

Suppose, you have a following dataset for which you would like to calculate the quartiles,

Calculate the quartiles using quantile() function,

From the output, you can see that Q1, Q2, and Q3 quartile values are 53, 62.5, and 67, respectively.

Suppose, you have a dataset in a data frame format.

You can also visualize the quartiles using the boxplot. The boxplot helps to visualize the spread and distribution of the data.

Create a boxplot,

quartiles from boxplot in R

Using the boxplot, we can estimate the quartile locations.

Minimum value or Q0 (43) is indicated by the bottom whisker, Q1 (53) is indicated by the bottom line of box, median value or Q2 (62.5) is indicated by the middle dark line, and maximum value or Q4 (82) is indicated by the top whisker.

Related : Calculate quartiles in Python

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Quantile,Percentile and Decile Rank in R using dplyr

Quantile, Decile and Percentile rank can be calculated using ntile() Function in R. Dplyr package is provided with mutate() function and ntile() function. The ntile() function is used to divide the data into N bins there by providing ntile rank. If the data is divided into 100 bins by ntile() , percentile rank in R is calculated on a particular column. similarly if the data is divided into 4 and 10 bins by ntile() function it will result in quantile and decile rank in R. In this example we will be creating the column with percentile, decile and quantile rank in R by descending order and by group.

  • Decile rank of the column in R using ntile() function
  • Quantile rank of the column in R
  • Percentile rank in R of the particular column using ntile().
  • Decile rank, quantile rank and percentile rank by descending order in R
  • Percentile rank, quantile rank and decile rank of a group in R.

Let’s First Create the dataframe

we will be using the following my_basket data frame

Calculate percentile, quantile, decile rank of the column in R N tile 1

Quantile rank in R:

We will be using my_basket data to depict the example of ntile() function. ntile() function takes column name and 4 as argument which in turn calculates the quantile ranking of the column in R.(i.e. the ranking ranges from 1 to 4)

So in the resultant data frame quantile rank is calculated and populated across

Calculate percentile, quantile, decile rank of the column in R N tile 5

Quantile rank of the column in descending order in R:

 ntile() function along with the descending() function, takes column name and 4 as argument which inturn calculates the quantile ranking of the column in descending order in R.(i.e. the ranking ranges from 1 to 4)

So the resultant data frame with quantile rank calculated in descending order will be

Calculate percentile, quantile, decile rank of the column in R N tile 6

Quantile rank of the column by group in R:

 ntile() function along with the group_by() function of dplyr package, groups the column and provides quantile ranking of the “Price” column within that group as shown below.

So, the resultant data frame with quantile rank calculated by group will be

Calculate percentile, quantile, decile rank of the column in R N tile 7

Decile rank in R:

ntile() function takes column name and 10 as argument which inturn calculates the decile ranking of the column in R.(i.e. the ranking ranges from 1 to 10)

So in the resultant data frame decile rank is calculated and populated across

Calculate percentile, quantile, decile rank of the column in R N tile 2

Decile rank of the column in descending order in R:

 ntile() function along with the descending() function, takes column name and 10 as argument which inturn calculates the decile ranking of the column in descending order in R.(i.e. the ranking ranges from 1 to 10)

So the resultant data frame with decile rank calculated in descending order will be

Calculate percentile, quantile, decile rank of the column in R N tile 3

Decile rank of the column by group in R:

 ntile() function along with the group_by() function of dplyr package, groups the column and provides decile ranking of the “Price” column within that group as shown below.

So, the resultant data frame with decile rank calculated by group will be

Calculate percentile, quantile, decile rank of the column in R N tile 4

Percentile rank in R:

We will be using my_basket data to depict the example of ntile() function. ntile() function takes column name and 100 as argument which in turn calculates the percentile ranking of the column in R.(i.e. the ranking ranges from 1 to 100)

So in the resultant data frame percentile rank is calculated and populated across

Calculate percentile, quantile, decile rank of the column in R N tile 8

percentile rank of the column in descending order in R:

 ntile() function along with the descending() function, takes column name and 100 as argument which inturn calculates the percentile ranking of the column in descending order in R.(i.e. the ranking ranges from 1 to 100)

So the resultant data frame with percentile rank calculated in descending order will be

Calculate percentile, quantile, decile rank of the column in R N tile 9

Percentile rank of the column by group in R:

 ntile() function along with the group_by() function of dplyr package, groups the column and provides percentile ranking of the “Price” column within that group as shown below.

So, the resultant data frame with percentile rank calculated by group will be

Calculate percentile, quantile, decile rank of the column in R N tile 10

With close to 10 years on Experience in data science and machine learning Have extensively worked on programming languages like R, Python (Pandas), SAS, Pyspark.

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Frequently asked questions

How do i find quartiles in r.

You can use the quantile() function to find quartiles in R. If your data is called “data”, then “quantile(data, prob=c(.25,.5,.75), type=1)” will return the three quartiles.

Frequently asked questions: Statistics

As the degrees of freedom increase, Student’s t distribution becomes less leptokurtic , meaning that the probability of extreme values decreases. The distribution becomes more and more similar to a standard normal distribution .

The three categories of kurtosis are:

  • Mesokurtosis : An excess kurtosis of 0. Normal distributions are mesokurtic.
  • Platykurtosis : A negative excess kurtosis. Platykurtic distributions are thin-tailed, meaning that they have few outliers .
  • Leptokurtosis : A positive excess kurtosis. Leptokurtic distributions are fat-tailed, meaning that they have many outliers.

Probability distributions belong to two broad categories: discrete probability distributions and continuous probability distributions . Within each category, there are many types of probability distributions.

Probability is the relative frequency over an infinite number of trials.

For example, the probability of a coin landing on heads is .5, meaning that if you flip the coin an infinite number of times, it will land on heads half the time.

Since doing something an infinite number of times is impossible, relative frequency is often used as an estimate of probability. If you flip a coin 1000 times and get 507 heads, the relative frequency, .507, is a good estimate of the probability.

Categorical variables can be described by a frequency distribution. Quantitative variables can also be described by a frequency distribution, but first they need to be grouped into interval classes .

A histogram is an effective way to tell if a frequency distribution appears to have a normal distribution .

Plot a histogram and look at the shape of the bars. If the bars roughly follow a symmetrical bell or hill shape, like the example below, then the distribution is approximately normally distributed.

Frequency-distribution-Normal-distribution

You can use the CHISQ.INV.RT() function to find a chi-square critical value in Excel.

For example, to calculate the chi-square critical value for a test with df = 22 and α = .05, click any blank cell and type:

=CHISQ.INV.RT(0.05,22)

You can use the qchisq() function to find a chi-square critical value in R.

For example, to calculate the chi-square critical value for a test with df = 22 and α = .05:

qchisq(p = .05, df = 22, lower.tail = FALSE)

You can use the chisq.test() function to perform a chi-square test of independence in R. Give the contingency table as a matrix for the “x” argument. For example:

m = matrix(data = c(89, 84, 86, 9, 8, 24), nrow = 3, ncol = 2)

chisq.test(x = m)

You can use the CHISQ.TEST() function to perform a chi-square test of independence in Excel. It takes two arguments, CHISQ.TEST(observed_range, expected_range), and returns the p value.

Chi-square goodness of fit tests are often used in genetics. One common application is to check if two genes are linked (i.e., if the assortment is independent). When genes are linked, the allele inherited for one gene affects the allele inherited for another gene.

Suppose that you want to know if the genes for pea texture (R = round, r = wrinkled) and color (Y = yellow, y = green) are linked. You perform a dihybrid cross between two heterozygous ( RY / ry ) pea plants. The hypotheses you’re testing with your experiment are:

  • This would suggest that the genes are unlinked.
  • This would suggest that the genes are linked.

You observe 100 peas:

  • 78 round and yellow peas
  • 6 round and green peas
  • 4 wrinkled and yellow peas
  • 12 wrinkled and green peas

Step 1: Calculate the expected frequencies

To calculate the expected values, you can make a Punnett square. If the two genes are unlinked, the probability of each genotypic combination is equal.

The expected phenotypic ratios are therefore 9 round and yellow: 3 round and green: 3 wrinkled and yellow: 1 wrinkled and green.

From this, you can calculate the expected phenotypic frequencies for 100 peas:

Step 2: Calculate chi-square

Χ 2 = 8.41 + 8.67 + 11.6 + 5.4 = 34.08

Step 3: Find the critical chi-square value

Since there are four groups (round and yellow, round and green, wrinkled and yellow, wrinkled and green), there are three degrees of freedom .

For a test of significance at α = .05 and df = 3, the Χ 2 critical value is 7.82.

Step 4: Compare the chi-square value to the critical value

Χ 2 = 34.08

Critical value = 7.82

The Χ 2 value is greater than the critical value .

Step 5: Decide whether the reject the null hypothesis

The Χ 2 value is greater than the critical value, so we reject the null hypothesis that the population of offspring have an equal probability of inheriting all possible genotypic combinations. There is a significant difference between the observed and expected genotypic frequencies ( p < .05).

The data supports the alternative hypothesis that the offspring do not have an equal probability of inheriting all possible genotypic combinations, which suggests that the genes are linked

You can use the chisq.test() function to perform a chi-square goodness of fit test in R. Give the observed values in the “x” argument, give the expected values in the “p” argument, and set “rescale.p” to true. For example:

chisq.test(x = c(22,30,23), p = c(25,25,25), rescale.p = TRUE)

You can use the CHISQ.TEST() function to perform a chi-square goodness of fit test in Excel. It takes two arguments, CHISQ.TEST(observed_range, expected_range), and returns the p value .

Both correlations and chi-square tests can test for relationships between two variables. However, a correlation is used when you have two quantitative variables and a chi-square test of independence is used when you have two categorical variables.

Both chi-square tests and t tests can test for differences between two groups. However, a t test is used when you have a dependent quantitative variable and an independent categorical variable (with two groups). A chi-square test of independence is used when you have two categorical variables.

The two main chi-square tests are the chi-square goodness of fit test and the chi-square test of independence .

A chi-square distribution is a continuous probability distribution . The shape of a chi-square distribution depends on its degrees of freedom , k . The mean of a chi-square distribution is equal to its degrees of freedom ( k ) and the variance is 2 k . The range is 0 to ∞.

As the degrees of freedom ( k ) increases, the chi-square distribution goes from a downward curve to a hump shape. As the degrees of freedom increases further, the hump goes from being strongly right-skewed to being approximately normal.

To find the quartiles of a probability distribution, you can use the distribution’s quantile function.

You can use the QUARTILE() function to find quartiles in Excel. If your data is in column A, then click any blank cell and type “=QUARTILE(A:A,1)” for the first quartile, “=QUARTILE(A:A,2)” for the second quartile, and “=QUARTILE(A:A,3)” for the third quartile.

You can use the PEARSON() function to calculate the Pearson correlation coefficient in Excel. If your variables are in columns A and B, then click any blank cell and type “PEARSON(A:A,B:B)”.

There is no function to directly test the significance of the correlation.

You can use the cor() function to calculate the Pearson correlation coefficient in R. To test the significance of the correlation, you can use the cor.test() function.

You should use the Pearson correlation coefficient when (1) the relationship is linear and (2) both variables are quantitative and (3) normally distributed and (4) have no outliers.

The Pearson correlation coefficient ( r ) is the most common way of measuring a linear correlation. It is a number between –1 and 1 that measures the strength and direction of the relationship between two variables.

This table summarizes the most important differences between normal distributions and Poisson distributions :

When the mean of a Poisson distribution is large (>10), it can be approximated by a normal distribution.

In the Poisson distribution formula, lambda (λ) is the mean number of events within a given interval of time or space. For example, λ = 0.748 floods per year.

The e in the Poisson distribution formula stands for the number 2.718. This number is called Euler’s constant. You can simply substitute e with 2.718 when you’re calculating a Poisson probability. Euler’s constant is a very useful number and is especially important in calculus.

The three types of skewness are:

  • Right skew (also called positive skew ) . A right-skewed distribution is longer on the right side of its peak than on its left.
  • Left skew (also called negative skew). A left-skewed distribution is longer on the left side of its peak than on its right.
  • Zero skew. It is symmetrical and its left and right sides are mirror images.

Skewness of a distribution

Skewness and kurtosis are both important measures of a distribution’s shape.

  • Skewness measures the asymmetry of a distribution.
  • Kurtosis measures the heaviness of a distribution’s tails relative to a normal distribution .

Difference between skewness and kurtosis

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (“ x affects y because …”).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses . In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).

The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).

The t distribution was first described by statistician William Sealy Gosset under the pseudonym “Student.”

To calculate a confidence interval of a mean using the critical value of t , follow these four steps:

  • Choose the significance level based on your desired confidence level. The most common confidence level is 95%, which corresponds to α = .05 in the two-tailed t table .
  • Find the critical value of t in the two-tailed t table.
  • Multiply the critical value of t by s / √ n .
  • Add this value to the mean to calculate the upper limit of the confidence interval, and subtract this value from the mean to calculate the lower limit.

To test a hypothesis using the critical value of t , follow these four steps:

  • Calculate the t value for your sample.
  • Find the critical value of t in the t table .
  • Determine if the (absolute) t value is greater than the critical value of t .
  • Reject the null hypothesis if the sample’s t value is greater than the critical value of t . Otherwise, don’t reject the null hypothesis .

You can use the T.INV() function to find the critical value of t for one-tailed tests in Excel, and you can use the T.INV.2T() function for two-tailed tests.

You can use the qt() function to find the critical value of t in R. The function gives the critical value of t for the one-tailed test. If you want the critical value of t for a two-tailed test, divide the significance level by two.

You can use the RSQ() function to calculate R² in Excel. If your dependent variable is in column A and your independent variable is in column B, then click any blank cell and type “RSQ(A:A,B:B)”.

You can use the summary() function to view the R²  of a linear model in R. You will see the “R-squared” near the bottom of the output.

There are two formulas you can use to calculate the coefficient of determination (R²) of a simple linear regression .

R^2=(r)^2

The coefficient of determination (R²) is a number between 0 and 1 that measures how well a statistical model predicts an outcome. You can interpret the R² as the proportion of variation in the dependent variable that is predicted by the statistical model.

There are three main types of missing data .

Missing completely at random (MCAR) data are randomly distributed across the variable and unrelated to other variables .

Missing at random (MAR) data are not randomly distributed but they are accounted for by other observed variables.

Missing not at random (MNAR) data systematically differ from the observed values.

To tidy up your missing data , your options usually include accepting, removing, or recreating the missing data.

  • Acceptance: You leave your data as is
  • Listwise or pairwise deletion: You delete all cases (participants) with missing data from analyses
  • Imputation: You use other data to fill in the missing data

Missing data are important because, depending on the type, they can sometimes bias your results. This means your results may not be generalizable outside of your study because your data come from an unrepresentative sample .

Missing data , or missing values, occur when you don’t have data stored for certain variables or participants.

In any dataset, there’s usually some missing data. In quantitative research , missing values appear as blank cells in your spreadsheet.

There are two steps to calculating the geometric mean :

  • Multiply all values together to get their product.
  • Find the n th root of the product ( n is the number of values).

Before calculating the geometric mean, note that:

  • The geometric mean can only be found for positive values.
  • If any value in the data set is zero, the geometric mean is zero.

The arithmetic mean is the most commonly used type of mean and is often referred to simply as “the mean.” While the arithmetic mean is based on adding and dividing values, the geometric mean multiplies and finds the root of values.

Even though the geometric mean is a less common measure of central tendency , it’s more accurate than the arithmetic mean for percentage change and positively skewed data. The geometric mean is often reported for financial indices and population growth rates.

The geometric mean is an average that multiplies all values and finds a root of the number. For a dataset with n numbers, you find the n th root of their product.

Outliers are extreme values that differ from most values in the dataset. You find outliers at the extreme ends of your dataset.

It’s best to remove outliers only when you have a sound reason for doing so.

Some outliers represent natural variations in the population , and they should be left as is in your dataset. These are called true outliers.

Other outliers are problematic and should be removed because they represent measurement errors , data entry or processing errors, or poor sampling.

You can choose from four main ways to detect outliers :

  • Sorting your values from low to high and checking minimum and maximum values
  • Visualizing your data with a box plot and looking for outliers
  • Using the interquartile range to create fences for your data
  • Using statistical procedures to identify extreme values

Outliers can have a big impact on your statistical analyses and skew the results of any hypothesis test if they are inaccurate.

These extreme values can impact your statistical power as well, making it hard to detect a true effect if there is one.

No, the steepness or slope of the line isn’t related to the correlation coefficient value. The correlation coefficient only tells you how closely your data fit on a line, so two datasets with the same correlation coefficient can have very different slopes.

To find the slope of the line, you’ll need to perform a regression analysis .

Correlation coefficients always range between -1 and 1.

The sign of the coefficient tells you the direction of the relationship: a positive value means the variables change together in the same direction, while a negative value means they change together in opposite directions.

The absolute value of a number is equal to the number without its sign. The absolute value of a correlation coefficient tells you the magnitude of the correlation: the greater the absolute value, the stronger the correlation.

These are the assumptions your data must meet if you want to use Pearson’s r :

  • Both variables are on an interval or ratio level of measurement
  • Data from both variables follow normal distributions
  • Your data have no outliers
  • Your data is from a random or representative sample
  • You expect a linear relationship between the two variables

A correlation coefficient is a single number that describes the strength and direction of the relationship between your variables.

Different types of correlation coefficients might be appropriate for your data based on their levels of measurement and distributions . The Pearson product-moment correlation coefficient (Pearson’s r ) is commonly used to assess a linear relationship between two quantitative variables.

There are various ways to improve power:

  • Increase the potential effect size by manipulating your independent variable more strongly,
  • Increase sample size,
  • Increase the significance level (alpha),
  • Reduce measurement error by increasing the precision and accuracy of your measurement devices and procedures,
  • Use a one-tailed test instead of a two-tailed test for t tests and z tests.

A power analysis is a calculation that helps you determine a minimum sample size for your study. It’s made up of four main components. If you know or have estimates for any three of these, you can calculate the fourth component.

  • Statistical power : the likelihood that a test will detect an effect of a certain size if there is one, usually set at 80% or higher.
  • Sample size : the minimum number of observations needed to observe an effect of a certain size with a given power level.
  • Significance level (alpha) : the maximum risk of rejecting a true null hypothesis that you are willing to take, usually set at 5%.
  • Expected effect size : a standardized way of expressing the magnitude of the expected result of your study, usually based on similar studies or a pilot study.

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

Statistical analysis is the main method for analyzing quantitative research data . It uses probabilities and models to test predictions about a population from sample data.

The risk of making a Type II error is inversely related to the statistical power of a test. Power is the extent to which a test can correctly detect a real effect when there is one.

To (indirectly) reduce the risk of a Type II error, you can increase the sample size or the significance level to increase statistical power.

The risk of making a Type I error is the significance level (or alpha) that you choose. That’s a value that you set at the beginning of your study to assess the statistical probability of obtaining your results ( p value ).

The significance level is usually set at 0.05 or 5%. This means that your results only have a 5% chance of occurring, or less, if the null hypothesis is actually true.

To reduce the Type I error probability, you can set a lower significance level.

In statistics, a Type I error means rejecting the null hypothesis when it’s actually true, while a Type II error means failing to reject the null hypothesis when it’s actually false.

In statistics, power refers to the likelihood of a hypothesis test detecting a true effect if there is one. A statistically powerful test is more likely to reject a false negative (a Type II error).

If you don’t ensure enough power in your study, you may not be able to detect a statistically significant result even when it has practical significance. Your study might not have the ability to answer your research question.

While statistical significance shows that an effect exists in a study, practical significance shows that the effect is large enough to be meaningful in the real world.

Statistical significance is denoted by p -values whereas practical significance is represented by effect sizes .

There are dozens of measures of effect sizes . The most common effect sizes are Cohen’s d and Pearson’s r . Cohen’s d measures the size of the difference between two groups while Pearson’s r measures the strength of the relationship between two variables .

Effect size tells you how meaningful the relationship between variables or the difference between groups is.

A large effect size means that a research finding has practical significance, while a small effect size indicates limited practical applications.

Using descriptive and inferential statistics , you can make two types of estimates about the population : point estimates and interval estimates.

  • A point estimate is a single value estimate of a parameter . For instance, a sample mean is a point estimate of a population mean.
  • An interval estimate gives you a range of values where the parameter is expected to lie. A confidence interval is the most common type of interval estimate.

Both types of estimates are important for gathering a clear idea of where a parameter is likely to lie.

Standard error and standard deviation are both measures of variability . The standard deviation reflects variability within a sample, while the standard error estimates the variability across samples of a population.

The standard error of the mean , or simply standard error , indicates how different the population mean is likely to be from a sample mean. It tells you how much the sample mean would vary if you were to repeat a study using new samples from within a single population.

To figure out whether a given number is a parameter or a statistic , ask yourself the following:

  • Does the number describe a whole, complete population where every member can be reached for data collection ?
  • Is it possible to collect data for this number from every member of the population in a reasonable time frame?

If the answer is yes to both questions, the number is likely to be a parameter. For small populations, data can be collected from the whole population and summarized in parameters.

If the answer is no to either of the questions, then the number is more likely to be a statistic.

The arithmetic mean is the most commonly used mean. It’s often simply called the mean or the average. But there are some other types of means you can calculate depending on your research purposes:

  • Weighted mean: some values contribute more to the mean than others.
  • Geometric mean : values are multiplied rather than summed up.
  • Harmonic mean: reciprocals of values are used instead of the values themselves.

You can find the mean , or average, of a data set in two simple steps:

  • Find the sum of the values by adding them all up.
  • Divide the sum by the number of values in the data set.

This method is the same whether you are dealing with sample or population data or positive or negative numbers.

The median is the most informative measure of central tendency for skewed distributions or distributions with outliers. For example, the median is often used as a measure of central tendency for income distributions, which are generally highly skewed.

Because the median only uses one or two values, it’s unaffected by extreme outliers or non-symmetric distributions of scores. In contrast, the mean and mode can vary in skewed distributions.

To find the median , first order your data. Then calculate the middle position based on n , the number of values in your data set.

\dfrac{(n+1)}{2}

A data set can often have no mode, one mode or more than one mode – it all depends on how many different values repeat most frequently.

Your data can be:

  • without any mode
  • unimodal, with one mode,
  • bimodal, with two modes,
  • trimodal, with three modes, or
  • multimodal, with four or more modes.

To find the mode :

  • If your data is numerical or quantitative, order the values from low to high.
  • If it is categorical, sort the values by group, in any order.

Then you simply need to identify the most frequently occurring value.

The interquartile range is the best measure of variability for skewed distributions or data sets with outliers. Because it’s based on values that come from the middle half of the distribution, it’s unlikely to be influenced by outliers .

The two most common methods for calculating interquartile range are the exclusive and inclusive methods.

The exclusive method excludes the median when identifying Q1 and Q3, while the inclusive method includes the median as a value in the data set in identifying the quartiles.

For each of these methods, you’ll need different procedures for finding the median, Q1 and Q3 depending on whether your sample size is even- or odd-numbered. The exclusive method works best for even-numbered sample sizes, while the inclusive method is often used with odd-numbered sample sizes.

While the range gives you the spread of the whole data set, the interquartile range gives you the spread of the middle half of a data set.

Homoscedasticity, or homogeneity of variances, is an assumption of equal or similar variances in different groups being compared.

This is an important assumption of parametric statistical tests because they are sensitive to any dissimilarities. Uneven variances in samples result in biased and skewed test results.

Statistical tests such as variance tests or the analysis of variance (ANOVA) use sample variance to assess group differences of populations. They use the variances of the samples to assess whether the populations they come from significantly differ from each other.

Variance is the average squared deviations from the mean, while standard deviation is the square root of this number. Both measures reflect variability in a distribution, but their units differ:

  • Standard deviation is expressed in the same units as the original values (e.g., minutes or meters).
  • Variance is expressed in much larger units (e.g., meters squared).

Although the units of variance are harder to intuitively understand, variance is important in statistical tests .

The empirical rule, or the 68-95-99.7 rule, tells you where most of the values lie in a normal distribution :

  • Around 68% of values are within 1 standard deviation of the mean.
  • Around 95% of values are within 2 standard deviations of the mean.
  • Around 99.7% of values are within 3 standard deviations of the mean.

The empirical rule is a quick way to get an overview of your data and check for any outliers or extreme values that don’t follow this pattern.

In a normal distribution , data are symmetrically distributed with no skew. Most values cluster around a central region, with values tapering off as they go further away from the center.

The measures of central tendency (mean, mode, and median) are exactly the same in a normal distribution.

Normal distribution

The standard deviation is the average amount of variability in your data set. It tells you, on average, how far each score lies from the mean .

In normal distributions, a high standard deviation means that values are generally far from the mean, while a low standard deviation indicates that values are clustered close to the mean.

No. Because the range formula subtracts the lowest number from the highest number, the range is always zero or a positive number.

In statistics, the range is the spread of your data from the lowest to the highest value in the distribution. It is the simplest measure of variability .

While central tendency tells you where most of your data points lie, variability summarizes how far apart your points from each other.

Data sets can have the same central tendency but different levels of variability or vice versa . Together, they give you a complete picture of your data.

Variability is most commonly measured with the following descriptive statistics :

  • Range : the difference between the highest and lowest values
  • Interquartile range : the range of the middle half of a distribution
  • Standard deviation : average distance from the mean
  • Variance : average of squared distances from the mean

Variability tells you how far apart points lie from each other and from the center of a distribution or a data set.

Variability is also referred to as spread, scatter or dispersion.

While interval and ratio data can both be categorized, ranked, and have equal spacing between adjacent values, only ratio scales have a true zero.

For example, temperature in Celsius or Fahrenheit is at an interval scale because zero is not the lowest possible temperature. In the Kelvin scale, a ratio scale, zero represents a total lack of thermal energy.

A critical value is the value of the test statistic which defines the upper and lower bounds of a confidence interval , or which defines the threshold of statistical significance in a statistical test. It describes how far from the mean of the distribution you have to go to cover a certain amount of the total variation in the data (i.e. 90%, 95%, 99%).

If you are constructing a 95% confidence interval and are using a threshold of statistical significance of p = 0.05, then your critical value will be identical in both cases.

The t -distribution gives more probability to observations in the tails of the distribution than the standard normal distribution (a.k.a. the z -distribution).

In this way, the t -distribution is more conservative than the standard normal distribution: to reach the same level of confidence or statistical significance , you will need to include a wider range of the data.

A t -score (a.k.a. a t -value) is equivalent to the number of standard deviations away from the mean of the t -distribution .

The t -score is the test statistic used in t -tests and regression tests. It can also be used to describe how far from the mean an observation is when the data follow a t -distribution.

The t -distribution is a way of describing a set of observations where most observations fall close to the mean , and the rest of the observations make up the tails on either side. It is a type of normal distribution used for smaller sample sizes, where the variance in the data is unknown.

The t -distribution forms a bell curve when plotted on a graph. It can be described mathematically using the mean and the standard deviation .

In statistics, ordinal and nominal variables are both considered categorical variables .

Even though ordinal data can sometimes be numerical, not all mathematical operations can be performed on them.

Ordinal data has two characteristics:

  • The data can be classified into different categories within a variable.
  • The categories have a natural ranked order.

However, unlike with interval data, the distances between the categories are uneven or unknown.

Nominal and ordinal are two of the four levels of measurement . Nominal level data can only be classified, while ordinal level data can be classified and ordered.

Nominal data is data that can be labelled or classified into mutually exclusive categories within a variable. These categories cannot be ordered in a meaningful way.

For example, for the nominal variable of preferred mode of transportation, you may have the categories of car, bus, train, tram or bicycle.

If your confidence interval for a difference between groups includes zero, that means that if you run your experiment again you have a good chance of finding no difference between groups.

If your confidence interval for a correlation or regression includes zero, that means that if you run your experiment again there is a good chance of finding no correlation in your data.

In both of these cases, you will also find a high p -value when you run your statistical test, meaning that your results could have occurred under the null hypothesis of no relationship between variables or no difference between groups.

If you want to calculate a confidence interval around the mean of data that is not normally distributed , you have two choices:

  • Find a distribution that matches the shape of your data and use that distribution to calculate the confidence interval.
  • Perform a transformation on your data to make it fit a normal distribution, and then find the confidence interval for the transformed data.

The standard normal distribution , also called the z -distribution, is a special normal distribution where the mean is 0 and the standard deviation is 1.

Any normal distribution can be converted into the standard normal distribution by turning the individual values into z -scores. In a z -distribution, z -scores tell you how many standard deviations away from the mean each value lies.

The z -score and t -score (aka z -value and t -value) show how many standard deviations away from the mean of the distribution you are, assuming your data follow a z -distribution or a t -distribution .

These scores are used in statistical tests to show how far from the mean of the predicted distribution your statistical estimate is. If your test produces a z -score of 2.5, this means that your estimate is 2.5 standard deviations from the predicted mean.

The predicted mean and distribution of your estimate are generated by the null hypothesis of the statistical test you are using. The more standard deviations away from the predicted mean your estimate is, the less likely it is that the estimate could have occurred under the null hypothesis .

To calculate the confidence interval , you need to know:

  • The point estimate you are constructing the confidence interval for
  • The critical values for the test statistic
  • The standard deviation of the sample
  • The sample size

Then you can plug these components into the confidence interval formula that corresponds to your data. The formula depends on the type of estimate (e.g. a mean or a proportion) and on the distribution of your data.

The confidence level is the percentage of times you expect to get close to the same estimate if you run your experiment again or resample the population in the same way.

The confidence interval consists of the upper and lower bounds of the estimate you expect to find at a given level of confidence.

For example, if you are estimating a 95% confidence interval around the mean proportion of female babies born every year based on a random sample of babies, you might find an upper bound of 0.56 and a lower bound of 0.48. These are the upper and lower bounds of the confidence interval. The confidence level is 95%.

The mean is the most frequently used measure of central tendency because it uses all values in the data set to give you an average.

For data from skewed distributions, the median is better than the mean because it isn’t influenced by extremely large values.

The mode is the only measure you can use for nominal or categorical data that can’t be ordered.

The measures of central tendency you can use depends on the level of measurement of your data.

  • For a nominal level, you can only use the mode to find the most frequent value.
  • For an ordinal level or ranked data, you can also use the median to find the value in the middle of your data set.
  • For interval or ratio levels, in addition to the mode and median, you can use the mean to find the average value.

Measures of central tendency help you find the middle, or the average, of a data set.

The 3 most common measures of central tendency are the mean, median and mode.

  • The mode is the most frequent value.
  • The median is the middle number in an ordered data set.
  • The mean is the sum of all values divided by the total number of values.

Some variables have fixed levels. For example, gender and ethnicity are always nominal level data because they cannot be ranked.

However, for other variables, you can choose the level of measurement . For example, income is a variable that can be recorded on an ordinal or a ratio scale:

  • At an ordinal level , you could create 5 income groupings and code the incomes that fall within them from 1–5.
  • At a ratio level , you would record exact numbers for income.

If you have a choice, the ratio level is always preferable because you can analyze data in more ways. The higher the level of measurement, the more precise your data is.

The level at which you measure a variable determines how you can analyze your data.

Depending on the level of measurement , you can perform different descriptive statistics to get an overall summary of your data and inferential statistics to see if your results support or refute your hypothesis .

Levels of measurement tell you how precisely variables are recorded. There are 4 levels of measurement, which can be ranked from low to high:

  • Nominal : the data can only be categorized.
  • Ordinal : the data can be categorized and ranked.
  • Interval : the data can be categorized and ranked, and evenly spaced.
  • Ratio : the data can be categorized, ranked, evenly spaced and has a natural zero.

No. The p -value only tells you how likely the data you have observed is to have occurred under the null hypothesis .

If the p -value is below your threshold of significance (typically p < 0.05), then you can reject the null hypothesis, but this does not necessarily mean that your alternative hypothesis is true.

The alpha value, or the threshold for statistical significance , is arbitrary – which value you use depends on your field of study.

In most cases, researchers use an alpha of 0.05, which means that there is a less than 5% chance that the data being tested could have occurred under the null hypothesis.

P -values are usually automatically calculated by the program you use to perform your statistical test. They can also be estimated using p -value tables for the relevant test statistic .

P -values are calculated from the null distribution of the test statistic. They tell you how often a test statistic is expected to occur under the null hypothesis of the statistical test, based on where it falls in the null distribution.

If the test statistic is far from the mean of the null distribution, then the p -value will be small, showing that the test statistic is not likely to have occurred under the null hypothesis.

A p -value , or probability value, is a number describing how likely it is that your data would have occurred under the null hypothesis of your statistical test .

The test statistic you use will be determined by the statistical test.

You can choose the right statistical test by looking at what type of data you have collected and what type of relationship you want to test.

The test statistic will change based on the number of observations in your data, how variable your observations are, and how strong the underlying patterns in the data are.

For example, if one data set has higher variability while another has lower variability, the first data set will produce a test statistic closer to the null hypothesis , even if the true correlation between two variables is the same in either data set.

The formula for the test statistic depends on the statistical test being used.

Generally, the test statistic is calculated as the pattern in your data (i.e. the correlation between variables or difference between groups) divided by the variance in the data (i.e. the standard deviation ).

  • Univariate statistics summarize only one variable  at a time.
  • Bivariate statistics compare two variables .
  • Multivariate statistics compare more than two variables .

The 3 main types of descriptive statistics concern the frequency distribution, central tendency, and variability of a dataset.

  • Distribution refers to the frequencies of different responses.
  • Measures of central tendency give you the average for each response.
  • Measures of variability show you the spread or dispersion of your dataset.

Descriptive statistics summarize the characteristics of a data set. Inferential statistics allow you to test a hypothesis or assess whether your data is generalizable to the broader population.

In statistics, model selection is a process researchers use to compare the relative value of different statistical models and determine which one is the best fit for the observed data.

The Akaike information criterion is one of the most common methods of model selection. AIC weights the ability of the model to predict the observed data against the number of parameters the model requires to reach that level of precision.

AIC model selection can help researchers find a model that explains the observed variation in their data while avoiding overfitting.

In statistics, a model is the collection of one or more independent variables and their predicted interactions that researchers use to try to explain variation in their dependent variable.

You can test a model using a statistical test . To compare how well different models fit your data, you can use Akaike’s information criterion for model selection.

The Akaike information criterion is calculated from the maximum log-likelihood of the model and the number of parameters (K) used to reach that likelihood. The AIC function is 2K – 2(log-likelihood) .

Lower AIC values indicate a better-fit model, and a model with a delta-AIC (the difference between the two AIC values being compared) of more than -2 is considered significantly better than the model it is being compared to.

The Akaike information criterion is a mathematical test used to evaluate how well a model fits the data it is meant to describe. It penalizes models which use more independent variables (parameters) as a way to avoid over-fitting.

AIC is most often used to compare the relative goodness-of-fit among different models under consideration and to then choose the model that best fits the data.

A factorial ANOVA is any ANOVA that uses more than one categorical independent variable . A two-way ANOVA is a type of factorial ANOVA.

Some examples of factorial ANOVAs include:

  • Testing the combined effects of vaccination (vaccinated or not vaccinated) and health status (healthy or pre-existing condition) on the rate of flu infection in a population.
  • Testing the effects of marital status (married, single, divorced, widowed), job status (employed, self-employed, unemployed, retired), and family history (no family history, some family history) on the incidence of depression in a population.
  • Testing the effects of feed type (type A, B, or C) and barn crowding (not crowded, somewhat crowded, very crowded) on the final weight of chickens in a commercial farming operation.

In ANOVA, the null hypothesis is that there is no difference among group means. If any group differs significantly from the overall group mean, then the ANOVA will report a statistically significant result.

Significant differences among group means are calculated using the F statistic, which is the ratio of the mean sum of squares (the variance explained by the independent variable) to the mean square error (the variance left over).

If the F statistic is higher than the critical value (the value of F that corresponds with your alpha value, usually 0.05), then the difference among groups is deemed statistically significant.

The only difference between one-way and two-way ANOVA is the number of independent variables . A one-way ANOVA has one independent variable, while a two-way ANOVA has two.

  • One-way ANOVA : Testing the relationship between shoe brand (Nike, Adidas, Saucony, Hoka) and race finish times in a marathon.
  • Two-way ANOVA : Testing the relationship between shoe brand (Nike, Adidas, Saucony, Hoka), runner age group (junior, senior, master’s), and race finishing times in a marathon.

All ANOVAs are designed to test for differences among three or more groups. If you are only testing for a difference between two groups, use a t-test instead.

Multiple linear regression is a regression model that estimates the relationship between a quantitative dependent variable and two or more independent variables using a straight line.

Linear regression most often uses mean-square error (MSE) to calculate the error of the model. MSE is calculated by:

  • measuring the distance of the observed y-values from the predicted y-values at each value of x;
  • squaring each of these distances;
  • calculating the mean of each of the squared distances.

Linear regression fits a line to the data by finding the regression coefficient that results in the smallest MSE.

Simple linear regression is a regression model that estimates the relationship between one independent variable and one dependent variable using a straight line. Both variables should be quantitative.

For example, the relationship between temperature and the expansion of mercury in a thermometer can be modeled using a straight line: as temperature increases, the mercury expands. This linear relationship is so certain that we can use mercury thermometers to measure temperature.

A regression model is a statistical model that estimates the relationship between one dependent variable and one or more independent variables using a line (or a plane in the case of two or more independent variables).

A regression model can be used when the dependent variable is quantitative, except in the case of logistic regression, where the dependent variable is binary.

A t-test should not be used to measure differences among more than two groups, because the error structure for a t-test will underestimate the actual error when many groups are being compared.

If you want to compare the means of several groups at once, it’s best to use another statistical test such as ANOVA or a post-hoc test.

A one-sample t-test is used to compare a single population to a standard value (for example, to determine whether the average lifespan of a specific town is different from the country average).

A paired t-test is used to compare a single population before and after some experimental intervention or at two different points in time (for example, measuring student performance on a test before and after being taught the material).

A t-test measures the difference in group means divided by the pooled standard error of the two group means.

In this way, it calculates a number (the t-value) illustrating the magnitude of the difference between the two group means being compared, and estimates the likelihood that this difference exists purely by chance (p-value).

Your choice of t-test depends on whether you are studying one group or two groups, and whether you care about the direction of the difference in group means.

If you are studying one group, use a paired t-test to compare the group mean over time or after an intervention, or use a one-sample t-test to compare the group mean to a standard value. If you are studying two groups, use a two-sample t-test .

If you want to know only whether a difference exists, use a two-tailed test . If you want to know if one group mean is greater or less than the other, use a left-tailed or right-tailed one-tailed test .

A t-test is a statistical test that compares the means of two samples . It is used in hypothesis testing , with a null hypothesis that the difference in group means is zero and an alternate hypothesis that the difference in group means is different from zero.

Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test . Significance is usually denoted by a p -value , or probability value.

Statistical significance is arbitrary – it depends on the threshold, or alpha value, chosen by the researcher. The most common threshold is p < 0.05, which means that the data is likely to occur less than 5% of the time under the null hypothesis .

When the p -value falls below the chosen alpha value, then we say the result of the test is statistically significant.

A test statistic is a number calculated by a  statistical test . It describes how far your observed data is from the  null hypothesis  of no relationship between  variables or no difference among sample groups.

The test statistic tells you how different two or more groups are from the overall population mean , or how different a linear slope is from the slope predicted by a null hypothesis . Different test statistics are used in different statistical tests.

Statistical tests commonly assume that:

  • the data are normally distributed
  • the groups that are being compared have similar variance
  • the data are independent

If your data does not meet these assumptions you might still be able to use a nonparametric statistical test , which have fewer requirements but also make weaker inferences.

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How to Calculate Quartiles in R?

In this article, we will discuss how to calculate quartiles in the R programming language. 

Quartiles are just special percentiles that occur after a certain percent of data has been covered.

  • First quartile: Refers to 25th percentile of the data. This depicts that 25% percent of data is under the produced value.
  • Second quartile: Refers to 50th percentile of the data. This depicts that 50% percent of data is under the produced value. This is also the median of the data.
  • Third quartile: Refers to the 75th percentile of the data. This predicts that 75% percent of the data is under the produced value.

To obtain the required quartiles, the quantile() function is used.

Syntax: quantile( data, probs) Parameter: data: data whose percentiles are to be calculated probs: percentile value

Example 1: Calculate quartile in vector

Example 2: calculate quartile in dataframe.

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Statistics Made Easy

How to Use the quantile() Function in R

In statistics,  quantiles  are values that divide a ranked dataset into equal groups.

The quantile() function in R can be used to calculate sample quantiles of a dataset.

This function uses the following basic syntax:

quantile(x, probs = seq(0, 1, 0.25), na.rm = FALSE)

  • x : Name of vector
  • probs : Numeric vector of probabilities
  • na.rm : Whether to remove NA values

The following examples show how to use this function in practice.

Example 1: Calculate Quantiles of a Vector

The following code shows how to calculate quantiles of a vector in R:

Example 2: Calculate Quantiles of Columns in Data Frame

The following code shows how to calculate the quantiles of a specific column in a data frame:

We can also use the sapply() function to calculate the quantiles of multiple columns at once:

Example 3: Calculate Quantiles by Group

The following code shows how to use functions from the dplyr package to calculate quantiles by a grouping variable:

Additional Resources

The following tutorials show how to use the quantile() function to calculate other common quantile values:

How to Calculate Percentiles in R How to Calculate Deciles in R How to Calculate Quartiles in R

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Assign Quantiles using phe_quantile

Description.

Assigns data to quantiles based on numeric data rankings.

When type = "full", returns the original data.frame with quantile (quantile value), nquantiles (number of quantiles requested), groupvars (grouping sets quantiles assigned within) and invert (indicating direction of quantile assignment) fields appended.

See OHID Technical Guide - Assigning Deprivation Categories for methodology. In particular, note that this function strictly applies the algorithm defined but some manual review, and potentially adjustment, is advised in some cases where multiple small areas with equal rank fall across a natural quantile boundary.

Other PHEindicatormethods package functions: assign_funnel_significance () , calculate_ISRate () , calculate_ISRatio () , calculate_funnel_limits () , calculate_funnel_points () , phe_dsr () , phe_life_expectancy () , phe_mean () , phe_proportion () , phe_rate () , phe_sii ()

IMAGES

  1. Quartile in Statistics: Detailed overview with solved examples

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  2. How to Find and Visualize Quartiles in R

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  3. Finden von Quartilen in R

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  4. R Tutorial : Median Quartiles and IQR

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  5. Comparing Medians and Inter-Quartile Ranges Using the Box Plot

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  6. How to Find the First Quartile

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  1. Quartile of Grouped Data [ Measures of Position ]

  2. Inter quartile range, quartile deviation & coefficient|3rd Sem| BBA|BCOM Calicut University| BNM|

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COMMENTS

  1. r

    11 Answers Sorted by: 125 There's a handy ntile function in package dplyr. It's flexible in the sense that you can very easily define the number of *tiles or "bins" you want to create. Load the package (install first if you haven't) and add the quartile column: library (dplyr) temp$quartile <- ntile (temp$value, 4)

  2. Quartile in R

    To calculate a quartile in R, set the percentile as parameter of the quantile function. You can use many of the other features of the quantile function which we described in our guide on how to calculate percentile in R. In the example below, we're going to use a single line of code to get the quartiles of a distribution using R.

  3. Create a quartile column for each value in an R data frame column

    If we want to create a quartile (1 to 4) column for each value in an R data frame column then we can use the quantile function and cut function as shown in the below Examples. Example 1 Following snippet creates a sample data frame − x<-sample (1:50,20) df1<-data.frame (x) df1 The following dataframe is created

  4. quantile Function in R (6 Examples)

    The quantile function computes the sample quantiles of a numeric input vector. In the following R tutorial, I'll explain in six examples how to use the quantile function to compute metrics such as quartiles, quintiles, deciles, or percentiles. Let's dive in! Example 1: Basic Application of quantile () in R

  5. Quartile

    An R tutorial on computing the quartiles of an observation variable in statistics. There are several quartiles of an observation variable. The first quartile, or lower quartile, is the value that cuts off the first 25% of the data when it is sorted in ascending order.The second quartile, or median, is the value that cuts off the first 50%.The third quartile, or upper quartile, is the value ...

  6. Quartile in Statistics: Detailed overview with solved examples

    First of all, arrange the given set of data in ascending order then take the middlemost value that is the median. The lower half of the set is the first quartile and the upper half is the third quartile. The difference between the lower and upper half can be identified by using the interquartile range. Formulas of quartile

  7. DECILES, QUARTILES and PERCENTILES in R [quantile function]

    Syntax Quartiles Remove missing values Quantile algorithms Visual representation Deciles Percentiles Considering a value p p, being 0 < p < 1 0 < p < 1 the quantile of order p p is the value that leaves a proportion of the data below ( p p) and the rest (1-p) (1 − p) above that value.

  8. How to Find and Visualize Quartiles in R

    We can easily calculate the quartiles of a given dataset in R by using the quantile () function. This tutorial provides examples of how to use this function in practice. Calculating Quartiles in R The following code shows how to calculate the quartiles of a given dataset in R:

  9. Learn R: Quartiles, Quantiles, and Interquartile Range

    The three dividing points (or quantiles) that split data into four equally sized groups are called quartiles. For example, in the figure, the three dividing points Q1, Q2, Q3 are quartiles. Median in Quantiles. The median is the divider between the upper and lower halves of a dataset. It is the 50%, 0.5 quantile, also known as the 2-quantile.

  10. quartile_ function

    The formula is the following: Examples Run this code { #data creation data = c(1:20) quartile_ (data) } <p>Calculates the 3 Quartiles of a vector of data</p>

  11. Calculate Quartiles in R

    Calculate Quartiles in R Quartiles are values which divide a dataset into four equal parts, each of which contains 25% of the data. It is helpful to understand the spread and distribution of a dataset by using quartiles. In general, there are three quartiles (Q1, Q2, and Q3) used.

  12. Quantile,Percentile and Decile Rank in R using dplyr

    Percentile rank, quantile rank and decile rank of a group in R. Let's First Create the dataframe 1 2 3 4 my_basket = data.frame(ITEM_GROUP = c("Fruit","Fruit","Fruit","Fruit","Fruit","Vegetable","Vegetable","Vegetable","Vegetable","Dairy","Dairy","Dairy","Dairy","Dairy"),

  13. quantiles

    R 's median function calculates this. The index of the middle value is m = (n + 1)/2. When it is not an integer, (xl +xu)/2 is the median, where l and u are m rounded down and up. Otherwise when m is an integer, xm is the median. In that case take l = m − 1 and u = m + 1.

  14. How do I find quartiles in R?

    Χ 2 = 8.41 + 8.67 + 11.6 + 5.4 = 34.08. Step 3: Find the critical chi-square value. Since there are four groups (round and yellow, round and green, wrinkled and yellow, wrinkled and green), there are three degrees of freedom.. For a test of significance at α = .05 and df = 3, the Χ 2 critical value is 7.82.. Step 4: Compare the chi-square value to the critical value

  15. R quantile by groups with assignments

    R quantile by groups with assignments Ask Question Asked 7 years, 11 months ago Modified 7 years, 11 months ago Viewed 8k times Part of R Language Collective 5 I have the following df: group = rep (seq (1,3),30) variable = runif (90, 5.0, 7.5) df = data.frame (group,variable)

  16. How to Calculate Quartiles in R?

    How to Calculate Quartiles in R? Read Courses Practice In this article, we will discuss how to calculate quartiles in the R programming language. Quartiles are just special percentiles that occur after a certain percent of data has been covered. First quartile: Refers to 25th percentile of the data.

  17. How to Use the quantile() Function in R

    The quantile () function in R can be used to calculate sample quantiles of a dataset. This function uses the following basic syntax: quantile (x, probs = seq (0, 1, 0.25), na.rm = FALSE) where: x: Name of vector probs: Numeric vector of probabilities na.rm: Whether to remove NA values

  18. R: Assign Quantiles using phe_quantile

    Assigns data to quantiles based on numeric data rankings. Usage phe_quantile ( data, values, nquantiles = 10L, invert = TRUE, inverttype = "logical", type = "full" ) Arguments Value

  19. r

    2 Answers Sorted by: 12 ggplot2 has a nice utility function, cut_number (), which does just what you want. library (ggplot2) as.numeric (cut_number (annual_exp [ [1]], n = 5)) # [1] 3 3 5 1 4 2 4 2 1 5 3 4 1 4 1 2 3 5 Share Follow answered Aug 28, 2012 at 15:42 Josh O'Brien 160k 28 368 457

  20. r

    11 Answers Sorted by: 97 In dplyr 1.0, summarise can return multiple values, allowing the following: library (tidyverse) mtcars %>% group_by (cyl) %>% summarise (quantile = scales::percent (c (0.25, 0.5, 0.75)), mpg = quantile (mpg, c (0.25, 0.5, 0.75))) Or, you can avoid a separate line to name the quantiles by going with enframe:

  21. r

    1 You might be better off using cut rather than a loop: Data = data.frame (Avg = runif (100)) quantpoints <- seq (0.1, 0.9, 0.1) quants <- quantile (Data$Avg, quantpoints) cutpoints <- c (-Inf, quants, Inf) cut (Data$Avg, breaks = cutpoints, labels = seq (1, length (cutpoints) - 1)) Share Follow answered Sep 13, 2016 at 12:26 sebastian-c