Lesson 2: Central Tendency

1 Introduction

Welcome. In this lesson, you will learn about central tendency. See the home page for more details on how to use this tutorial and for troubleshooting tips!

2 Analyzing Central Tendency

When analyzing a set of data values, it is often important to summarize where the numerical ‘center’ of the data values lies.

Start by creating a variable called mydata that contains the values 4, 4, 2, 6, 3, 0, and 3

Did you remember to use the assignment arrow (<-) and the concatenate operator (c)?

mydata <- c(4, 4, 2, 6, 3, 0, 3)

2.1 Calculating Mean and Median

2.1.1 Mean

The mean (also known as the average) is a commonly-used measure of the center of a set of values. To compute the mean, you can use the command mean(data), substituting the name of your variable for the placeholder shown here.

Find the mean of your mydata variable.

Put the name of your variable in the parentheses. You don’t need to use quotes.

mean(mydata)

2.1.2 Arguments in a function

When you use functions like mean() in R, the thing you put in the parentheses is called an argument. In this case, our argument for the mean function is mydata. Sometimes, functions will require multiple different arguments.

For some data sets, the mean is not a very good measure of the data’s central tendency. For example, what if your data values were 2, 2, 2, 3, 1, 3, 12?

To analyze this data, create a new variable, data_two, containing these values.

Don’t forget to use the assignment arrow and the concatenate function.

data_two<-c(2, 2, 2, 3, 1, 3, 12)

Next, compute the mean of data_two.

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Answer

mean(data_two)

Notice that all but one of the data values are lower than the mean, because there is one really high value. For data like these, the median is a better measure of central tendency.

2.1.3 Median

The median is the middle value when the data are ordered from lowest to highest.

The function median(dataName) computes the median for a variable.

Compute the median of data_two.

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Answer

median(data_two)

Notice that the median is not unduly influenced by the single large data value, and so is a more accurate measure of the center of these data values.

If you are unsure whether to use mean or median to describe the center of data values, a histogram can be very helpful. In lesson 4, you will learn how to create a histogram but for now it is worth noting that a histogram can help you see how data is distributed. When a histogram’s peak is not near the center, and/or its two tails are quite asymetrical, it’s best to use the median, not the mean, to summarize the central tendency.

2.1.4 Basic Arithmetic with mean/median

Next, write an expression for the difference between the mean and the median of data_two. R uses the symbols + for addition, - for subtraction, * for multiplication, and / for division. Subtract the smaller value from the larger value.

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Answer

mean(data_two)-median(data_two)

2.1.5 Finding mean and median in a dataset

In lesson 1, we used a dataframe called data with the variables RQ, Condition, heart_rate, Individual.

Type data to recall what it contains.

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Answer

Now, try to find the mean of the heart_rate column in data.

Use the syntax yourDataName$columnName to refer to a specific column. You can type your column name directly in the parentheses in the mean() function.

mean(data$heart_rate)

You should get an answer that says ‘NA’. This is because there are NA values in the heart rate column, which prevent R from calculating the mean. So, we have to add one more thing to our code which tells R to ignore the ‘NA’ values: na.rm=TRUE. We add this in the parentheses in the mean() and median() functions. Use a comma to seperate yourDataName and na.rm=TRUE.

For example:

mean(data$heart_rate, na.rm=TRUE)
median(data$heart_rate, na.rm=TRUE)

Try finding the mean of the heart rate column again, but add na.rm=TRUE this time.

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Answer

mean(data$heart_rate, na.rm=TRUE)

3 Summarizing central tendency

In this section, we’ll learn how you can easily display the mean and median of multiple different groups in a data set.

The functions group_by() and summarize() help us do this.

First, though, we’ll learn about a function called the pipe.

3.1 The Pipe

A very useful function in the tidyverse (our current set of R packages) is called the pipe. It’s a way to string together functions and data. You can read it as “AND THEN”.

The pipe can be written as %>% or |>

Both do the same thing! Let’s look at an example. Click “run code” below.

Let’s break down what is happening:

First, we call up the data we want with data$RQ.
Then we use the pipe (|>) to say “AND THEN”.
Finally, we calculate the mean, but we don’t need to input our data because we already did in the first line and used the pipe to string them together.

The pipe will be very helpful to string together functions!

3.2 Summarizing data with `group_by()` and `summarize()`

We use the functions group_by() and summarize() to generate summary statistics (mean, median, etc) for specific groups in our data set.

group_by(): a function that takes a data set and groups it by a variable/column
summarize(): uses the grouped data set from group_by, and lets you define summary statistics columns for that group.
- The syntax is: summarize(yourStatName = formula(columnName, na.RM=TRUE))

Let’s see how these work with an example:

Recall, the Condition variable specifies whether the measurements were made under exercise (1) or resting (2) conditions. Suppose we want to compute the mean and median heart rate for the individuals in our sample, but we want separate values for the exercise and resting conditions.

We can use group_by() and summarize() to do this.

You should see a table that displays the average and median heart rate for each condition (1 and 2).

Let’s break down how this works:

First, we specify our data and pipe to the next function.
Then we group by our conditions, listed in the column Condition.
Next, use summarize() and specify that we want to find the mean and median. summarize() creates new columns for each statistic we define. We also rename the columns averageHearRate and medianHeartRate.
- Define mean: averageHeartRate = mean(heart_rate, na.rm=TRUE)
- Define median: medianHeartRate = median(heart_rate, na.rm=TRUE)

We use the pipe |> to string all these functions together.

Now you try! Find the mean and median for the RQ (respiratory quotiont), and find separate values for each condition (resting or exercise).

What is the mean of the RQ for condition 1?

✓0.876

✗0.87

✗0.973

Use the pipe to string together your functions.

Here’s how you set up the summarize() function:

summarize(averageRQ = mean(________, na.rm=TRUE), 
          medianRQ = median(_______, na.rm=TRUE))

data |>
  group_by(________) |>
  summarize(averageRQ = mean(________, na.rm=TRUE), 
            medianRQ = median(_______, na.rm=TRUE))

data |>
  group_by(Condition) |>
  summarize(averageRQ = mean(RQ, na.rm=TRUE), 
            medianRQ = median(RQ, na.rm=TRUE))

4 Congratulations!

You have finished this tutorial. You learned how to calculate the mean and median of data and how to use group_by() and summarize() to calculate the mean and median of specific groups within a column.

Another important kind of descriptive statistic is variability; that’s the topic of lesson three.