Slides 📊

3.2. Measures of Central Tendency

When analyzing data, one of the first questions we ask is: “What’s the typical value?” In this chapter, we’ll explore three fundamental measures of central tendency that can be used to answer this qustion.

For simplicity, we will denote the variable under study as $x$ throughout this section.

Road Map 🧭

Define sample mode, sample mean, and sample median. Learn their mathematical notations and properties.
Describe the strengths and weaknesses of each measure, and recognize which measure is the most appropriate given a data set.

3.2.1. Sample Mode, $M$

The sample mode is the simplest measure of central tendency—it’s the value that occurs most frequently in a dataset. We denote the sample mode with $M$.

Properties

A data set might have zero, one, or more than one mode.
- Zero when all values occur with equal frequency
- One when exactly one value occurs most frequently
- “Multimodal” when multiple values tie for the most frequent
- A dataset with one mode is called unimodal. Datasets with two and three modes are called bimodal and trimodal, respectively.
The sample mode is most useful when working with discrete numerical data with a limited set of possible values.
It is less informative when the data contains many unique values with few repetitions. For this reason, the sample mode is not often used unless the data set is very simple.

Example 💡: Finding the Sample Mode

Consider this simple dataset: {1, 1, 2, 3, 4}

Computing the mode by hand

Since 1 occurs with the highest frequency, $M = 1$.

Computing the mode on R

R does not have a built-in function for computing the sample mode. Therefore, we create a new function:

# define a new function:
getmode <- function(v) {
   uniqv <- unique(v)
   uniqv[which.max(tabulate(match(v, uniqv)))]
   }

# define data
x <- c(1,1,2,3,4)

# use the newly defined function
getmode(x) # returns 1

🛑 Note: You may copy and paste the function above for your own use. If you are interested in learning how to create your own R functions, a general guide is provided in the appendix. However, you are not expected to write custom R functions in this course.

3.2.2. Sample Mean, $\bar{x}$

The sample mean, denoted as $\bar{x}$ (pronounced “x-bar”), is what most people think of as the “average.” The sample mean is calculated by summing all observations and dividing by the sample size:

\[\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i = \frac{x_1 + x_2 + \ldots + x_n}{n}\]

Properties

It is influenced by each observation in the dataset.
It can be affected significantly by outliers or extreme values.

Example 💡: Finding the Sample Mean

For the dataset $\{2, 4, 6, 8\}$,

\[\bar{x} = \frac{2 + 4 + 6 + 8}{4} = \frac{20}{4} = 5.\]

R provides a built-in function for calculating the mean.

x <- c(2, 4, 6, 8)
mean(x)  # Returns 5

3.2.3. Sample Median, $\tilde{x}$

The sample median, denoted as $\tilde{x}$ (pronounced “x-tilde”), is the middle value of the data arranged in order. It splits the data set so that 50% of the observations fall below and another 50% above it.

How to find the sample median

Arrange all observations in ascending order.
If there is an odd number of observations, the median is the middle value.
If there is an even number of observations, the median is the average of the two middle values.

Mathematically,

\[\begin{split}\tilde{x} = \begin{cases} x_{\left(\frac{n+1}{2}\right)}, & \text{if } n \text{ is odd} \\[4pt] \frac{x_{\left(\frac{n}{2}\right)} + x_{\left(\frac{n}{2}+1\right)}}{2}, & \text{if } n \text{ is even} \end{cases}\end{split}\]

where $x_{(i)}$ represents the $i$-th smallest value in the data set.

Properties

It depends only on the order (ranks) of most of the data, not the exact values.
It is robust to outliers—few extreme values don’t usually change a median.

Example 💡: Finding the sample median when $n$ is odd

Take the data set $\{1, 2, 4, 6, 8\}$. The data is already sorted with $n = 5$.

\[\frac{n+1}{2} = \frac{5+1}{2} = 3.\]

The median is $\tilde{x} = x_{(3)} = 4$.

y <- c(1, 2, 4, 6, 8)
median(y)  # Returns 4

Example 💡: Finding the sample median when $n$ is even

The dataset $\{2, 4, 6, 8\}$ has $n=4$, which is even. $\frac{n}{2} = \frac{4}{2} = 2$. The median is the average of the 2nd and 3rd values:

\[\tilde{x} = \frac{x_{(2)} + x_{(3)}}{2} = \frac{4 + 6}{2} = 5.\]

x <- c(2, 4, 6, 8)
median(x)  # Returns 5

3.2.4. Comparing the Measures: An Example

To better understand how the different measures of central tendency work in practice, let’s explore an example in depth.

Part 1

Miss Claridge asked her preschool class of nine students how many pets each had in their household. The results are recorded in the table below. A dot plot of the data is shown in Fig. 3.1.

Unsorted pet counts data
4	8	7	9	4	3	5	1	4

We will analyze this data using the three measures of central tendency. Let us begin by expressing the data using formal notion. Denote the variable of pet counts with $x$. We have $n=9$. The individual data points are then denoted as $x_1, x_2, \cdots, x_9$.

Sample Mode

The value 4 occurs most frequently (three times). Therefore, $M = 4$.
Sample Mean

\[\bar{x} = \frac{1}{9}\sum_{i=1}^9 x_i = \frac{1}{9}(4 + 8 + 7 + 9 + 4 + 3 + 5 + 1 + 4) = \frac{45}{9} = 5\]
Sample Median

First, sort the data: $\{1, 3, 4, 4, 4, 5, 7, 8, 9\}$. Since $n$ is odd, compute $(9+1)/2 = 5$. The median is the 5th value. $\tilde{x} = 4$.

Part 2

Miss Claridge discovers that there was an error during the data recording process; the student who was recorded as having nine pets actually had nineteen. How will the measures change due to this update?

New sorted pet counts data
1	3	4	4	4	5	7	8	9 → 19

Sample Mode

The value 4 still occurs most frequently. $M = 4$.
Sample Mean

\[\bar{x} = \frac{1}{9}\sum_{i=1}^9 x_i = \frac{1}{9}(1+3+4+4+4+5+7+8+19) = \frac{55}{9} = 6.11\]
Sample Median

$n$ did not change. $(9+1)/2 = 5$. The median is still the 5th value. $\tilde{x} = 4$.

Mode, median, and mean of pet counts data before and after update — Fig. 3.2 Upper graph: central tendency measures before update. Lower: after update.

In Part 2, both the sample mode and median remained at 4, while the sample mean increased from 5 to 6.11. This illustrates how the sample mean is pulled towards all data points, including outliers and extreme values. In contrast, the sample median and mode are more resistant to such influences.

3.2.5. Bringing It All Together

Key Takeaways 📝

Measures of central tendency indicate where the majority of the data is centered, bunched, or clustered.
The sample mode ($M$) finds the most frequent value(s) and works best for discrete data with few unique values.
The sample mean ($\bar{x}$) calculates the arithmetic average.
The sample median ($\tilde{x}$) finds the middle value and is robust against extreme values.

Appendix: Defining Your Own Functions in R

As we’ve seen with our sample mode function, R allows us to create custom functions for statistical operations that aren’t built into the language. The basic structure of a function in R is:

function_name <- function(argument1, argument2, ...) {
  # Code goes between brackets
  return(return_value) # Can return a single entity
}

Let’s break down the components:

Function name: The name you’ll use when calling the function
Arguments: The list of input values
- Arguments can have default values: function(argument = 0)
- You can create functions with no arguments: function()
Function body (Everything between the curly brackets { }): code describing the steps to be taken when the function is called
Return value: What the function outputs after computation

3.2.6. Exercises

These exercises build your skills in calculating and interpreting the three measures of central tendency: mode, mean, and median.

Exercise 1: Calculating Measures of Central Tendency

A quality control engineer records the number of defects found in 12 randomly inspected circuit boards:

\[\{2, 0, 1, 3, 2, 1, 2, 4, 1, 2, 0, 2\}\]

Calculate the sample mode ($M$).
Calculate the sample mean ($\bar{x}$). Show your work using summation notation.
Calculate the sample median ($\tilde{x}$). Show the sorted data and your calculation.
Which measure best represents the “typical” number of defects for this data? Justify your answer.
Verify your calculations using R.

Exercise 2: Effect of Outliers on Central Tendency

A startup tracks the daily number of app downloads for 10 consecutive days:

\[\{45, 52, 48, 51, 47, 49, 53, 46, 50, 48\}\]

Calculate the mean, median, and mode for this data.
On day 11, the app goes viral and receives 500 downloads. Add this value to the dataset and recalculate all three measures.
Create a table comparing the measures before and after adding the viral day. Which measure changed the most (in percentage terms)? Which changed the least?
If you were reporting “typical daily downloads” to investors, which measure would you use after the viral day? Explain your reasoning.

Solution

Part (a): Original Data (n = 10)

Sorted: $\{45, 46, 47, 48, 48, 49, 50, 51, 52, 53\}$

Mode: $M = 48$ (appears twice; all others appear once)
Mean: $\bar{x} = \frac{45+52+48+51+47+49+53+46+50+48}{10} = \frac{489}{10} = 48.9$
Median: $\tilde{x} = \frac{x_{(5)} + x_{(6)}}{2} = \frac{48 + 49}{2} = 48.5$

Part (b): With Viral Day (n = 11)

New data: $\{45, 52, 48, 51, 47, 49, 53, 46, 50, 48, 500\}$

Sorted: $\{45, 46, 47, 48, 48, 49, 50, 51, 52, 53, 500\}$

Mode: $M = 48$ (still appears twice)
Mean: $\bar{x} = \frac{489 + 500}{11} = \frac{989}{11} = 89.9$
Median: $\tilde{x} = x_{(6)} = 49$ (6th value when n = 11)

Part (c): Comparison Table

Measure	Before	After	% Change
Mode	48	48	0%
Median	48.5	49	1.0%
Mean	48.9	89.9	83.8%

Changed most: Mean (increased by 83.8%)
Changed least: Mode (no change)

Part (d): Which Measure to Report?

After the viral day, the median (49) would be the best choice for reporting “typical daily downloads.”

Reasons:

The mean (89.9) is heavily distorted by the single outlier and doesn’t represent any typical day
The mode (48) only reflects frequency, not central tendency in the traditional sense
The median (49) represents what investors could expect on a “normal” day
For skewed data with outliers, the median is more representative of typical behavior

You might report both: “Typical daily downloads are around 49, with one viral day reaching 500.”

Exercise 3: Interpreting Differences Between Measures

For each scenario below, determine the likely relationship between the mean and median (mean > median, mean < median, or mean ≈ median). Explain your reasoning.

Home sale prices in a neighborhood where most homes are modest but a few luxury mansions exist.
Exam scores on a test that most students found easy, with a few students performing poorly.
Heights of adult males in a large random sample.
Time to complete an online checkout process, where most transactions are quick but some have technical difficulties causing long delays.
A company reports that their employees’ mean salary is $85,000 and median salary is $62,000. What does this suggest about the salary distribution?

Exercise 4: Choosing the Appropriate Measure

For each situation, identify which measure of central tendency (mode, mean, or median) is most appropriate. Justify your choice.

A shoe store wants to know which size to stock most heavily.
A professor wants to report the typical performance on an exam where scores are approximately normally distributed.
An economist wants to describe the typical household income in a city with significant wealth inequality.
A server administrator wants to report typical response times, knowing that occasional network issues cause very long delays.
A survey asks customers to rate satisfaction on a 1-5 scale. The company wants to report the most common rating.
A manufacturing process produces parts where nearly all measurements cluster tightly around the target, with no outliers.

Exercise 5: Working with Grouped Data

A software company has three development teams. The number of bugs fixed by each developer last month is recorded:

Team A (4 developers): 12, 15, 14, 13
Team B (6 developers): 8, 10, 9, 11, 7, 9
Team C (5 developers): 18, 22, 19, 20, 21

Calculate the mean number of bugs fixed for each team separately.
Calculate the overall mean for all 15 developers combined. (Hint: You must account for different team sizes.)
If you simply averaged the three team means from part (a), would you get the same answer as part (b)? Explain why or why not.
Which team shows the most variability in bugs fixed? (Just estimate from looking at the data—you’ll learn to calculate this precisely in the next section.)

Exercise 6: Conceptual Understanding

Answer the following conceptual questions about measures of central tendency.

A dataset has $\bar{x} = \tilde{x} = M = 50$. What can you conclude about the shape of this distribution?
Can a dataset have more than one mode? Can it have more than one median? Explain.
A professor says: “The median exam score was 78, but after adding 5 points to everyone’s score, the median became 83.” Verify whether this statement makes sense mathematically.
If every value in a dataset is multiplied by 2, what happens to the mean? What happens to the median? What happens to the mode?
True or False: “The mean of a dataset must be equal to one of the values in the dataset.” Explain.
A dataset of 99 values has a median of 50. If a 100th value is added, and the new median is 52, what can you conclude about the value that was added?

Solution

Part (a): When All Three Measures Are Equal

When $\bar{x} = \tilde{x} = M$, the distribution is likely:

Symmetric (or very close to symmetric)
Unimodal (single peak)
No significant outliers

This is characteristic of a symmetric, bell-shaped distribution like the normal distribution. However, you cannot conclude it’s perfectly normal—only that it’s reasonably symmetric with a single mode.

Part (b): Multiple Modes and Medians

Mode: Yes, a dataset can have multiple modes.

Bimodal: Two values tie for highest frequency
Multimodal: Three or more values tie
Example: {1, 1, 2, 3, 3} has modes M = 1 and M = 3

Median: Technically, when n is even, the median is defined as the average of the two middle values, so there is always exactly one median value. However, both middle values could be considered “median positions.”

In practice, we report a single median value
Example: {1, 2, 3, 4} has median = (2+3)/2 = 2.5

Part (c): Adding a Constant

Yes, this makes sense.

When you add a constant $c$ to every value:

The median increases by exactly $c$
Original median: 78
After adding 5: 78 + 5 = 83 ✓

This is because the median depends only on position (order), and adding the same amount to all values shifts every value—including the middle one—by that amount.

Part (d): Multiplying by a Constant

If every value is multiplied by 2:

Mean: Multiplied by 2 → $\bar{x}_{new} = 2\bar{x}_{old}$
Median: Multiplied by 2 → $\tilde{x}_{new} = 2\tilde{x}_{old}$
Mode: Multiplied by 2 → $M_{new} = 2M_{old}$

All three measures scale proportionally because multiplication is a linear transformation that preserves the relative ordering and proportions of values.

Part (e): Mean Must Equal a Data Value?

False.

The mean often does not equal any value in the dataset.

Example: {1, 2, 3, 4} has mean $\bar{x} = \frac{10}{4} = 2.5$, but 2.5 is not in the dataset.

The mean is a mathematical calculation that can fall anywhere within (or even outside, if negative values exist) the range of the data.

Part (f): Effect of Adding One Value on Median

Original: 99 values with median = $x_{(50)} = 50$ (the 50th value when sorted)

After adding one value (n = 100, even):

New median = $\frac{x_{(50)} + x_{(51)}}{2} = 52$

For the median to increase from 50 to 52:

The average of the new 50th and 51st values must be 52
This means $x_{(50)} + x_{(51)} = 104$

The added value must be greater than 50 (the old median). Specifically:

If the new value is inserted and becomes the 51st value or higher, it pushes the median up
The new value is likely ≥ 54 (to make the new 51st value at least 54, giving average ≥ 52 with 50)

We can conclude the added value was significantly larger than the original median.

3.2.7. Additional Practice Problems

True/False Questions (1 point each)

The sample mean is always affected by every value in the dataset.

Ⓣ or Ⓕ
The sample median is the value that occurs most frequently in a dataset.

Ⓣ or Ⓕ
For a symmetric distribution, the mean and median are approximately equal.

Ⓣ or Ⓕ
A dataset can have no mode if all values occur with equal frequency.

Ⓣ or Ⓕ
If the mean is greater than the median, the distribution is likely left-skewed.

Ⓣ or Ⓕ
The median is calculated by finding $\frac{1}{n}\sum_{i=1}^{n} x_i$.

Ⓣ or Ⓕ

Multiple Choice Questions (2 points each)

A real estate website reports that the median home price in a city is $350,000 and the mean is $425,000. This suggests that:

Ⓐ Most homes cost exactly $350,000

Ⓑ The distribution of home prices is left-skewed

Ⓒ Some expensive homes are pulling the mean above the median

Ⓓ The mode is $425,000
For the dataset {2, 2, 3, 3, 3, 4, 4, 100}, the mean is 15.125 and the median is 3. Which statement best explains this difference?

Ⓐ The calculation of the mean is incorrect

Ⓑ The dataset has two modes, which affects the mean

Ⓒ The outlier (100) pulls the mean much higher than the median

Ⓓ The median should be recalculated using the mean

3.2. Measures of Central Tendency

3.2.1. Sample Mode, \(M\)

Properties

3.2.2. Sample Mean, \(\bar{x}\)

Properties

3.2.3. Sample Median, \(\tilde{x}\)

How to find the sample median

Properties

3.2.4. Comparing the Measures: An Example

Part 1

Part 2

3.2.5. Bringing It All Together

Appendix: Defining Your Own Functions in R

3.2.6. Exercises

3.2.7. Additional Practice Problems