2.1: Measures of Central Tendency- Mean, Median, and Mode
- Page ID
- 58862
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Imagine a local high school is holding a fundraiser to support a new science lab. Each student donates what they can; some give a few dollars, while others contribute much more. At the end of the campaign, the principal wants to share the results with the school community. But what number should she report? Should she present the average donation? The amount most students gave? The amount exactly in the middle?
These are all ways of summarizing data using a single number; what statisticians call measures of center. These values help us describe what is "typical" in a dataset and give us a meaningful way to compare data, report insights, or support decisions. Choosing the right measure often depends on the shape of the data and on what story you want to tell.
In this section, you’ll explore the three most common summary statistics for center: the mean (average) [opens in new window], the median (middle value) [opens in new window], and the mode (most frequent value) [opens in new window]. Each of these captures a different perspective on what might be considered “typical” and contributes to understanding the greater story.
You’ll learn what each measure means, how to calculate it, when it’s useful, and when it can be misleading. After exploring interactive summary cards for each statistic, you'll examine the school fundraiser in more detail and compare how each measure helps tell the story. Finally, you’ll see how to calculate these statistics in Excel; a skill you can apply in your own data projects.
Mean (Arithmetic Average)
Description: Add up all the values and divide by how many there are. This gives us the “center of gravity” of the data.
Symbols:
- \( \bar{x} \): Sample mean
- \( x_1, x_2, \dots, x_n \): Individual data values
- \( n \): Sample size (number of values)
Formula:
\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \]Notes: The mean is not robust. It is sensitive to outliers and skewed values.
Example: What is the mean of the values {5, 10, 12}?
- Sum: \( 5 + 10 + 12 = 27 \)
- Count: \( n = 3 \)
- Mean: \( \bar{x} = \frac{27}{3} = 9 \)
Quick Check: What is the mean of {3, 4, 8, 10}?
Median (Middle Value)
Description: The median is the middle number when the data is sorted. If there are an odd number of values, it’s the one in the center. If there’s an even number, it’s the average of the two middle numbers.
Notation: Usually referred to as “median”. Abbreviation: \( \text{Med} \)
How to Find the Median:
- Sort the values from least to greatest
- If \( n \) is odd: take the middle value
- If \( n \) is even: average the two middle values
Notes: The median is robust. It is not affected by outliers or extreme values.
Example: What is the median of {4, 6, 9, 10, 11}?
- Ordered: {4, 6, 9, 10, 11}
- Middle value = 9
Another Example: What is the median of {6, 8, 10, 14}?
- Ordered: {6, 8, 10, 14}
- Middle two values: 8 and 10
- Median: \( \frac{8 + 10}{2} = 9 \)
Quick Check: What is the median of: {1, 3, 3, 7, 10, 12, 13}?
Mode (Most Frequent)
Description: The mode is the value that appears most often in a dataset. A set can have one mode (unimodal), more than one (bimodal or multimodal), or no mode at all. On a smooth distribution (shown below), the mode is the highest point on the graph.
Notation: Mode has no standard symbol; just use the word “mode.”
How to Find:
- Tally the frequency of each value
- Pick the value(s) that occur(s) the most
Notes: Mode is useful for categorical data and discrete data. It may not describe “center” well unless repeated values dominate the dataset.
Example 1: Mode of {1, 2, 2, 3, 4} is 2
Example 2: Mode of {5, 6, 7, 8} → No repetitions → No mode
Example 3: Mode of {2, 3, 3, 4, 4, 5} → Bimodal: 3 and 4
Quick Check: What is the mode of {6, 2, 6, 3, 3, 3, 6}?
Visual Comparison: Mean, Median, and Mode
This visual shows a powerful comparison of how the mean (blue), median (green), and mode (red) behave within two different probability distributions. Both distributions are unimodal (one peak) but have different shapes and spreads.
The taller, narrower curve represents a dataset with smaller variability (standard deviation \( \sigma = 0.25 \)) — the values are tightly centered. Notice how in this case, the mean, median, and mode are all very close together, reinforcing how symmetric distributions tend to align these three statistics.
The shorter, wider curve shows a more spread-out dataset (\( \sigma = 1 \)). In this less symmetric shape, the mean, median, and mode become notably separated. The mode remains near the peak (most frequent value), the median stays near the center of mass, and the mean shifts toward the longer tail — pulled by extreme values.
This visualization helps illustrate why it's important to examine the shape of your data before choosing which measure of center to report. When your data is skewed, has outliers, or is irregularly distributed, these summary values can tell very different stories. We will go into this more in the next few sections.
As you analyze sample data (like in our upcoming example), remember to reflect on the overall distribution — not just compute a summary automatically. These differences also affect how we generalize from samples back to the larger **population**. Understanding the shape and the behavior of each summary statistic helps us make deeper, more credible conclusions from our data.
Example: Telling a Data Story with Measures of Center
Olive Grove High School recently held a district-wide fundraiser: each student was asked to contribute any amount they could toward a new science lab. The funds were collected anonymously, and the principal received a list of how much each of the 20 students donated. As the school newspaper reports on the results, they want to highlight a summary of the donations made.
Here’s the dataset showing dollars donated by each student:
[0, 5, 5, 5, 10, 10, 10, 10, 15, 15, 20, 20, 20, 25, 30, 35, 40, 50, 90, 200]
Calculate the Measures
- Mean = 32.75. This is pulled upward by a few very large donations, especially the $200 gift (an outlier).
- Median = 17.5. Half the students gave less than $17.50 and half gave more. This better reflects what most students gave.
- Mode = 10. More students gave $10 than any other amount. This is the most “popular” donation level.
Storytelling with the Right Statistic
Each of these measures tells a slightly different story:
- The newspaper might highlight the median to report what a “typical” student gave and avoid distortion from unusually large donations.
- The principal might focus on the mean to emphasize overall generosity and total impact.
- A student campaign reflection might mention the mode to show which amount students most commonly felt comfortable donating.
Which measure is best? That depends on your audience and purpose. Choosing between mean, median, and mode is often less about math, and more about what story you're trying to tell. Usually, multiple measures of center should be reported to give a clearer picture of the data.
Example: Computing Measures of Center in Excel
Let’s apply what we’ve learned using Microsoft Excel, a useful tool for quickly computing summary statistics. Suppose you’re tracking the number of hours 10 students reported studying during the past week. You’ve entered the data in an Excel column like this:
A Hours Studied 1 8 2 5 3 7 4 10 5 4 6 6 7 12 8 8 9 7 10 9 11 20
Goal:
Use Excel to compute the mean, median, and mode of the data in column A.
Step-by-Step Instructions
- Open Excel and type your data into cells A2 through A11 (under a header like "Hours Studied").
- In any empty cell, compute the mean using a
=AVERAGE(A2:A11)
- Compute the median using:
=MEDIAN(A2:A11)
- Compute the mode using:
=MODE.SNGL(A2:A11)
Note: Use=MODE.MULTfor multiple modes (in array formulas)
Interpretation:
Results you might get (based on the above data):
- Mean: 8.8 (skewed slightly upward by the 20-hour outlier)
- Median: 7.5 (a better “typical” value)
- Mode: 7 (most common number of study hours reported)
Which one would you highlight when reporting class study habits and why?
Optional: Add a Chart
- Select your data in A2:A11.
- Go to the Insert tab → choose a Histogram or Column Chart.
- Add a title and observe how the visual distribution supports your values.
You can apply these same steps to your semester-long housing dataset to compute summary statistics for price, square footage, number of bedrooms, and more!
Next Up: We'll explore how data spreads out, including how to calculate the range, interquartile range (IQR), standard deviation, and more.