Measures of Central Tendency
Learning Objectives
- Find the mean of a set of data
- Find the mean from a frequency table
- Find the median of a set of data
- Find the mode of a set of data
- Determine whether the mean, median, or mode is the best measure of center for a data set
Why This Matters
When a company posts "average salary: $95,000" on a job listing, that number might include executive compensation pulling the mean up. The median salary at the same company could be $62,000. One number gets you excited, the other tells you what you'd actually earn. Mean, median, and mode aren't interchangeable -- picking the wrong one can cost you a negotiation or mislead an entire report.
How to Use This Simulation
- Drag the outlier slider and watch how the mean, median, and mode respond differently to extreme values.
- Switch between preset datasets to see how data shape changes which measure best represents center.
- Open the Frequency Table tab to practice computing the mean from grouped data using the weighted formula.
- Check the Explanation Panel below the chart -- it updates as you interact and tells you which measure fits best.
Drag to move the last data point and observe how each measure of center responds.
Mean from a Frequency Table
When data is already grouped, multiply each value by its frequency, sum those products, then divide by the total number of observations. The result is identical to computing the mean from raw data -- this is a shortcut for pre-grouped data.
| Value (x) | Frequency (f) | x · f |
|---|
Practice: Streaming Subscriptions
A survey asked 45 college students how many streaming subscriptions they pay for. Calculate the mean number of subscriptions using the weighted formula.
| Subscriptions (x) | Students (f) | x · f |
|---|---|---|
| 0 | 5 | 0 |
| 1 | 12 | 12 |
| 2 | 18 | 36 |
| 3 | 8 | 24 |
| 4 | 2 | 8 |
| Total | 45 | 80 |
What's Happening
Quick Check
A company reports that the "average" employee salary is $94,000. You discover that the CEO earns $4.2 million and 49 other employees earn between $48,000 and $72,000. Which measure of center did the company most likely use, and which measure would better represent a typical employee's pay?
Try This
Load the "Gig Work Earnings" preset dataset. Before touching the outlier slider, predict: if the highest earner's trip jumps from $16 to $80, will the mean or the median change more? By how much?
Now drag the slider to $80 and check your prediction. Were you right? Write one sentence explaining why the measure you picked moved more.
A ride-share driver records earnings for 7 trips: $8, $12, $10, $45, $11, $9, $13. One trip was a long-distance airport fare. Calculate all three measures of center. The driver wants to tell a friend "here's what a typical trip pays." Which measure should they report, and why would the other two be misleading?
Now switch to the Frequency Table tab and verify the mean using the weighted formula. Does seeing the frequency of each value change how you think about "typical"?
You're a data analyst at a streaming company. Viewer watch times (in minutes) for a new show across 12 users: 5, 88, 91, 90, 87, 3, 92, 89, 2, 90, 88, 91. Enter this data into the simulation using the Edit Data panel.
Calculate mean, median, and mode. The content team wants one number for the report -- what do you recommend and why? Before you answer, look at the dot plot. What does the shape tell you about viewer behavior that no single measure of center captures? Write a two-sentence recommendation to the content team that names the measure you'd report AND acknowledges what it hides.
Instructor Notes
Teaching Notes
This simulation is most effective when you let students discover the outlier effect before explaining it. Have them predict what will happen when the slider moves, then drag it. The visual gap between the mean line jumping and the median line holding still creates an immediate conceptual anchor for resistance to outliers.
The frequency table tab addresses a separate but related objective. Students who can compute the mean from raw data sometimes struggle with the weighted formula. The practice problem (streaming subscriptions) is intentionally simple so they can focus on the method, not the arithmetic.
Common Student Errors
- Forgetting to sort data before finding the median -- the simulation sorts automatically, but ask students to practice sorting by hand.
- Averaging Q2 and Q3 to find the median in an even-sized dataset instead of averaging the two middle values at positions n/2 and (n/2)+1 in the sorted list.
- Claiming "no mode" when multiple values tie for highest frequency. A dataset can be bimodal or multimodal.
- Using the mean by default without considering data shape. The misconception that "average = mean" is deeply ingrained.
Discussion Questions
- Why do news reports almost always use "median household income" instead of "mean household income"? What would happen if they switched?
- Can you think of a dataset where the mode is actually more useful than both the mean and the median? (Hint: think about shoe sizes in a store's inventory.)
- If someone told you their class average was 82, what questions would you want to ask before deciding whether that's a "good" class?
Exam Connection
Typical exam questions present a dataset with an obvious outlier and ask students to identify the best measure of center. Emphasize that students must justify their choice based on data shape, not just pick the median every time. Some exam items give a frequency table and ask for the mean using the weighted formula -- the Frequency Table tab directly prepares students for this format.