9.1Measures of Center
Calculate and interpret mean, median, and mode. Understand when each measure best represents a data set and how outliers affect each one.
Why This Matters
Mean, median, and mode are the most commonly reported statistics in the real world — from average test scores to median household income. You'll use these in AP Statistics, Biology labs, and any data-driven career.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How do we choose the best single number to represent an entire data set, and what happens when extreme values are present?
Lesson Overview
A measure of center is a single value that summarizes a data set by describing its middle or most typical value. The three measures we use are mean (the arithmetic average), median (the middle value when data is ordered), and mode (the value that appears most often). Each measure tells a different story about the data, and choosing the right one depends on the shape of the distribution and whether outliers — unusually high or low values — are present.
Measures of Center — Formulas & Rules
| Measure | How to Find | Best Used When |
|---|---|---|
| Mean | Sum of all values ÷ number of values | Data is symmetric with no outliers |
| Median | Order data; find the middle value (or average of two middle values) | Data is skewed or has outliers |
| Mode | The most frequently occurring value(s) | Categorical data or finding most common value |
Distribution Shape & Measure of Center
Symmetric
Mean ≈ Median
No outliers — either measure works
Skewed Right
Mean > Median
High outlier pulls mean up — use median
Skewed Left
Mean < Median
Low outlier pulls mean down — use median
Worked Examples
Find the mean, median, and mode of: 4, 7, 2, 9, 7, 5, 3
Mean: 4 + 7 + 2 + 9 + 7 + 5 + 3 = 37. Divide by 7: 37 ÷ 7 ≈ 5.29.
Median: Order the data: 2, 3, 4, 5, 7, 7, 9. Seven values — the 4th is the middle: 5.
Mode: 7 appears twice; all others appear once. Mode = 7.
Find the mean and median of: 12, 15, 14, 13, 16, 14. Which better represents the data?
Mean: 12 + 15 + 14 + 13 + 16 + 14 = 84. Divide by 6: 84 ÷ 6 = 14.
Median: Order: 12, 13, 14, 14, 15, 16. Even count — two middle values are 14 and 14. Median = (14 + 14) ÷ 2 = 14.
Mean = Median = 14. Data is symmetric with no outliers — either measure works well.
A student scored: 82, 85, 79, 88, 91, 14. Find the mean and median. Which better represents typical performance?
Mean: 82 + 85 + 79 + 88 + 91 + 14 = 439. Divide by 6: 439 ÷ 6 ≈ 73.2.
Median: Order: 14, 79, 82, 85, 88, 91. Two middle values: 82 and 85. Median = (82 + 85) ÷ 2 = 83.5.
14 is an outlier — it pulls the mean down to 73.2, which does not reflect the student's typical score.
The median (83.5) better represents typical performance.
Find the mode(s): (a) 3, 5, 5, 7, 8, 8, 9 (b) 2, 4, 6, 8, 10 (c) red, blue, red, green, blue, red
(a) 5 appears twice, 8 appears twice. Bimodal: modes are 5 and 8.
(b) Every value appears exactly once. No mode.
(c) Red appears 3 times, blue 2 times, green once. Mode = red.
Salaries (thousands): 35, 38, 40, 42, 45, 48, 200. Find the mean and median. Which better represents the typical salary?
Mean: 35 + 38 + 40 + 42 + 45 + 48 + 200 = 448. Divide by 7: 448 ÷ 7 ≈ 64.0.
Median: 7 values ordered — the 4th value is 42.
200 is an outlier (likely the CEO). It inflates the mean to $64,000 even though 6 of 7 employees earn $35,000–$48,000.
The median ($42,000) better represents the typical employee salary.
Guided Practice
Guided Practice Video: Measures of Center
Review mean, median, and mode — including how to handle even-count data sets and the effect of outliers — before completing the guided problems below.
Video by Sang Real Math
Watch on YouTube ↗Find the mean of: 6, 10, 4, 8, 12
Hint: Add all five values, then divide by 5.
Find the median of: 11, 3, 7, 15, 9, 5, 13
Hint: Order the data first. There are 7 values — the median is the 4th.
Find the median of: 20, 14, 18, 22, 16, 24
Hint: Order the data. There are 6 values (even) — average the 3rd and 4th values.
Find the mode of: 5, 3, 8, 5, 9, 3, 5, 7
Hint: Count how many times each value appears. Which appears most often?
Data: 50, 52, 55, 53, 51, 98. Find the mean and median. Which better represents the data?
Hint: 98 is an outlier. Calculate both, then compare — which is pulled toward 98?
Key Vocabulary
Mean
The arithmetic average. Found by adding all values and dividing by the number of values. Sensitive to outliers.
Median
The middle value of an ordered data set. If there are two middle values, the median is their average. Resistant to outliers.
Mode
The value (or values) that appear most frequently. A data set can have no mode, one mode, or multiple modes.
Outlier
A data value significantly higher or lower than the rest. Outliers strongly affect the mean but not the median.
Skewed Right
Most data clusters on the left; a few high values pull the tail right. Mean > Median.
Skewed Left
Most data clusters on the right; a few low values pull the tail left. Mean < Median.
Interactive Practice — 5 Questions
What is the mean of: 4, 8, 6, 10, 2?
What is the median of: 3, 9, 5, 1, 7?
A data set has mean = 40 and median = 55. What does this suggest?
Which measure of center is MOST resistant to outliers?
Find the mode of: 7, 3, 5, 7, 9, 3, 7, 5
Independent Practice
Independent Practice
Find all three measures of center for: 9, 3, 7, 5, 11, 7. Show your work for each — order the data first.
Outlier effect: Data set A is 10, 12, 11, 13, 10. Data set B adds an outlier: 10, 12, 11, 13, 10, 50. Find the mean and median for both. Describe how the outlier changed each measure.
Even data set: Find the median of 30, 20, 50, 40, 10, 60. Explain the two-step process for an even number of values.
Choose the best measure: (a) Test scores: 72, 85, 90, 88, 76, 91, 83 — which measure best represents the class? (b) Shoe sizes sold: 8, 9, 9, 10, 8, 9, 11 — which measure is most useful for a store manager?
Real-world: Seven friends' weekly screen times (hours): 12, 15, 10, 14, 13, 11, 42. Find the mean and median. Identify the outlier and explain which measure better describes the group's typical screen time.
Common Mistakes
Not ordering the data before finding the median — picking a middle-looking value from the unordered list.
Always sort the data from least to greatest first. The median is the positional middle of the ordered list.
For an even-sized data set, picking one of the two middle values instead of averaging them.
When there are two middle values, the median is their mean: add them and divide by 2.
Writing 0 as the mode when no value repeats.
If no value appears more than once, there is no mode — 'no mode' is the correct answer, not 0.
Using the mean when the data contains a strong outlier.
Outliers pull the mean toward them. When outliers are present, the median is the more representative measure of center.
Math Tips
Always order the data first before finding the median — skipping this step is the most common error.
Even number of values: the median is the average of the two middle values — add them and divide by 2.
Outliers pull the mean toward them but do not affect the median. Use the median when outliers are present.
Mean vs. Median comparison: mean > median → skewed right (high outlier); mean < median → skewed left (low outlier).
Mode works for any data type — numbers, colors, categories. Mean and median only work for numerical data.
No mode is a valid answer — if every value appears exactly once, write "no mode," not 0.