Mean, Median, Mode, and Range

Learning Objectives

By the end of this lesson, you’ll be able to:

Compute and interpret the mean, median, mode, and range of a dataset.
Choose the most appropriate measure of center for a given data context.
Understand how outliers affect mean and median and when to prefer each.

Key Ideas

Mean (arithmetic mean): the sum of all values divided by the number of values.
\[ \bar{x} = \frac{1}{n}\sum_{i=1}^n x_i \]
Median: the middle value after sorting the data.
- Odd \(n\) → the center value
- Even \(n\) → average of the two center values
Mode: the value(s) occurring most frequently. Can be none, one, or many.
Range:
\[ \text{range} = \max(x) - \min(x) \]

Choosing a Measure of Center

The mean uses all values and is sensitive to outliers.
Use it when the distribution is roughly symmetric.
The median is resistant to outliers and skew.
Use it for income data, reaction times, or anything skewed.
The mode is useful for categorical data or identifying the most common value.

Common Problem Types

Computing All Four Statistics for a Small Dataset

These problems ask for mean, median, mode, and range directly from a short list.

Example:
Data: [4, 8, 6, 6]
- Mean uses sum ÷ count
- Median uses sorted values
- Mode = 6
- Range = 8 − 4 = 4

Finding the Median for Even vs. Odd Number of Values

Always sort first, then check whether \(n\) is even or odd.

Example (odd):
[2, 5, 9] → median = 5

Example (even):
[1, 4, 6, 7] → median = (4 + 6)/2 = 5

Identifying Mode(s) or No Mode

Some datasets have one mode, multiple modes, or no mode.

Example:
[3, 3, 8, 8, 10] → two modes (3 and 8)
[5, 7, 9] → no mode

Determining Influence of Outliers on Mean vs. Median

Outliers pull the mean but leave the median largely unchanged.

Example:
[10, 11, 12, 100]
- Mean increases greatly due to 100
- Median remains near the middle two values (11.5)

Choosing Between Mean and Median in Context

Decide which measure better reflects a “typical” value.

Example:
Incomes: [40k, 42k, 45k, 1.2M] → median better describes the typical income.

Comparing Centers or Spreads Between Two Datasets

SAT problems often ask which group is “higher” or “more variable.”

Example:
Group A median = 60
Group B median = 72
→ Group B has higher center

Interpreting Changes in the Dataset

Adding/removing numbers may affect mean, median, mode, or range differently.

Example:
Original: [4, 4, 6]
Add 100 → mean increases, median stays 4, mode stays 4, range increases.

Understanding Range as a Measure of Spread

Range uses only highest and lowest values; sensitive to outliers.

Example:
[10, 12, 14, 100] → range = 90
Removing 100 → range becomes 4

Strategies

Sort first — the median and mode are easier to identify in sorted data.
Check for outliers — decide if mean or median is more appropriate.
Match statistic to data type — mode for categories, median for skew, mean for symmetry.
Show work — list sorted values, show sums, identify min/max.
Use units — report answers with context (points, dollars, etc.).

Calculating Each Statistic

Mean

Example for \([4,7,5,6]\): \[ \bar{x} = \frac{4+7+5+6}{4} = \frac{22}{4} = 5.5 \]

Median

Sort first:
\([3,8,2,5,7] \to [2,3,5,7,8]\)
Median \(= 5\)

Even \(n\):
\([1,4,6,9]\) → centers \(4\) and \(6\) → median \(=5\)

Mode

Example: \([2,3,3,4,5]\) → mode \(=3\)
If all values are unique → no mode.

Range

Example: \([10,14,9,11]\) → \(14 - 9 = 5\)

Worked Examples

Example 1 — Small numeric list

Data: \([12, 15, 12, 18, 20]\)

Mean: \(\bar{x}=(12+15+12+18+20)/5 = 77/5 = 15.4\)
Median: sort → \([12,12,15,18,20]\), median \(= 15\)
Mode: \(12\)
Range: \(20 - 12 = 8\)

Example 2 — Even count and an outlier

Data: \([5, 7, 8, 40]\)

Mean: \((5+7+8+40)/4 = 60/4 = 15\)
Median: sort → \([5,7,8,40]\), median \(=(7+8)/2 = 7.5\)
Mode: none
Range: \(40 - 5 = 35\)

Note the outlier 40 pulls the mean up, but the median stays near the center of the smaller values.

Interpreting Results

Use median when data are skewed or influenced by extreme values.
Use mean for symmetric, numeric data where every value matters.
Report both mean and median if the context benefits (e.g., salaries).

Common Mistakes

Confusing mean and median — they behave differently with outliers.
Forgetting to sort before finding the median.
Averaging the wrong two center values when \(n\) is even.
Reporting a mode when no value repeats.
Omitting units when describing a summary statistic.

Practice Problems

Compute mean, median, mode, and range for \([3, 7, 7, 2, 9, 10]\).
For \([18, 24, 22, 30, 22, 21]\), compute all four statistics and decide which measure best describes the center.
Test scores: \([86, 92, 75, 92, 88, 100, 58]\). Compute the four statistics and describe how the low score affects mean vs median.
Which measure of center is better for household income: mean or median? Explain why.

Answers

Sorted: \([2,3,7,7,9,10]\)
Mean \(= 38/6 \approx 6.33\)
Median \(=7\)
Mode \(=7\)
Range \(= 10 - 2 = 8\)
Sorted: \([18,21,22,22,24,30]\)
Mean \(= 137/6 \approx 22.83\)
Median \(= (22+22)/2 = 22\)
Mode \(=22\)
Range \(=12\)
Interpretation: distribution roughly symmetric → median and mean similar.
Sorted: \([58,75,86,88,92,92,100]\)
Mean \(= 591/7 \approx 84.43\)
Median \(= 88\)
Mode \(= 92\)
Range \(=42\)
The very low 58 drags the mean down; median stays stable → median better describes typical score.
Median — income distributions are right-skewed with large high outliers.

Summary

Mean: uses all values, sensitive to outliers.
Median: middle value, robust to outliers.
Mode: most frequent value, useful for categorical data.
Range: max − min, sensitive to extremes.

Quick Tips

Sort the data before finding median or mode.
Check for outliers — they distort the mean, not the median.
For income, prices, times, and skewed data: pick the median.
If no value repeats, there is no mode.
Use units to interpret your results (“average score,” “typical income,” etc.).