Mean, Median, Mode, and Range
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
- Odd \(n\) → the center value
- 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).
- 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.
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.
- 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.).