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Learn · Mathematics · Grade 9 · Data

Data: mean, median, mode and spread

Reading tables and charts, the three averages and when each is the honest one, range and standard deviation, and misleading graphs.

High school Lesson 13 of 31 Interactive figure Practice

First, the rules

Three averages and two spreads

A single number can summarise a list — but which number depends on what you want to know, and a few extreme values can drag one of them far from the rest.

★ Mean
Examples 1, 2
x̄ = sum of the valuesnumber of values
Uses every value; pulled towards extremes.
★ Median
Example 1
Sort the values; the middle one (odd count) or the average of the two middle ones (even count). Not moved by extremes.
Mode
Example 1
The most frequent value; there may be none, one or several. The only average that works for categories (favourite colour).
★ Which one?
Example 2
Symmetric data: mean. Skewed data or outliers (salaries, house prices): median. Categories: mode.
Range
Example 1
largest − smallest — quick, but decided by two values only.
★ Standard deviation
Example 3
σ = √Σ(x − x̄)²n
The typical distance of a value from the mean. Square the deviations, average, square-root. (With n − 1 for a sample.)
Honest charts
Start bar charts at zero, keep equal intervals, label the axes. A truncated axis makes a small difference look huge.
Larger data sets
Paste the numbers into the statistics tool: it gives every value here plus a histogram and box plot.

See it

Drag the bars

Pull a bar up or down and see the mean, median, mode and standard deviation react. Send the last value to 40 and see which average follows it.

Step by step

Worked examples

1

Four summaries of one list

mean · median · mode · range

Test scores: 7, 9, 4, 9, 6, 8, 9, 4. Find the mean, median, mode and range.

Ask yourself: Sort first — the median and the range both need the order; the mode needs a count.

Show the solution
  1. Sorted: 4, 4, 6, 7, 8, 9, 9, 9. Sum = 56, n = 8.
  2. mean = 56 / 8 = 7.
  3. Eight values: the median is the average of the 4th and 5th: (7 + 8) / 2 = 7.5.
  4. Mode = 9 (three times). Range = 9 − 4 = 5.

Answer mean 7, median 7.5, mode 9, range 5

2

When the mean lies

choosing the average

Monthly rents in a street (in hundreds): 5, 5, 6, 6, 7, 7, 8, 40. Which average describes a typical rent?

Ask yourself: Compute both the mean and the median and look at the gap.

Show the solution
  1. Mean: sum 84, n = 8: 84 / 8 = 10.5 — higher than seven of the eight rents.
  2. Median: sorted already; average of the 4th and 5th: (6 + 7) / 2 = 6.5.
  3. The single value 40 drags the mean up. The median (6.5) is the honest "typical" rent; report the mean only together with the outlier.

Answer median 6.5 (the mean 10.5 is distorted by the outlier)

3

Standard deviation by hand

spread

Find the standard deviation of 2, 4, 4, 4, 5, 5, 7, 9 (treat it as the whole population).

Ask yourself: Mean first; then each deviation squared; average; square root.

Show the solution
  1. Mean: 40 / 8 = 5.
  2. Deviations: −3, −1, −1, −1, 0, 0, 2, 4. Squares: 9, 1, 1, 1, 0, 0, 4, 16, sum 32.
  3. σ = √32 / 8 = √4 = 2. A typical score sits about 2 away from the mean of 5.

Answer σ = 2

Practice: summarise the data

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A short list of numbers — find its mean, median, mode or range. Your best score is kept in this browser only.

About this lesson

Reading and summarising data: tables and bar charts, the mean, median and mode and when each is the honest average, range and standard deviation as measures of spread, and how a chart can mislead. Practice questions on small data sets; the statistics tool does the heavy lifting for larger ones.

Part of Learn mathematics · Grade 9 · Data. Check a number with the scientific calculator or draw a function with the function plotter. Everything on this page runs in your browser; nothing you type is sent or stored, and your practice score stays in this browser only. Spotted an error? Tell us.