Statistics & Data · Grades 6–10
Mean, Median and Mode Explained (and When Each Misleads)
Three ways to say “typical”
- Mean — add everything up and divide by how many values there are.
- Median — sort the values and take the middle one (or the average of the middle two).
- Mode — the value that appears most often; there may be none, or several.
Worked example: 4, 8, 6, 5, 3, 8, 9
- Mode: 8
- Range: 6
- Sample SD: 2.2678
- Population SD: 2.0996
Step-by-step working 7 steps
- 1Sort the data
3, 4, 5, 6, 8, 8, 9
7 values.
- 2Mean = sum ÷ count
Sum = 43 Mean = 43 ÷ 7 = 6.142857
- 3Median = middle value
Value 4 of 7 = 6
- 4Mode = most frequent value
8 (appears 2 times)
- 5Range = largest − smallest
9 − 3 = 6
- 6Squared deviations from the mean
(4 − 6.142857)² = 4.5918 (8 − 6.142857)² = 3.449 (6 − 6.142857)² = 0.0204 (5 − 6.142857)² = 1.3061 (3 − 6.142857)² = 9.8776 (8 − 6.142857)² = 3.449 (9 − 6.142857)² = 8.1633 Sum of squares = 30.8571
Sum these for every value to get the total used below.
- 7Variance and standard deviation
Population: σ² = 30.8571 ÷ 7 = 4.4082, σ = 2.0996 Sample: s² = 30.8571 ÷ 6 = 5.1429, s = 2.2678
Use the sample version (÷ n−1) when your data is a sample from a larger group; use the population version when it is the whole group.
How an outlier changes the picture
Five similar values give Mean 11.6 · Median 12. Add one extreme value and it becomes Mean 25.5 · Median 12. The mean jumped; the median barely moved. That is why house prices and incomes are usually reported as medians.
With an outlier
- Mode: 12
- Range: 85
- Sample SD: 34.0632
- Population SD: 31.0953
Step-by-step working 7 steps
- 1Sort the data
10, 11, 12, 12, 13, 95
6 values.
- 2Mean = sum ÷ count
Sum = 153 Mean = 153 ÷ 6 = 25.5
- 3Median = middle value
Average of values 3 and 4: (12 + 12) ÷ 2 = 12
- 4Mode = most frequent value
12 (appears 2 times)
- 5Range = largest − smallest
95 − 10 = 85
- 6Squared deviations from the mean
(10 − 25.5)² = 240.25 (12 − 25.5)² = 182.25 (11 − 25.5)² = 210.25 (13 − 25.5)² = 156.25 (12 − 25.5)² = 182.25 (95 − 25.5)² = 4830.25 Sum of squares = 5801.5
Sum these for every value to get the total used below.
- 7Variance and standard deviation
Population: σ² = 5801.5 ÷ 6 = 966.9167, σ = 31.0953 Sample: s² = 5801.5 ÷ 5 = 1160.3, s = 34.0632
Use the sample version (÷ n−1) when your data is a sample from a larger group; use the population version when it is the whole group.
Spread: range and standard deviation
The range is largest minus smallest. Standard deviation measures how far values typically sit from the mean. Divide by n for a whole population, or by n − 1 for a sample, which corrects for the sample’s tendency to understate spread.
Common mistakes
Forgetting to sort for the median
The median is the middle of the ordered list.
Reporting a mean on skewed data
A few extreme values drag the mean; report the median too.
Using the wrong standard deviation
Sample data → divide by n − 1. Complete population → divide by n.
Frequently asked questions
Can a data set have two modes?
Yes. If two values tie for the highest frequency, both are modes. If no value repeats, there is no mode.
What if there is an even number of values?
Average the two middle values after sorting.
