Averages summarise a dataset with a single representative value. Measures of spread tell us how consistent or variable the data is.
Mean
Sum of all values ÷ number of values
Median
Middle value when sorted in order
Mode
Most frequently occurring value
Range
Largest − Smallest value
📖 Learn
Mean, Median, Mode, Range
Mean = Σx / n = (sum of all values) ÷ (number of values) Median = middle value (odd n) or average of middle two (even n) Mode = most frequent value (there can be more than one, or none) Range = max − min
For the median: sort the data first, then find position (n+1)/2.
💡 For even n=8, median is the average of the 4th and 5th values after sorting.
Mean from a Frequency Table
Mean = Σ(f × x) / Σf
Multiply each value by its frequency, sum, then divide by total frequency.
x
f
f × x
1
3
3
2
5
10
3
2
6
Total
10
19
Mean = 19 ÷ 10 = 1.9
Estimated Mean from Grouped Data
Use the midpoint of each class as x. Estimated mean = Σ(f × midpoint) / Σf
This is an estimate because we don't know the actual values within each class — we assume they're spread evenly around the midpoint.
💡 The estimated mean from grouped data is NOT the same as the true mean — it's an estimate.
Comparing Distributions
When comparing two datasets, always comment on:
Average (mean or median): which group has a higher/lower typical value?
Spread (range or IQR): which group is more consistent/variable?
💡 A good comparison uses data: "Class A has a higher mean (7.2 vs 6.5) but greater range (8 vs 5), suggesting Class A scores were higher on average but less consistent."
✏️ Worked Examples
Example 1: Mean, median, mode, range
Data: 3, 7, 4, 3, 9, 4, 3, 8
Sort: 3, 3, 3, 4, 4, 7, 8, 9
Mean = (3+3+3+4+4+7+8+9) ÷ 8 = 41 ÷ 8 = 5.125
Median (n=8): average of 4th and 5th = (4+4)/2 = 4
Mode = 3 (appears 3 times)
Range = 9 − 3 = 6
Example 2: Mean from frequency table
Values 1,2,3,4 with frequencies 3,5,2,1. Find mean.
Σ(f × x) = 1×3 + 2×5 + 3×2 + 4×1 = 3+10+6+4 = 23
Σf = 3+5+2+1 = 11
Mean = 23 ÷ 11 = 2.09 (2 d.p.)
Example 3: Estimated mean from grouped data
Classes 0–10, 10–20, 20–30 with frequencies 4, 8, 3. Estimate the mean.