Averages & Spread: Mean, Median, Mode & Range
Averages & Spread
An average is a single value that represents a data set. Spread measures how varied the data is.
The Three Averages
Mean = sum of all values ÷ number of values
Median = the middle value when data is arranged in order
Mode = the most frequent value (there can be more than one, or none)
Worked Example: Find the mean, median and mode of: 3, 7, 7, 4, 9, 2, 7, 5, 8
- Mean = (3+7+7+4+9+2+7+5+8) ÷ 9 = 52 ÷ 9 = 5.78 (to 2 d.p.)
- Ordered: 2, 3, 4, 5, 7, 7, 7, 8, 9 → Median = 7 (5th value of 9)
- Mode = 7 (appears 3 times)
For an even number of values, the median is the mean of the two middle values.
Range and Interquartile Range
Range = highest value − lowest value (a simple measure of spread)
Interquartile range (IQR) = upper quartile (Q3) − lower quartile (Q1)
The IQR measures the spread of the middle 50% of the data, ignoring extremes.
Finding quartiles for a data set of n values in order:
- Q1 is at position (n+1)/4
- Q2 (median) is at position (n+1)/2
- Q3 is at position 3(n+1)/4
Worked Example: 12 values in order: 2, 4, 5, 7, 8, 9, 11, 13, 15, 18, 20, 24
- Q1 position = 13/4 = 3.25 → between 3rd and 4th values = (5+7)/2 = 6
- Q2 = (9+11)/2 = 10
- Q3 position = 9.75 → between 9th and 10th = (15+18)/2 = 16.5
- IQR = 16.5 − 6 = 10.5
Mean from a Frequency Table
When data is in a frequency table, use: Mean = Σ(fx) ÷ Σf
Worked Example:
| Score (x) | Frequency (f) | fx |
|---|---|---|
| 1 | 3 | 3 |
| 2 | 7 | 14 |
| 3 | 12 | 36 |
| 4 | 5 | 20 |
| 5 | 3 | 15 |
| Total | 30 | 88 |
Mean = 88 ÷ 30 = 2.93 (to 2 d.p.)
Estimated Mean from Grouped Data
For grouped data, use the midpoint of each class as the x value.
Worked Example:
| Time (t mins) | Frequency | Midpoint | fx |
|---|---|---|---|
| 0 < t ≤ 10 | 4 | 5 | 20 |
| 10 < t ≤ 20 | 9 | 15 | 135 |
| 20 < t ≤ 30 | 15 | 25 | 375 |
| 30 < t ≤ 40 | 7 | 35 | 245 |
| Total | 35 | 775 |
Estimated mean = 775 ÷ 35 = 22.1 minutes (to 1 d.p.)
This is an estimate because we do not know the exact values within each group — we assume they are at the midpoint.
Which Average to Use?
| Average | Advantage | Disadvantage |
|---|---|---|
| Mean | Uses all data | Affected by outliers |
| Median | Not affected by outliers | Ignores most values |
| Mode | Shows most common value | May not exist or may be several |
Comparing Data Sets
When comparing, always comment on an average (centre) AND a measure of spread.
Example comparison: "The mean score for class A (67) is higher than class B (54), suggesting class A performed better on average. However, the IQR for class B (8) is smaller than class A (15), indicating class B's results were more consistent."
Exam Tips
- For grouped data, the mean is always an ESTIMATE — use that word
- The modal class is the class with the highest frequency, not the highest midpoint
- When finding the median from a frequency table, use cumulative frequency to locate the correct value
- The range is easily distorted by a single outlier — the IQR is more reliable