Measures of central tendency and spread
Statistics questions appear on every OCR J560 paper. Higher-tier extends to estimated mean from grouped data, interquartile range from cumulative frequency, and comparing two distributions using both a measure of average and a measure of spread.
Measures of central tendency
Mean
Mean = Sum of all values ÷ Number of values
Mean is affected by outliers (extreme values).
Example: 3, 7, 7, 8, 10, 11, 14. Mean = (3+7+7+8+10+11+14)/7 = 60/7 ≈ 8.57.
Median
The middle value when data is ordered. For n values:
- Odd n: median is the ((n+1)/2)th value.
- Even n: median is the mean of the (n/2)th and (n/2+1)th values.
Example: 3, 7, 7, 8, 10, 11, 14. n=7 → 4th value → median = 8.
Median is resistant to outliers.
Mode
The most frequent value. There can be more than one mode, or no mode.
Example: 3, 7, 7, 8, 10, 11, 14. Mode = 7.
Mean from a frequency table
Mean = Σ(fx) / Σf where f = frequency, x = data value (or midpoint for grouped data).
Example:
| Score x | Frequency f | fx |
|---|---|---|
| 2 | 3 | 6 |
| 4 | 5 | 20 |
| 6 | 2 | 12 |
| Total | 10 | 38 |
Mean = 38/10 = 3.8.
Estimated mean from grouped data
Use the midpoint of each class as x.
| Class | Midpoint x | Frequency f | fx |
|---|---|---|---|
| 0–10 | 5 | 4 | 20 |
| 10–20 | 15 | 8 | 120 |
| 20–30 | 25 | 3 | 75 |
| Total | 15 | 215 |
Estimated mean = 215/15 ≈ 14.3.
Note: this is an ESTIMATE because we assume all data in a class is at the midpoint.
Measures of spread
Range
Range = Maximum − Minimum. Simple but affected by outliers.
Interquartile range (IQR)
IQR = Q3 − Q1. Measures spread of the middle 50% of data. More robust than range.
Standard deviation (Higher, calculator)
Measures average distance of each value from the mean. Larger SD = more spread.
Comparing distributions
To compare two distributions, always comment on both:
- A measure of average (mean or median): "Distribution A has a higher mean, so on average..."
- A measure of spread (range, IQR): "Distribution A has a larger IQR, so the data is more spread out / less consistent."
OCR mark schemes require a comparison in context — reference what the numbers represent.
Common OCR exam mistakes
- Using midpoints incorrectly: 0–10 midpoint is 5 (not 10 or 0).
- Confusing mean and median — always state which you're using.
- For comparison questions: not referencing context ("students scored higher on average" is better than just "the mean is higher").
- Finding median with even n: averaging only one value instead of two.
AI-generated · claude-opus-4-7 · v3-ocr-maths