Applying statistics to describe a population
OCR J560 expects students to interpret numerical summaries of a population — using the right average, the right spread, and choosing between them in context.
The three averages
| Average | Definition | Best when… |
|---|---|---|
| Mean | Sum / count | All values typical (no outliers) |
| Median | Middle value when ordered | Skewed data; outliers present |
| Mode | Most frequent value | Categorical data; bimodal cases |
Mean
Mean = (sum of values) / (number of values).
Sensitive to outliers — one extreme value pulls the mean strongly.
Median
To find the median: order the data; the median is the middle value.
- Odd n: median is the (n+1)/2-th value.
- Even n: median is the average of n/2-th and (n/2 + 1)-th values.
Median is resistant to outliers — adding one huge value barely shifts it.
Mode
The most-occurring value. Possibly more than one (bimodal) or none if all values appear once.
Range
Range = max − min.
Crude measure of spread, very sensitive to outliers. OCR often asks "compare the range" alongside an average.
Interquartile range (IQR)
(Higher tier; covered in S3 detail).
- Q1 = lower quartile (median of lower half).
- Q3 = upper quartile (median of upper half).
- IQR = Q3 − Q1. Resistant to outliers.
Choosing the right average
OCR Foundation papers often ask: "Which average best describes…?"
- Use median for skewed data (incomes, house prices, exam scores with floor/ceiling effects).
- Use mean for symmetric distributions (heights, exam scores roughly normal).
- Use mode for categorical data (favourite colour, favourite sport).
Comparing two populations
Always state both an average and a measure of spread when comparing.
Example: "The mean for class A is higher than for class B (so on average A scored more), and the range for class A is smaller (so A's scores were more consistent)."
OCR mark scheme conventions
- B1 for the correct calculation.
- B1 for stating the comparison in context (mention "on average", "more consistent", etc.).
- "Use a measure of average AND a measure of spread" — both required for full marks.
⚠Common mistakes
- Forgetting to order data before finding the median.
- Computing the mean by averaging the median and mode (wrong!).
- Comparing two datasets with only an average (no spread) — loses 1 mark.
- Stating "the range of class A is bigger so class A is better" — bigger range usually means LESS consistent.
AI-generated · claude-opus-4-7 · v3-ocr-maths-leaves