Statistics

Averages and spread, grouped data and frequency density, and reading scatter graphs and correlation.

7 min read

Statistics questions are about interpreting data correctly: choosing the right average, reading a histogram by area, and not confusing correlation with causation.

Mean $= \frac{sum of values}{number of values}$
Median $=$ the middle value when ordered
Mode $=$ the most frequent value
Range $=$ largest $-$ smallest; the interquartile range $= Q_{3} - Q_{1}$ ignores extremes.

Worked example

Find the median of $4, 8, 6, 10, 6, 2, 9$ .

Order: $2, 4, 6, 6, 8, 9, 10$ . With $7$ values the median is the $4$ th $= 6.$

Exam tip

The median resists outliers; the mean does not. If a data set has an extreme value, expect the question to favour the median.

For grouped data you estimate the mean using midpoints: $mean \approx \frac{\sum ( f \times midpoint )}{\sum f}$ . The exact values are lost, so it is only an estimate.

On a histogram with unequal class widths, the bar height is frequency density $= \frac{frequency}{class width}$ , so frequency is the area of the bar, not its height.

Worked example

A histogram bar spans $10$ – $30$ (width $20$ ) at frequency density $1.5$ . How many values?

Frequency $=$ area $= 20 \times 1.5 = 30.$

A scatter graph shows correlation (positive, negative or none) between two variables. A line of best fit lets you estimate, but correlation does not imply causation, and extrapolating beyond the data is risky.

Reading frequency off the height of a histogram bar instead of its area.
Treating the estimated grouped mean as exact.
Concluding one variable *causes* another from correlation alone.

Statistics

Overview

Averages & spread

Grouped data & histograms

Scatter graphs & correlation

Common mistakes