Averages, Spread and Charts

The three averages + range

• Mean = sum of values ÷ how many. Uses every value; affected by outliers.
• Median = the middle value when ordered. Best with outliers.
• Mode = most common value. Only average for categories.
• Range = highest − lowest (a measure of spread, not an average).

Median position

For n values, the median is at position (n + 1) ÷ 2 in the ordered list. For an even n, average the two middle values.

Mean from a frequency table

        Σ(value × frequency)
mean = ──────────────────────
            Σ frequency

Estimated mean (grouped data)

Use the midpoint of each class as the value, then apply the formula above. The answer is an estimate because exact values are unknown.

Scatter graphs & correlation

• Positive – both increase. Negative – one up, one down. None – no pattern.
• A line of best fit lets you estimate values. Predicting outside the data range = extrapolation (unreliable).

Other charts

• Cumulative frequency → running total; read off the median (½n) and quartiles (¼n, ¾n). IQR = UQ − LQ.
• Box plots show min, LQ, median, UQ, max.

Exam tip

"Estimate the mean" signals grouped data → use class midpoints. Don't confuse the mean with the median.