Mathematical Ability - Statistics and Data Interpretation: Averages, Spread and Chart Reading
Measures of central tendency
Mean = sum ÷ count; it shifts with every value. Median is the middle value after sorting, or the average of the two middle values for an even count, and it resists outliers. Mode is the most frequent value; a data set can be bimodal or have no mode. For grouped data, use class midpoints for the mean and the median-class formula for the median.
Dispersion
Range = maximum − minimum. Variance is the mean of squared deviations; standard deviation is its square root. Adding a constant c to every observation raises the mean and median by c but leaves the standard deviation, variance and range unchanged. Multiplying every observation by k multiplies the standard deviation and range by |k| and the variance by k².
Data interpretation
Read the axes and units before the numbers; a graph whose baseline is not zero exaggerates differences. Pie charts total 360°, so a sector of 72° is 20%. Percentage change = (new − old)/old × 100. Ratio questions ask for part-to-part, not part-to-whole. When a table supplies row and column totals, cross-check them - a value is often obtainable only by subtraction.
Worked example
Data: 4, 7, 7, 9, 13. Sum = 40, so mean = 8; sorted, the median is 7; the mode is 7. Add 5 to every value to get 9, 12, 12, 14, 18: the mean becomes 13 and the median becomes 12, but the standard deviation is unchanged, because the data set was only shifted, not stretched.
Traps
- Forgetting to sort before finding the median.
- Expecting the standard deviation to change when a constant is added.
- Reading a pie-chart percentage as a degree value.
- Averaging percentages without weighting by their bases.