Statistical Representation of Data and Measures of Central Tendency and Dispersion
This domain covers how data are organised and summarised: frequency distributions, class boundaries and midpoints, histograms and ogives, and the measures that describe centre and spread. The exam tests both calculation and interpretation, especially the effect of skewness and the choice between population and sample formulas.
Core rules
- For grouped data, use class midpoints as x-values; class width = upper boundary − lower boundary.
- Mean = Σfx/Σf; median position = (n+1)/2; modal class is the class with highest frequency.
- Quartiles: Q1 at (n+1)/4, Q3 at 3(n+1)/4; IQR = Q3 − Q1; range = max − min.
- Population variance = Σ(x − μ)²/N; sample variance = Σ(x − x̄)²/(n − 1). Standard deviation is the square root.
- Coefficient of variation = (standard deviation/mean) × 100%; it compares relative dispersion and is meaningful only for a positive mean.
- If mean > median the distribution is positively skewed; if mean < median it is negatively skewed.
Worked example
Data: 2, 4, 4, 6, 9. Mean = 25/5 = 5; median = 4; mode = 4; range = 7. Deviations from 5: −3, −1, −1, 1, 4; squares sum to 28. Population variance = 28/5 = 5.6, SD = 2.37. Sample variance = 28/4 = 7, SD = 2.65. CV (population) = 2.37/5 × 100 = 47.3%. Mean exceeds median, so the data are positively skewed; the single value 9 pulls the mean upward but leaves the median at 4.
Traps
- Dividing by n when the question says sample.
- Using class limits instead of boundaries for grouped data.
- Reporting variance in the same units as the data; variance is squared.
- Comparing SDs across data sets with very different means instead of using CV.