Data can be distributed in different shapes: symmetric (normal), positively skewed, negatively skewed, uniform, bimodal/multimodal and J-shaped; they also differ in peakedness (kurtosis).
- Normal (symmetric, bell-shaped): most values cluster around the centre; mean = median = mode. About 68% of values lie within \(\pm1\sigma\), 95% within \(\pm2\sigma\), 99.7% within \(\pm3\sigma\). Example: heights of adults, measurement errors.
- Positively (right) skewed: long tail to the right; mean > median > mode. Example: incomes, house prices, waiting times.
- Negatively (left) skewed: long tail to the left; mean < median < mode. Example: marks in an easy exam, age at retirement.
- Uniform (rectangular): all values roughly equally frequent; flat histogram. Example: outcome of a fair die, random number generators.
- Bimodal / multimodal: two or more peaks, often meaning the data mix two groups. Example: heights of a mixed group of men and women; restaurant customers at lunch and dinner.
- J-shaped / reverse J (exponential-like): frequencies keep rising or keep falling. Example: time between failures of a device, number of people with \(k\) accidents.
Kurtosis (peakedness)
- Mesokurtic: normal peak.
- Leptokurtic: sharper peak and heavier tails (e.g. daily stock returns).
- Platykurtic: flatter than normal.
Why it matters: the shape decides which average is suitable (median for skewed data), whether parametric tests that assume normality can be used, and whether transformations (such as log) are needed before modelling. Shape is checked using histograms, box plots and coefficients of skewness, e.g. Karl Pearson's \(S_k = \dfrac{\text{Mean} - \text{Mode}}{\sigma}\).