Shapes of distributions
- A symmetrical distribution has mean = median = mode, and looks the same on both sides.
- Positive skew: the tail is on the right (high values). Mean > median > mode. On a box plot, UQ − median is greater than median − LQ.
- Negative skew: the tail is on the left (low values). Mean < median < mode. On a box plot, median − LQ is greater than UQ − median.
- Incomes are usually positively skewed: most earn moderate amounts, and a few earn a lot.
Measuring skewness
- Pearson's coefficient of skewness = 3 × (mean − median) ÷ standard deviation.
- A positive answer means positive skew; a negative answer means negative skew; close to 0 means roughly symmetrical.
- Example: mean 30, median 27 and standard deviation 6 give 3 × 3 ÷ 6 = 1.5, so the data is positively skewed.
- The quartile measure uses (UQ − 2 × median + LQ) ÷ (UQ − LQ).
Outliers
- An outlier is a value that is very different from the rest.
- A value is an outlier if it is below LQ − 1.5 × IQR or above UQ + 1.5 × IQR. With LQ 20 and UQ 30, the IQR is 10, so outliers are below 5 or above 45.
- Another rule treats values more than 3 standard deviations from the mean as outliers.
- If an outlier is a mistake, remove it when cleaning the data. If it is genuine, keep it and explain it.
Key terms
- Symmetrical distribution
- A distribution where mean, median and mode are equal.
- Positive skew
- A distribution with a tail of high values.
- Negative skew
- A distribution with a tail of low values.
- Pearson's coefficient of skewness
- 3 × (mean − median) ÷ standard deviation.
- Outlier
- A value very different from the rest of the data.