Range and interquartile range
- The range = largest value − smallest value. It is easy to find but affected by extreme values.
- The interquartile range (IQR) = UQ − LQ. It measures the spread of the middle 50% and is not affected by extreme values.
- The interdecile range = 9th decile − 1st decile. It covers the middle 80%. Interpercentile ranges work the same way.
Standard deviation
- Standard deviation measures how spread out values are around the mean. It uses every value.
- Standard deviation = √(Σx² ÷ n − mean²). Variance is the standard deviation squared.
- Example: 2, 4, 4, 4, 5, 5, 7, 9. The mean is 5 and Σx² = 232. 232 ÷ 8 = 29, then 29 − 25 = 4, and √4 = 2.
- From summary values: n = 10, Σx = 50 and Σx² = 340 give a mean of 5 and a standard deviation of √(34 − 25) = 3.
- For a frequency table, use √(Σfx² ÷ Σf − mean²). Calculators in statistics mode can find it directly.
Interpreting spread
- A bigger standard deviation or IQR means the data is more spread out, so it is less consistent.
- Always compare in context, such as "Team A's scores were more consistent than Team B's because their standard deviation was smaller."
Key terms
- Range
- The largest value minus the smallest value.
- Interquartile range
- Upper quartile minus lower quartile.
- Interdecile range
- The 9th decile minus the 1st decile.
- Standard deviation
- A measure of how spread out values are around the mean.
- Variance
- The square of the standard deviation.
- Consistent
- Having values close together, with little spread.