Measures of Dispersion

Definition: Difference between the highest and lowest score; simple to interpret
Formula: $\text{Range} = \text{Highest score} - \text{Lowest score}$
How to work: Example: Highest 35, Lowest 8 → Range = 27
Caution: Do not report the min and max values as a pair; report only the range. Example: Highest 85, Lowest 8 → Range = 77

Definition: The dispersion of the central 50% of values in the dataset
Quartiles: Q1 is the lower quartile (25th percentile); Q3 is the upper quartile (75th percentile)
IQR: $\text{IQR} = Q<em>3 - Q</em>1$
Position notes: Q1 lies at 25th percentile; Q3 lies at 75th percentile
Example: Median lies between 60 and 62; Q1 between 52 and 53 (Q1 = 52.5); Q3 between 70 and 71 (Q3 = 70.5); IQR = 70.5 - 52.5 = 18; Range = 80 - 43 = 37

Definition: Indicates dispersion around the mean; considers distance of each score from the mean; depends on sample size; used to compare variability across samples
Formula: $s = \sqrt{\frac{1}{n-1}\sum<em>{i=1}^n (x</em>i - \bar{x})^2}$
Calculation steps: compute mean, compute deviations (x_i - \bar{x}), square deviations, sum, divide by (n-1) to get variance, then take the square root to get the standard deviation
Example: If the sum of squared deviations is 10 and n = 5, $s = \sqrt{\frac{10}{5-1}} = \sqrt{2.5} \approx 1.58$
Understanding:
- About 2/3 of data fall within \pm 1 SD of the mean (assuming normal distribution)
- SD = 0 means no variability; all scores are the same
- SD is sensitive to outliers
- The size of SD depends on dispersion and the units of measurement (e.g., metres vs kilometres)

Range: definition, formula, interpretation; do not report min/max values—report only the range
Interquartile Range: definition, Q1 and Q3, IQR formula; central 50% dispersion
Standard Deviation: definition, formula, interpretation; relation to normal distribution; outlier sensitivity; depends on dispersion and units