Measures of Spread: Range, IQR, and Standard Deviation

Measures of Spread

By the end of this lesson, you will be able to calculate and interpret the range, IQR, standard deviation, and variance, and choose the correct measure for a given distribution.

Why Spread Matters

Two distributions can have the same center but very different spreads. A class where every student scores 80 and a class where scores range from 20 to 100 both have a mean of 80 — but they're describing entirely different situations.

Range

$$\text{Range} = \text{Max} - \text{Min}$$

Simple, but heavily influenced by outliers. Two datasets with the same range can have very different distributions.

Interquartile Range (IQR)

$$\text{IQR} = Q_3 - Q_1$$

The range of the middle 50% of the data.

Computing quartiles:

1. Find the median (splits data into lower and upper halves)
2. Q₁ = median of the lower half
3. Q₃ = median of the upper half

Example: Data: 2, 5, 7, 8, 10, 12, 15, 18

Median = $\frac{8+10}{2} = 9$

Lower half: 2, 5, 7, 8 → Q₁ = $\frac{5+7}{2} = 6$

Upper half: 10, 12, 15, 18 → Q₃ = $\frac{12+15}{2} = 13.5$

IQR = 13.5 − 6 = 7.5

IQR is resistant to outliers — like the median, it doesn't use the extreme values.

1.5 × IQR Outlier Rule:

• Below Q₁ − 1.5(IQR): outlier
• Above Q₃ + 1.5(IQR): outlier

Standard Deviation

$$s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}$$

The standard deviation measures the typical distance of values from the mean.

Computing s:

1. Find the mean $\bar{x}$
2. Find each deviation: $x_i - \bar{x}$
3. Square each deviation: $(x_i - \bar{x})^2$
4. Sum the squared deviations
5. Divide by $n - 1$ (sample variance $s^2$)
6. Take the square root

Why n − 1? Using n would systematically underestimate the population variance. Dividing by n − 1 corrects this bias (this is called Bessel's correction).

Example: Data: 2, 4, 6, 8, 10; $\bar{x} = 6$