Lesson 2: Spread — range, variance, standard deviation, IQR

Knowing the mean is only half the story. You also need to know how widely the data are spread around the center: two groups can have the same mean but very different spread. The range is the simplest measure: maximum minus minimum. For [2, 4, 4, 4, 5, 5, 7, 9]: range = 9 − 2 = 7. It's easy to calculate, but it depends only on the two extreme points and is very sensitive to outliers. The variance takes every point into account. Steps: 1) find the mean; 2) subtract it from each value and square; 3) add up the squared deviations; 4) divide by the number of values (here N, the whole population). For [2, 4, 4, 4, 5, 5, 7, 9], mean = 5. Squared deviations: (2−5)²=9, (4−5)²=1, (4−5)²=1, (4−5)²=1, (5−5)²=0, (5−5)²=0, (7−5)²=4, (9−5)²=16. Sum = 32. Variance = 32 / 8 = 4. Standard deviation (SD) = √variance = √4 = 2. SD is in the same units as the data — its main advantage. Quartiles and the IQR. Q1 is the median of the lower half of the data (the 25th percentile), Q3 the median of the upper half (the 75th percentile). The interquartile range IQR = Q3 − Q1 describes the spread of the “middle” 50% of the data. For [2, 4, 4, 4, 5, 5, 7, 9]: lower half [2, 4, 4, 4] → Q1 = (4+4)/2 = 4; upper half [5, 5, 7, 9] → Q3 = (5+7)/2 = 6; IQR = 6 − 4 = 2. Like the median, the IQR is resistant to outliers.