Descriptive statistics: study guide

Key ideas

Descriptive statistics summarizes a set of values by its center and its spread. The mean, median and mode describe the center, while the range, variance, standard deviation and coefficient of variation describe the spread.

• The mean is the sum divided by the count. The median is the middle value of the sorted list, and the mode is the most frequent value.
• The sample standard deviation divides the sum of squared deviations by n − 1, and the population standard deviation divides by N.
• The range is the largest value minus the smallest. The coefficient of variation CV = s ÷ mean × 100 compares spread between data sets of different size.
• A few extreme values pull the mean, but the median barely moves.

Key equations

• mean = Σx ÷ n
• s = √(Σ(x − x̄)² ÷ (n − 1))
• σ² = Σ(x − μ)² ÷ N
• range = max − min
• CV = s ÷ mean × 100

Worked example

For 4, 6, 8, 10, 12 the mean is 8 and the median is 8. The squared deviations are 16, 4, 0, 4, 16, which sum to 40, so the sample variance is 40 ÷ 4 = 10 and s = 3.16. The range is 12 − 4 = 8 and CV = 3.16 ÷ 8 × 100 = 39.5 %.

Common mistakes

• Dividing by n instead of n − 1 for a sample standard deviation.
• Taking the median without sorting the values first.
• Reporting the variance in the units of the data. Variance is in squared units; the standard deviation is back in the original units.

Practice

Test yourself with practice questions on this topic: Descriptive statistics