Summary Statistics, Boxplots, and Outliers

Measures of center

• Mean x̄ = Σx / n. Sensitive to outliers (not resistant).
• Median: the middle value. Resistant to outliers.

Measures of spread

• Range = max − min.
• Interquartile range IQR = Q3 − Q1 (resistant).
• Standard deviation s: the typical distance of values from the mean (not resistant).

Interpreting s: The heights typically vary by about 4.2 cm from the mean of 170 cm.

Five-number summary and boxplots

Min, Q1, median, Q3, max. A boxplot draws the box from Q1 to Q3 with a line at the median.

The 1.5 × IQR rule for outliers

A value is an outlier if it is

• below Q1 − 1.5 × IQR, or
• above Q3 + 1.5 × IQR.

Example. Q1 = 20, Q3 = 30, IQR = 10. Fences: 5 and 45. A value of 50 is an outlier.

Effect of transformations

Operation on every value	Mean, median	SD, IQR
add a constant c	increase by c	unchanged
multiply by b	multiply by b	multiply by abs(b)

Converting °C to °F (multiply by 1.8, add 32): the mean is transformed fully, the SD is multiplied by 1.8 only.

Which summary to use

• Skewed data or outliers: median and IQR.
• Roughly symmetric data: mean and standard deviation.

Quick summary

• Median and IQR resist outliers; mean and SD do not.
• Outlier fences: Q1 − 1.5 IQR, Q3 + 1.5 IQR.
• Adding shifts center only; multiplying scales center and spread.