Box Plot & Five-Number Summary Calculator

What Is a Box Plot?

A box plot (also called a box-and-whisker plot) is a standardized way of displaying the distribution of data based on a five-number summary. Invented by statistician John Tukey in 1977, it gives you a quick visual snapshot of where data is centered, how spread out it is, and whether any values are unusually extreme.

The Five-Number Summary

The five-number summary consists of:

• Minimum — the smallest value in the data set (excluding outliers)
• Q1 (First Quartile) — the median of the lower half of the data; 25% of values fall below this
• Median (Q2) — the middle value; 50% of values fall on each side
• Q3 (Third Quartile) — the median of the upper half; 75% of values fall below this
• Maximum — the largest value (excluding outliers)

Interquartile Range (IQR)

The IQR = Q3 - Q1 measures the spread of the middle 50% of the data. It is a robust measure of variability — meaning it is not affected by extreme values the way the range or standard deviation are. A large IQR means high variability; a small IQR means the data is tightly clustered.

Outlier Detection (Tukey Fences)

John Tukey defined outliers using fences:

• Lower fence = Q1 - 1.5 × IQR
• Upper fence = Q3 + 1.5 × IQR

Any value below the lower fence or above the upper fence is considered an outlier. Outliers are plotted as individual points beyond the whiskers of a box plot.

Skewness Detection

Comparing the mean to the median reveals skewness:

• Mean ≈ Median → roughly symmetric distribution
• Mean > Median → right-skewed (positive skew) — a few high values pull the mean up
• Mean < Median → left-skewed (negative skew) — a few low values pull the mean down

Example: For the data set {3, 7, 8, 10, 12, 14, 15, 18, 22, 50}: Q1=8, Median=13, Q3=18, IQR=10. Lower fence = 8 - 15 = -7. Upper fence = 18 + 15 = 33. The value 50 is an outlier. Mean = 15.9 > Median = 13 → right-skewed.

Box plots are ideal for comparing distributions side by side, spotting outliers quickly, and understanding data shape without drawing a full histogram.