What does my box plot look like?

How it works

Sort the n numbers: x₍₁₎ ≤ x₍₂₎ ≤ … ≤ x₍ₙ₎. The minimum is x₍₁₎, the maximum is x₍ₙ₎, and the median is the middle value (the average of the two middle values when n is even). Q1 and Q3 follow the method you pick, the same four methods as the quartile calculator:

• Median of halves (TI-83), the default. The lower half is the first ⌊n ÷ 2⌋ sorted values and the upper half the last ⌊n ÷ 2⌋, so for odd n the median is in neither. Q1 and Q3 are the medians of the halves.
• Tukey's hinges. The same, but for odd n the median is in both halves.
• Inclusive (Excel QUARTILE.INC). The p-quantile is at position h = 1 + (n − 1)p; with j the whole part of h, it is x₍ⱼ₎ + (h − j)(x₍ⱼ₊₁₎ − x₍ⱼ₎).
• Exclusive (Excel QUARTILE.EXC). The same interpolation at h = (n + 1)p. A position below 1 or above n reads as the smallest or largest value.

Then:

• IQR = Q3 − Q1, the width of the box.
• Whiskers. "Min to max": the whiskers end at the minimum and maximum. "1.5 × IQR": they end at the smallest and largest values inside the fences Q1 − 1.5 × IQR and Q3 + 1.5 × IQR (a value on a fence is inside).
• Outliers are the values strictly outside the fences, smallest first (repeats listed each time), or "None". They are listed with either whisker style.
• The five-number summary and the outliers show each value rounded half up from its exact value to 10 significant digits, comma separated, with a true minus sign.

Rules:

• 1 to 10,000 numbers. With one number, every value of the summary is that number.
• The quartiles, IQR, fences and the outlier test are exact on the decimals you type.

Assumptions

• The dots under the box are your numbers; outliers sit beyond the whiskers when the whiskers stop at the fences.

Worked examples by hand

1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5 (the default, OpenStax §2.4, n = 14). The median is (6.8 + 7.2) ÷ 2 = 7. The lower half 1, 1, 2, 2, 4, 6, 6.8 has median Q1 = 2; the upper half 7.2, 8, 8.3, 9, 10, 10, 11.5 has median Q3 = 9. The summary is 1, 2, 7, 9, 11.5 and IQR = 7. The fences are 2 − 10.5 = −8.5 and 9 + 10.5 = 19.5, so there are no outliers.

10, 20, 30, 40, 50, 60, 70, 80, 90, 1000, whiskers at the fences. Q1 = 30, median 55, Q3 = 80, IQR = 50. The upper fence is 80 + 75 = 155, so 1000 is an outlier and the upper whisker stops at 90. The lower fence is 30 − 75 = −45, so the lower whisker stops at 10.

1 to 10, inclusive method. h = 1 + 9 × 0.25 = 3.25 gives Q1 = 3.25 and h = 7.75 gives Q3 = 7.75; the median is 5.5. The summary is 1, 3.25, 5.5, 7.75, 10, IQR = 4.5.

1 to 10, exclusive method. h = 11 × 0.25 = 2.75 gives Q1 = 2.75 and h = 8.25 gives Q3 = 8.25, so the summary is 1, 2.75, 5.5, 8.25, 10, IQR = 5.5.

1 to 9, Tukey's hinges. The halves 1–5 and 5–9 have medians 3 and 7, so the summary is 1, 3, 5, 7, 9, IQR = 4.