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Box plot skewness is read from where the median line sits inside the box and how long the two whiskers are. If the median is closer to the lower quartile (Q1) and the upper whisker is longer, the distribution is right-skewed; if the median is closer to the upper quartile (Q3) and the lower whisker is longer, it is left-skewed; if the median is near the middle of the box and the whiskers are about equal, it is roughly symmetric. This guide shows all three on worked data, compares the visual read with a computed skewness coefficient, and explains when the box misleads.
Definition: skewness from a box plot
NIST defines skewness as “a measure of symmetry, or more precisely, the lack of symmetry” (NIST/SEMATECH e-Handbook, 1.3.5.11). A box plot never shows skewness directly. It shows five summary values: the minimum or lower whisker end, the lower quartile, the median, the upper quartile and the maximum or upper whisker end. NIST describes the quartiles as the 25th and 75th percentiles, so the box holds the middle 50% of the data. Skewness is inferred from how asymmetrically those five values are spread. The single-direction case is covered in right-skewed box plot; this page puts all three cases side by side.
Box and whisker plot skewness: the three cues
These cues are editorial interpretation derived from the quartile definitions, not rules stated in the NIST or OpenStax pages cited below.
| Cue | Symmetric | Right-skewed (positive) | Left-skewed (negative) |
|---|---|---|---|
| Median inside the box | Near the centre | Nearer Q1 (lower edge) | Nearer Q3 (upper edge) |
| Whiskers | About equal length | Upper whisker longer | Lower whisker longer |
| Outliers beyond the fences | Few, on either side | Typically on the high side | Typically on the low side |
| Mean relative to median | About the same | Mean often above the median | Mean below the median |
The last row follows OpenStax Introductory Statistics 2e, section 2.6: for a symmetric distribution the mean and median are the same, for a left-skewed one the mean is less than the median, and for a right-skewed one the mean is often greater than the median. For the mean-versus-median logic in more depth, see mean greater than median.
Formula: how the box and the skewness coefficient are computed
Box plot pieces. IQ = Q3 – Q1. NIST gives the fences as L1 = Q1 – 1.5 × IQ and U1 = Q3 + 1.5 × IQ; points beyond them are drawn individually as circles. A whisker is drawn to the most extreme value that is still inside a fence. For quartile background see percentiles and quartiles.
Median-position ratio (descriptive, used on this page). R = (Q3 – Median) / (Median – Q1). R near 1 means the median is centred in the box; R above 1 means the median sits nearer Q1 (right-skew cue); R below 1 means it sits nearer Q3 (left-skew cue). This is a plain arithmetic summary of the box, not a named statistic.
Moment skewness g1 (NIST). g1 = [ Σ(Yi – Ȳ)³ / N ] / s³, where Yi is each observation, Ȳ the mean, N the number of observations and s the standard deviation computed with N (not N – 1) in the denominator. NIST states that negative values indicate data skewed left and positive values data skewed right.
Worked example: three datasets, one box plot each
Each illustrative dataset has N = 11 values. Quartiles use the linear-interpolation method (R type 7, also Excel QUARTILE.INC and the NumPy default): the position of Q1 is 1 + 0.25 × (N – 1) = 3.5 and of Q3 is 1 + 0.75 × (N – 1) = 8.5 in the sorted data. Different quartile methods can give slightly different values, so name the method you used.
- A (left-skewed): 52, 70, 78, 82, 85, 87, 88, 90, 91, 92, 93
- B (symmetric): 40, 46, 50, 53, 55, 57, 59, 62, 64, 68, 74
- C (right-skewed): 3, 4, 4, 5, 6, 7, 9, 12, 16, 24, 41
Dataset A. Q1 is halfway between the 3rd and 4th values: (78 + 82) / 2 = 80. The median is the 6th value, 87. Q3 is halfway between the 8th and 9th: (90 + 91) / 2 = 90.5. IQ = 90.5 – 80 = 10.5, so the fences are 80 – 15.75 = 64.25 and 90.5 + 15.75 = 106.25. The value 52 lies below the lower fence and is plotted as an outlier; the lower whisker ends at 70, and the upper whisker at 93. R = (90.5 – 87) / (87 – 80) = 3.5 / 7 = 0.50.
Dataset B. Q1 = (50 + 53) / 2 = 51.5, median = 57, Q3 = (62 + 64) / 2 = 63. IQ = 11.5, fences 34.25 and 80.25, so there are no outliers and the whiskers run to 40 and 74. R = (63 – 57) / (57 – 51.5) = 6 / 5.5 ≈ 1.09.
Dataset C. Q1 = (4 + 5) / 2 = 4.5, median = 7, Q3 = (12 + 16) / 2 = 14. IQ = 9.5, fences -9.75 and 28.25. The value 41 is above the upper fence (an outlier); the upper whisker ends at 24 and the lower whisker at 3. R = (14 – 7) / (7 – 4.5) = 7 / 2.5 = 2.80.
Interpretation: box plot and skewness side by side
| Dataset | Min | Q1 | Median | Q3 | Max | Mean | R | g1 |
|---|---|---|---|---|---|---|---|---|
| A (left) | 52 | 80 | 87 | 90.5 | 93 | 82.55 | 0.50 | -1.57 |
| B (symmetric) | 40 | 51.5 | 57 | 63 | 74 | 57.09 | 1.09 | -0.02 |
| C (right) | 3 | 4.5 | 7 | 14 | 41 | 11.91 | 2.80 | 1.65 |
The box read and g1 agree here: R below 1 with negative g1 (A), R close to 1 with g1 near zero (B), R above 1 with positive g1 (C). Dataset A also shows the OpenStax pattern, with a mean (82.55) below the median (87), and C a mean (11.91) above its median (7). g1 was computed as defined by NIST, with N in the standard deviation (s = 11.65, 9.38 and 10.99 for A, B and C).
Box plot and skewness: when the box misleads
- Outliers drive g1, not the box. g1 cubes each deviation from the mean, so one extreme value such as 41 in dataset C or 52 in dataset A moves it strongly, while the box (built on quartiles) barely moves. A modest-looking box with a large g1 usually means a few extreme points.
- Quartiles ignore the tails. A box plot can look symmetric while the tails beyond the quartiles are lopsided, and the reverse. Use a histogram for the full shape; see left-skewed histogram.
- It cannot show multimodality. Two clusters can produce the same five numbers as one smooth peak.
- Small samples. With N = 11, moving one value can flip the quartile positions; treat visual skew in small samples as descriptive only.
- Quartile method. Tukey hinges, type 6, type 7 and others can disagree on Q1 and Q3, which changes the median-position read. State the method.
- Do not equate “outlier” with “error”. Fence-flagged points are candidates for checking; see how to find an outlier.
How to report it
Report the five-number summary, name the quartile method, and give a coefficient if you want a number. An illustrative sentence in APA-style reporting: “Scores were left-skewed (Mdn = 87.0, IQR = 10.5, g1 = -1.57), with one low outlier at 52.” Describe skewness in words from both the plot and the coefficient, and state N.
How to calculate it in Excel, R or Python
Excel (data in A1:A11): =QUARTILE.INC(A1:A11,1), =MEDIAN(A1:A11), =QUARTILE.INC(A1:A11,3) and =SKEW.P(A1:A11) for population skewness (N in the denominator, matching the NIST g1). The plain SKEW function applies a sample adjustment and gives a different value.
# R (quartile type 7 is the default) x <- c(52,70,78,82,85,87,88,90,91,92,93) fivenum_t7 <- quantile(x, c(0, .25, .5, .75, 1), type = 7) g1 <- mean((x - mean(x))^3) / (mean((x - mean(x))^2))^1.5 boxplot(x, horizontal = TRUE)# Python import numpy as np from scipy.stats import skew x = np.array([52,70,78,82,85,87,88,90,91,92,93]) q1, med, q3 = np.percentile(x, [25, 50, 75]) # linear method = type 7 g1 = skew(x) # bias=True, N in the denominatorMore on drawing and customising the plot is in box plots in R, and the five-number summary is explained in interquartile range and the five-number summary.
FAQ
How do you tell skewness from a box plot?
Look at the median inside the box and the whisker lengths. Median near Q1 with a longer upper whisker is right-skewed; median near Q3 with a longer lower whisker is left-skewed; a centred median with equal whiskers is roughly symmetric.
Is the skewness box plot read the same as a histogram read?
Usually, but not always. The box uses only quartiles and whisker ends, so it can disagree with a histogram or g1 when tails or outliers are lopsided or the data are multimodal.
Does a long upper whisker always mean positive skewness?
No. A long whisker with the median centred in the box may reflect a few high values rather than overall asymmetry. Check the median position and a coefficient such as g1 as well.
What does a negative skewness value mean on a box and whisker plot?
NIST states that negative skewness indicates data skewed left, meaning the left tail is long relative to the right. On the box plot this typically shows as a median nearer Q3 and a longer lower whisker.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods, Box Plot (read 2026-11-07).
- NIST/SEMATECH e-Handbook, 1.3.5.11 Measures of Skewness and Kurtosis (read 2026-11-07).
- OpenStax, Introductory Statistics 2e, 2.6 Skewness and the Mean, Median, and Mode (read 2026-11-07).
- OpenStax, Introductory Statistics 2e, 2.4 Box Plots (read 2026-11-07).
