Left-Skewed Histogram: How to Read It, Check It and Report It

A left-skewed histogram has its tallest bars toward the right and a long tail stretching to the left. It is also called a negatively skewed histogram, a skewed left histogram or a histogram skewed left. The word “left” names the side of the tail, not the side where the bars pile up.

Definition: what a left-skewed histogram looks like

A histogram summarises one variable by splitting its range into equal-width bins and counting how many observations fall in each (NIST/SEMATECH e-Handbook, section 1.3.3.14). In a left-skewed histogram the bars rise slowly across the low values, peak near the high end, then drop off sharply. NIST defines it directly: “By skewed left, we mean that the left tail is long relative to the right tail.” OpenStax puts it the same way: the distribution “is called skewed to the left because it is pulled out to the left.”

Skewed left histogram vs histogram skewed left

These are the same shape under different word orders. “Left skewed”, “skewed left” and “negatively skewed” all describe a long left tail. Typical sources are scores with a ceiling (an easy exam) or any measure bunched against an upper limit. For the reverse shape, see left-skewed vs right-skewed distributions and, for box plots, how to read a right-skewed box plot.

Formula: skewness g₁ and why it is negative

NIST gives the moment coefficient of skewness for data Y₁, …, Y_N as:

g₁ = [ Σ(Y_i − Ȳ)³ / N ] / s³

• Y_i: the i-th observation; Ȳ: the sample mean; N: the number of observations.
• s: the standard deviation, computed with N (not N − 1) in the denominator, as NIST notes for this formula.
• Σ: the sum over all N observations.

NIST states that “negative values for the skewness indicate data that are skewed left.” The reason is the cube: the few very low values lie far below the mean, and their cubed deviations are large and negative, so they outweigh the many small positive deviations on the high side. For the coefficient and the Pearson alternatives in more depth, see the skewness guide.

Worked example: 20 exam scores

An illustrative dataset (invented for this example, not real student data) of 20 exam scores out of 100, sorted: 48, 55, 62, 68, 72, 75, 77, 79, 81, 82, 84, 85, 86, 87, 88, 89, 90, 92, 93, 95.

Step 1: the histogram counts (bins of width 10, each bin includes its lower edge)

Bin	40–49	50–59	60–69	70–79	80–89	90–99
Count	1	1	2	4	8	4

The counts add to 20. The tallest bar (80–89) sits right of centre and the bars thin out toward 40: a left-skewed shape.

Step 2: mean, median, spread

• Mean: 1,588 / 20 = 79.4.
• Median: the 10th and 11th sorted values are 82 and 84, so the median is (82 + 84) / 2 = 83.
• Quartiles (inclusive method): Q1 = 74.25, Q3 = 88.25, so IQR = 14.
• Sum of squared deviations = 3,122.8; divided by N = 20 gives 156.14; s = √156.14 = 12.50 (with N − 1 it would be 12.82).

Step 3: skewness

The cubed deviations from the mean sum to −40,694.64. Dividing by N gives −2,034.73. The cube of s is 12.4956³ ≈ 1,951.1. So g₁ = −2,034.73 / 1,951.1 = −1.04.

The negative sign agrees with the picture: the mean (79.4) is pulled below the median (83) by the low scores. Software that applies a small-sample adjustment, such as Excel’s SKEW or scipy.stats.skew(..., bias=False), reports −1.13 for the same data. State which version you used.

Reading a histogram skewed left: mean, median and mode

In its left-skewed example, OpenStax notes: “the mean is less than the median, and they are both less than the mode.” It also hedges, saying the mean is “often” less than the median, and that the mean reflects the skew more than the median does. Treat the ordering as a tendency, not a law.

What you see or compute	What it indicates for a left-skewed histogram	Source
Long tail on the left, tallest bars on the right	Skewed left (negatively skewed)	NIST 1.3.5.11; OpenStax 2.6
g1 below 0	Skewed left	NIST 1.3.5.11
Mean below median	Common in left-skewed data; the mean is pulled toward the tail	OpenStax 2.6
Mean and median both below the mode	The typical ordering OpenStax shows for a distribution skewed left	OpenStax 2.6

Because the median is less affected by the tail than the mean, it is the usual summary to report beside the IQR for data like this. See median and interquartile range for when each is appropriate.

Assumptions and common mistakes

• Confusing the tail with the peak. A left-skewed histogram looks like it leans right because the bars are tallest on the right. The name follows the tail.
• Relying on one number. NIST calls the histogram “an effective graphical technique” for showing skewness; check the picture and the coefficient together.
• Bin edges change the look. Bin width and which edge each bin includes can alter the shape, especially for small samples. Try more than one width before concluding.
• Assuming mean < median always holds. It is a common pattern, not a guarantee.
• Treating skewed data as normal. Methods that assume a symmetric, bell-shaped distribution may fit poorly; compare against the normal curve overlay before choosing a test.

How to report it (APA example)

For the example above, one way to write it in APA style is:

Exam scores were negatively skewed (g₁ = −1.04), with a median of 83 (IQR = 14) and a mean of 79.4 (SD = 12.82).

Report the median and IQR alongside the mean when the distribution is clearly skewed, say which skewness formula you used, and include the histogram as a figure. The SD of 12.82 is the usual sample standard deviation (N − 1); the s of 12.50 above is the version the skewness formula uses.

How to calculate and plot it in Excel, R or Python

Excel

=SKEW.P(A2:A21)
=SKEW(A2:A21)
=AVERAGE(A2:A21)
=MEDIAN(A2:A21)

Results for the 20 scores:
SKEW.P = -1.04 (g1, N in the denominator)
SKEW = -1.13 (adjusted sample skewness)
AVERAGE = 79.4
MEDIAN = 83

To draw it, select the data and use Insert > Histogram, then set the bin width to 10.

R

y <- c(48,55,62,68,72,75,77,79,81,82,84,85,86,87,88,89,90,92,93,95)
hist(y, breaks = seq(40, 100, by = 10), right = FALSE)
m <- mean(y)
s <- sqrt(mean((y - m)^2))   # s uses N
g1 <- mean((y - m)^3) / s^3  # -1.043
c(mean = m, median = median(y), g1 = g1)

right = FALSE makes each bin include its lower edge; the default would move the score 90 into the 80–89 bar.

Python

import numpy as np
from scipy.stats import skew
import matplotlib.pyplot as plt

y = [48,55,62,68,72,75,77,79,81,82,84,85,86,87,88,89,90,92,93,95]
print(np.mean(y), np.median(y))   # 79.4 83.0
print(skew(y))                    # -1.0429 (g1)
print(skew(y, bias=False))        # -1.1294 (adjusted)
plt.hist(y, bins=range(40, 101, 10), edgecolor="black")
plt.show()

For more on drawing histograms, see histograms in R.

FAQ

Is a histogram skewed left the same as negatively skewed?

Yes. Both mean the left tail is the long one, and skewness is below zero.

Where is the peak in a left-skewed histogram?

Toward the right (the high values). The tail, not the peak, gives the direction its name.

Is the mean greater or less than the median?

Often less, because the low values pull the mean toward the tail. OpenStax says “often”, so check your own data instead of assuming it.

Which summary should I report?

The median with the IQR describes the centre and spread without being dragged by the tail; the mean and SD can be given as well.

Sources

• NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.5.11 Measures of Skewness and Kurtosis, itl.nist.gov/div898/handbook/eda/section3/eda35b.htm (checked 2026-10-29).
• NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.3.14 Histogram, itl.nist.gov/div898/handbook/eda/section3/histogra.htm (checked 2026-10-29).
• OpenStax, Introductory Statistics 2e, 2.6 Skewness and the Mean, Median, and Mode, openstax.org (checked 2026-10-29).