Skewness is a dimensionless measure of distributional asymmetry that indicates whether observations extend farther, or carry greater third-moment weight, on one side of the center than on the other. A distribution with a long or influential upper tail typically has positive skewness, a distribution extending more strongly toward smaller values typically has negative skewness, and a perfectly symmetric distribution with a finite third moment has skewness equal to zero. The standard population measure is the third standardized central moment, γ₁ = μ₃/σ³, so deviations retain their signs because they are cubed rather than squared; large positive deviations contribute positively, while equally large negative deviations contribute negatively. This signed third-power structure distinguishes skewness from variance, which measures nondirectional squared spread, and also explains why isolated extreme observations can materially alter an estimated skewness value. Skewness is therefore most useful when interpreted as a numerical summary of asymmetry alongside the underlying distribution rather than as a complete description of shape.
Within descriptive statistics and the broader Statistics & Probability framework, skewness helps explain why two data sets can have similar centers and spreads yet behave very differently in their tails. It is particularly useful for income, waiting-time, transaction-size, biological, environmental, and reliability data where the distribution may extend substantially farther in one direction than the other.
What Is Skewness?
Skewness measures the degree and direction of asymmetry around a distribution’s mean.
Three broad cases are commonly distinguished:
| Skewness | General Interpretation |
|---|---|
| Negative | Longer or more influential lower tail |
| Zero | No third-moment asymmetry |
| Positive | Longer or more influential upper tail |
A positive value means positive deviations from the mean dominate the standardized third moment, while a negative value means negative deviations dominate.
The numerical magnitude indicates the strength of that third-moment imbalance, although there is no universal threshold separating “small,” “moderate,” and “large” skewness for every application.
Population Skewness Formula
For a random variable X with population mean μ and standard deviation σ:
γ₁ = E[(X − μ)³]/σ³
If the third central moment is:
μ₃ = E[(X − μ)³]
then:
γ₁ = μ₃/σ³
For a finite population containing N observations:
μ₃ = Σ(xᵢ − μ)³/N
and:
σ² = Σ(xᵢ − μ)²/N.
Therefore:
γ₁ = [Σ(xᵢ − μ)³/N] / σ³
provided:
σ > 0.
Because both numerator and denominator have cubed measurement units, those units cancel and skewness is dimensionless.
Why the Third Power Is Used
The third power preserves direction.
Suppose two standardized deviations are:
z = −3
and:
z = +3.
Their squared values are identical:
(−3)² = 9
3² = 9.
Variance therefore treats them equally.
Their cubes differ in sign:
(−3)³ = −27
3³ = 27.
Skewness can therefore distinguish whether extreme standardized departures occur predominantly below or above the mean.
Positive Skewness
Positive skewness is commonly called:
right skewness.
A positively skewed distribution has greater third-moment influence on the high-value side.
For example, consider:
1, 2, 3, 4, 10
Most observations lie between:
1 and 4,
while one substantially larger observation extends the upper side of the distribution.
This structure creates positive skewness.
The terms “right skew” and “positive skew” refer to the direction of the tail, not to where most observations are located.
Negative Skewness
Negative skewness is commonly called:
left skewness.
Consider:
0, 6, 7, 8, 9.
Most observations occupy the upper part of the observed span, while:
0
extends the lower side.
Negative deviations therefore exert greater third-moment influence.
The resulting skewness is negative.
As with right skewness, the direction refers to the longer or more influential tail.
Zero Skewness
A symmetric distribution with a finite third central moment has:
γ₁ = 0.
For example:
1, 2, 3, 4, 5
has mean:
μ = 3.
Deviations are:
−2, −1, 0, 1, 2.
Their cubes are:
−8, −1, 0, 1, 8.
The total is:
Therefore:
μ₃ = 0
and:
γ₁ = 0.
The positive and negative third-power contributions balance exactly.
Zero Skewness Does Not Prove Symmetry
Although symmetry implies zero skewness when the relevant moments exist, the reverse statement is not generally true.
A distribution can be asymmetric while its positive and negative third-moment contributions happen to cancel.
Therefore:
Skewness = 0
does not prove:
perfectly symmetric distribution.
Skewness summarizes one moment property.
A graph or fuller distributional analysis is needed to establish symmetry.
Worked Population Skewness Example
Consider the finite population:
1, 2, 3, 4, 10.
First calculate the population mean:
μ = (1 + 2 + 3 + 4 + 10)/5
= 20/5
= 4.
The deviations are:
−3, −2, −1, 0, 6.
Calculate the Second Central Moment
Square the deviations:
9, 4, 1, 0, 36.
Their sum is:
Therefore:
μ₂ = 50/5
= 10.
Since:
σ² = 10,
we have:
σ = √10
≈ 3.1623.
Calculate the Third Central Moment
Cube the deviations:
(−3)³ = −27
(−2)³ = −8
(−1)³ = −1
0³ = 0
6³ = 216.
Their sum is:
−27 − 8 − 1 + 0 + 216
= 180.
Therefore:
μ₃ = 180/5
= 36.
Calculate Population Skewness
Use:
γ₁ = μ₃/σ³.
Since:
σ² = 10,
then:
σ³ = 10√10
≈ 31.6228.
Therefore:
γ₁ = 36/31.6228
≈ 1.138.
So:
Population skewness ≈ 1.14.
The positive sign confirms that the distribution has greater upper-tail third-moment influence.
Mirror-Image Negative Skewness Example
Now consider:
0, 6, 7, 8, 9.
The mean is:
μ = 6.
Its deviations are:
−6, 0, 1, 2, 3.
The squared deviations again sum to:
50,
so:
σ² = 10.
The cubed deviations are:
−216, 0, 1, 8, 27.
Their total is:
−180.
Therefore:
μ₃ = −36.
Hence:
γ₁ = −36/31.6228
≈ −1.138.
This is the mirror image of the previous distribution, so the magnitude remains the same while the sign reverses.
Sample Moment Skewness
For sample observations, define the central sample moments with denominator n:
m₂ = Σ(xᵢ − x̄)²/n
and:
m₃ = Σ(xᵢ − x̄)³/n.
A common moment coefficient of sample skewness is:
g₁ = m₃/m₂^(3/2).
This is the direct sample analogue of the population standardized third moment.
However, g₁ is biased in finite samples, so statistical software may report an adjusted version instead.
Adjusted Fisher–Pearson Sample Skewness
A frequently used bias-adjusted sample coefficient is:
G₁ = [√(n(n − 1))/(n − 2)]g₁
for:
n > 2.
This adjustment is often called the adjusted Fisher–Pearson standardized moment coefficient.
Because different programs may report:
g₁
or:
G₁,
two software packages can produce different sample skewness values from the same observations without either calculation necessarily being incorrect.
The estimator convention should always be checked.
Sample Skewness Example
Use the sample:
1, 2, 3, 4, 10.
We already have:
x̄ = 4
m₂ = 10
m₃ = 36.
Therefore:
g₁ = 36/10^(3/2)
≈ 1.1384.
For:
n = 5,
the adjusted coefficient is:
G₁ = √(5×4)/(5 − 2)
= √20/3
≈ 1.6971.
Thus, the same sample can legitimately be reported as approximately:
1.138
under the unadjusted moment convention
or:
1.697
under the adjusted Fisher–Pearson convention.
Why Software Skewness Values Differ
Differences often arise from estimator definitions rather than computational mistakes.
One application may calculate:
m₃/m₂^(3/2).
Another may apply a finite-sample correction.
A third may use a formulation expressed in terms of the sample standard deviation.
Therefore, comparisons should identify the exact definition before interpreting small numerical discrepancies.
This matters especially for small n, where bias corrections can be substantial.
Skewness and Sample Variance
Sample variance measures nondirectional squared dispersion around the sample mean, whereas skewness measures directional third-moment imbalance.
For a deviation:
−5
and one:
+5,
variance gives both:
Skewness gives:
−125
and:
+125.
Therefore, variance can be large while skewness remains zero if extreme observations balance symmetrically.
Likewise, a sample can have noticeable skewness even when its overall numerical variance is modest.
Skewness and Standard Deviation
Standard deviation appears in the denominator of moment skewness because the third central moment must be standardized for scale.
Without standardization, measuring the same variable in centimeters instead of meters would multiply the third central moment dramatically.
Dividing by:
σ³
or the corresponding sample scale removes the measurement units.
Thus, skewness describes distributional shape independently of ordinary positive unit conversions.
Skewness and Standard Error
A standard error describes sampling variability of an estimator, while skewness describes asymmetry in a distribution.
These quantities should not be confused.
For example, a sample mean can have a standard error even when the underlying distribution is symmetric.
Likewise, an estimated sample skewness has its own sampling variability, particularly when n is small.
A skewness coefficient is a point estimate of shape, not an uncertainty measure.
Sampling Variability of Skewness
Sample skewness can fluctuate substantially between random samples, especially when sample size is small.
The reason is that third powers amplify extreme deviations.
One unusually large observation can materially change:
m₃
and therefore:
g₁.
As sample size grows, sample skewness can become more stable under suitable population assumptions, but very heavy-tailed distributions may still produce considerable variation.
A precise-looking coefficient from a small sample should therefore be interpreted cautiously.
Standard Error of Skewness
Under normal-distribution assumptions, approximate formulas exist for the sampling variability of skewness estimators.
A commonly encountered rough large-sample approximation is:
SE(skewness) ≈ √(6/n).
For:
n = 100,
this gives:
SE ≈ √0.06
≈ 0.245.
However, exact finite-sample formulas depend on the estimator convention, and normal-theory approximations should not be treated as universal for heavily skewed or non-normal populations.
The reported skewness statistic and its uncertainty formula must be compatible.
Skewness and Relative Frequency
A relative frequency distribution can reveal the pattern that produces asymmetry.
For example, suppose most relative frequency lies in intervals:
0–10
and:
10–20,
while increasingly small proportions extend through:
20–30,
30–40,
40–50.
That pattern suggests an upper tail and therefore potential positive skewness.
Relative frequencies show where the observations are located, while skewness compresses directional asymmetry into one number.
Skewness From a Frequency Distribution
If exact values xᵢ occur with frequencies fᵢ, define:
n = Σfᵢ.
The mean is:
x̄ = Σfᵢxᵢ/n.
The moment quantities are:
m₂ = Σfᵢ(xᵢ − x̄)²/n
and:
m₃ = Σfᵢ(xᵢ − x̄)³/n.
Then:
g₁ = m₃/m₂^(3/2).
Frequency weighting therefore provides an exact calculation when the table contains exact values rather than grouped class intervals.
Frequency-Table Example
Suppose:
| Value | Frequency |
|---|---|
| 1 | 3 |
| 2 | 2 |
| 5 | 1 |
Total:
n = 6.
Mean:
x̄ = [1(3) + 2(2) + 5(1)]/6
= 12/6
= 2.
The large value:
5
creates an extended upper side relative to the concentration at 1 and 2.
The resulting third central moment is positive, so the distribution has positive moment skewness.
The frequency formulation avoids listing repeated observations individually.
Skewness From Grouped Data
When only class intervals are available, skewness can be approximated using class midpoints.
Let:
mᵢ = midpoint of class i
and:
fᵢ = class frequency.
Approximate mean:
x̄ ≈ Σfᵢmᵢ/n.
Then:
m₂ ≈ Σfᵢ(mᵢ − x̄)²/n
and:
m₃ ≈ Σfᵢ(mᵢ − x̄)³/n.
Approximate skewness is:
g₁ ≈ m₃/m₂^(3/2).
The result is approximate because every observation in a class is represented by its midpoint.
Why Grouped Skewness Can Be Inaccurate
Third powers make grouped-data approximation especially sensitive to how tail observations are represented.
Suppose the highest interval is:
100–200.
A value of:
101
and a value of:
199
would both be replaced by midpoint:
Their actual cubed deviations from the mean could differ substantially.
Wide tail classes can therefore distort estimated skewness.
Raw observations are preferable when precise shape measurement matters.
Skewness and Weighted Mean
When observations carry unequal importance or represent unequal population amounts, the center may be a weighted mean:
μ_w = Σwᵢxᵢ/Σwᵢ.
A weighted moment skewness can then be defined using the same weights:
m₂,w = Σwᵢ(xᵢ − μ_w)²/Σwᵢ
m₃,w = Σwᵢ(xᵢ − μ_w)³/Σwᵢ
and:
γ₁,w = m₃,w / m₂,w^(3/2).
The exact finite-sample bias corrections for complex survey or reliability weights require additional care, but the weighted-moment structure illustrates the basic principle.
Why Weights Can Change Skewness
Suppose a high observation occurs only once but receives a very large weight.
Its contribution to:
weighted mean,
weighted variance,
and:
weighted third moment
can become substantial.
The weighted distribution may therefore be strongly right-skewed even when an unweighted sample appears only mildly asymmetric.
Weights change the empirical distribution being summarized.
They should represent a genuine sampling, frequency, exposure, or analytical structure rather than being chosen merely to modify the result.
Skewness and Range
The range measures endpoint separation:
Maximum − Minimum.
Skewness measures directional asymmetry.
Consider two samples with range:
One can be symmetric around its center, while another can place nearly every observation near the minimum and only one value near the maximum.
Their ranges are identical, but their skewness values can be very different.
Endpoint span therefore cannot determine asymmetry.
Same Range, Different Skewness
Consider:
A = 0, 5, 10, 15, 20
and:
B = 0, 1, 2, 3, 20.
Both have:
Range = 20.
Sample A is symmetric around:
10,
so its moment skewness is zero.
Sample B has most observations near the lower endpoint and one distant upper value, producing substantial positive skewness.
This demonstrates why a spread statistic and a shape statistic should not be used interchangeably.
Skewness and Mean
The arithmetic mean is sensitive to extreme values.
In right-skewed data, large upper observations often pull the mean upward.
Therefore, many positively skewed distributions satisfy the rough relationship:
Mean > Median.
For negatively skewed distributions, a common pattern is:
Mean < Median.
These relationships can be useful clues, but they are not definitions of skewness and are not guaranteed for every possible distribution.
Mean, Median, and Mode Relationship
A commonly taught pattern for a unimodal right-skewed distribution is:
Mode < Median < Mean.
For a unimodal left-skewed distribution:
Mean < Median < Mode.
For a symmetric unimodal distribution:
Mean ≈ Median ≈ Mode.
These are useful conceptual patterns rather than universal mathematical identities.
Multimodal or irregular distributions can violate them substantially.
Pearson’s First Skewness Coefficient
One descriptive alternative to moment skewness is Pearson’s first coefficient:
Sk₁ = (Mean − Mode)/Standard deviation.
In sample notation:
Sk₁ = (x̄ − Mode)/s.
If:
Mean > Mode,
the coefficient is positive.
This can provide a rough indication of right skew in suitable unimodal data.
However, mode can be unstable or undefined, especially for continuous samples, limiting the usefulness of this formula.
Pearson’s Second Skewness Coefficient
Pearson’s second coefficient replaces mode with median:
Sk₂ = 3(Mean − Median)/Standard deviation.
In sample notation:
Sk₂ = 3(x̄ − Median)/s.
If:
x̄ > Median,
the coefficient is positive.
If:
x̄ < Median,
it is negative.
This measure is simple, but it is not identical to third-moment skewness.
Different skewness coefficients can produce different numerical values for the same data.
Pearson Skewness Example
Suppose:
Mean = 30
Median = 24
Standard deviation = 12.
Pearson’s second coefficient is:
Sk₂ = 3(30 − 24)/12
= 18/12
= 1.5.
The positive value suggests right asymmetry.
However, this:
1.5
should not be directly compared as though it were the same estimator as:
γ₁ = μ₃/σ³.
The formulas measure asymmetry differently.
Moment Skewness vs Pearson Skewness
Moment skewness uses every cubed standardized deviation.
Pearson coefficients summarize differences among central-tendency measures relative to standard deviation.
Moment skewness therefore responds directly to the entire distribution, including tails.
Pearson’s coefficients can be easier to calculate but may miss more complex shape information.
When reporting a numerical skewness value, identify which definition is being used.
Bowley Skewness
A robust quantile-based measure sometimes used for skewness is Bowley’s coefficient:
Sk_B = (Q₃ + Q₁ − 2Q₂)/(Q₃ − Q₁)
where:
Q₂ = Median.
An equivalent form is:
Sk_B = [(Q₃ − Q₂) − (Q₂ − Q₁)]/(Q₃ − Q₁).
This compares the spread above and below the median within the central half of the data.
Because it uses quartiles, it is more resistant to extreme tail values than moment skewness.
Bowley Skewness Interpretation
If:
Q₃ − Median
is larger than:
Median − Q₁,
Bowley skewness is positive.
If the lower central half extends farther:
Bowley skewness < 0.
For a quartile-symmetric distribution:
Q₃ − Q₂ = Q₂ − Q₁
so:
Sk_B = 0.
This measure focuses on central asymmetry and can therefore differ from moment skewness when extreme tails dominate.
Skewness Is Sensitive to Outliers
Because deviations are cubed, extreme observations can influence skewness strongly.
Suppose most standardized deviations are near:
1
but one observation lies:
6
standard deviations above the mean.
Its cubed standardized contribution is:
6³ = 216.
An observation one standard deviation above contributes:
Thus, one extreme upper observation can dominate the third-moment calculation.
This sensitivity is important both analytically and diagnostically.
Effect of an Extreme Upper Observation
Consider:
1, 2, 3, 4, 5.
This data set is symmetric and has:
skewness = 0.
Replace:
5
with:
Now the upper tail becomes much longer and the mean shifts upward.
The third central moment becomes strongly positive.
Therefore, estimated skewness rises sharply.
The statistic accurately records that the transformed data are no longer symmetric, but the reason may be one observation rather than a broadly skewed central distribution.
Effect of an Extreme Lower Observation
Similarly, replacing the smallest value with a much more negative number can generate strong negative skewness.
The cubed negative deviation remains negative:
(−d)³ = −d³.
Therefore, a distant lower observation pushes the skewness coefficient downward.
This signed sensitivity makes skewness useful for identifying tail direction but also makes small-sample estimates vulnerable to isolated extremes.
Outliers vs Genuine Skewness
A positive skewness coefficient can arise because:
many observations form a naturally long upper tail
or because:
one isolated upper observation is extremely distant.
Those situations can have very different substantive meanings.
The numerical coefficient alone cannot distinguish them.
Inspection of the observations or a suitable distribution graph is therefore important before describing an entire population as strongly skewed.
Skewness and Kurtosis
Skewness and kurtosis describe different standardized moments.
Skewness uses the third central moment and retains direction.
Kurtosis uses the fourth central moment and therefore gives the same sign-free contribution to equally distant positive and negative deviations.
A distribution can be:
symmetric with high kurtosis,
skewed with moderate kurtosis,
or exhibit both strong asymmetry and heavy-tail behavior.
Neither statistic substitutes for the other.
Skewness and Symmetry
A symmetric distribution around μ satisfies:
X − μ
and:
μ − X
with mirror-image behavior.
When the third moment exists, positive and negative cubic deviations cancel.
Therefore:
γ₁ = 0.
Examples of symmetric distributions include many normal, uniform, and Student-type distributions under appropriate parameters.
They can have very different variances and tail behavior despite sharing zero skewness.
Normal Distribution Skewness
A normal distribution is symmetric around its mean.
Therefore:
γ₁ = 0.
Its population:
mean = median = mode.
A random sample from a normal population will generally not have sample skewness exactly zero.
Sampling variation creates positive or negative estimated coefficients.
Thus, nonzero sample skewness by itself does not prove that the underlying population is non-normal.
Uniform Distribution Skewness
A continuous uniform distribution on:
[a, b]
is symmetric around:
(a + b)/2.
Therefore:
γ₁ = 0.
Its shape is completely different from a normal distribution, yet both have zero skewness.
This is another reminder that zero skewness describes symmetry of the third moment rather than a specific distributional shape.
Exponential Distribution Skewness
A standard exponential-type distribution is strongly right-skewed.
Its theoretical population skewness is:
γ₁ = 2.
The distribution has a lower boundary and a long upper tail.
This provides a useful benchmark showing that a positive skewness value can arise naturally from the probability model rather than from erroneous observations.
Bernoulli Distribution Skewness
For a Bernoulli variable with success probability p:
γ₁ = (1 − 2p)/√[p(1 − p)]
for:
0 < p < 1.
If:
p = 0.5,
then:
γ₁ = 0.
If:
p < 0.5,
the coefficient is positive because rare 1s extend the upper side.
If:
p > 0.5,
the coefficient is negative because rare 0s extend the lower side relative to the dominant 1s.
Skewness and Bounded Data
A variable can be skewed even when it has strict lower and upper limits.
For example, test scores bounded between:
0 and 100
may show negative skew when most students score highly and only a few receive low scores.
Likewise, a difficult test may produce positive skew with many low scores and a thin upper tail.
Skewness therefore concerns how probability mass is arranged within the available scale, not whether the variable is unbounded.
Floor Effects
A floor effect occurs when many observations accumulate near the lowest possible measurement value.
Because values cannot extend much farther downward, variation may extend predominantly upward.
This can create:
positive skewness.
For example, a very difficult assessment can produce many scores near zero with a smaller number extending upward.
The observed skewness then reflects both the underlying performance distribution and the measurement boundary.
Ceiling Effects
A ceiling effect occurs when observations accumulate near the highest possible value.
The remaining tail often extends downward.
This can create:
negative skewness.
For example, an easy examination might place many scores near 100 while a smaller group extends toward much lower scores.
Measurement design can therefore generate or magnify skewness.
Skewness Under Translation
Adding a constant does not change skewness.
Let:
Y = X + b.
The mean changes from:
μX
to:
μY = μX + b.
However:
Y − μY = X − μX.
Therefore, both the second and third centered moments remain unchanged.
Thus:
γ₁(Y) = γ₁(X).
Changing the zero point of the measurement scale does not alter asymmetry.
Positive Scaling
If:
Y = aX
with:
a > 0,
then the third central moment is multiplied by:
a³,
while standard deviation cubed is also multiplied by:
a³.
Therefore:
γ₁(Y) = γ₁(X).
Converting meters to centimeters, for example, leaves skewness unchanged.
The numerical magnitudes change, but the standardized shape does not.
Negative Scaling
If:
a < 0,
the distribution is reflected.
The third central moment is multiplied by:
a³,
which is negative, while:
σY³ = |a|³σX³
remains positive.
Therefore:
γ₁(aX + b) = sign(a)γ₁(X).
A negative scaling reverses the sign of skewness.
A right-skewed distribution becomes left-skewed when reflected.
Reflection Example
Suppose X has:
γ₁ = 1.4.
Define:
Y = −X.
Then:
γ₁(Y) = −1.4.
The magnitude remains unchanged because the shape is mirrored exactly.
Only the direction reverses.
This transformation property provides a useful check on skewness calculations.
Nonlinear Transformations
Nonlinear transformations can change both the magnitude and structure of skewness.
For positive right-skewed data, a logarithmic transformation:
Y = ln(X)
compresses large upper observations more strongly than small ones.
Therefore, log-transformed data often exhibit lower positive skewness than the raw data.
The transformation changes the scale and the substantive interpretation, so it should be justified by the underlying process rather than applied merely to force a desired coefficient.
Log Transformation Example
Suppose data span:
1, 10, 100, 1000.
On the raw scale, distances increase dramatically and the distribution is strongly right-oriented.
On a base-10 logarithmic scale:
0, 1, 2, 3.
The transformed observations are evenly spaced.
Skewness can therefore change substantially.
Raw-scale and log-scale analyses answer different questions about the same underlying observations.
Square-Root Transformations
For nonnegative count-like data, a square-root transformation:
Y = √X
compresses larger values while preserving ordering.
It can reduce positive skewness in some applications.
The effect is less aggressive than a logarithm for very large values.
Again, transformation should reflect a meaningful modeling or variance-stabilizing objective rather than an arbitrary attempt to make the data appear symmetric.
Skewness and the Median
The median is resistant to extreme observations, while moment skewness is highly tail-sensitive.
Suppose one large upper observation increases from:
100
to:
1,000,000.
The median may remain completely unchanged.
The skewness coefficient can change dramatically.
This contrast helps explain why median-centered summaries are useful companions to moment-based shape statistics.
Skewness and the Mode
The mode identifies the most common value or density peak.
For a simple right-skewed unimodal distribution, the mode is often below the median and mean.
However, skewness does not require the distribution to have one mode.
A bimodal distribution can have positive, negative, or zero third-moment skewness depending on the locations and weights of its peaks.
Therefore, modal structure should be examined separately.
Multimodal Distributions
A distribution with two or more peaks can make simple skewness interpretation difficult.
Suppose one cluster lies around:
20
and another around:
The overall skewness may be close to zero if the clusters balance, even though the distribution is far from a conventional symmetric single-peak shape.
A one-number skewness coefficient cannot reveal multimodality.
Graphical examination remains necessary.
Skewness and Mixture Distributions
Combining subgroups can create asymmetry that is absent within each subgroup.
For example, two individually symmetric groups with different means and unequal sample sizes can produce an overall skewed mixture.
Conversely, skewed subgroups can combine to produce nearly zero overall skewness.
This phenomenon shows why population structure matters.
Before attributing skewness to one homogeneous process, consider whether several distinct groups have been pooled.
Skewness and Missing Data
Missing values should not be replaced automatically with zero.
Suppose actual observations are mostly between:
50 and 100,
and missing records are coded as:
Those artificial zeros can create a long lower tail and substantial negative skewness.
The calculated coefficient would describe the coding scheme rather than the intended measurements.
Missing-data handling should therefore precede shape analysis.
Skewness and Rounding
Rounding can alter estimated skewness, especially when the sample is small or many values are close together.
For example, measurements:
1.42, 1.46, 1.49
may all round to:
1.5.
This creates ties and changes moment contributions.
With large samples and fine measurement resolution, the effect may be negligible.
With coarse categories or heavily rounded data, it can be substantial.
Skewness and Sample Size
A sample of:
n = 8
can exhibit high skewness because of one unusual observation.
A sample of:
n = 8,000
provides much more information about the population’s overall shape, although genuine heavy tails can still create unstable moments.
Therefore, the same reported skewness magnitude should not be interpreted identically across radically different sample sizes.
The underlying data and estimator uncertainty matter.
Can Skewness Be Greater Than 1?
Yes.
Moment skewness is not restricted to:
−1 ≤ γ₁ ≤ 1.
Values such as:
2,
5,
or even much larger magnitudes are possible.
The statistic has no simple finite universal upper or lower bound across all distributions with finite relevant moments.
Therefore, a value above 1 is not mathematically invalid.
Informal Magnitude Guidelines
Some descriptive conventions use approximate categories such as:
|skewness| < 0.5 → relatively mild asymmetry
0.5 to 1 → moderate asymmetry
|skewness| > 1 → substantial asymmetry.
These thresholds are only heuristics.
Acceptable asymmetry depends on the method being used, sample size, data type, model, and practical objective.
A regression procedure with thousands of observations may tolerate a distributional feature that matters greatly in a small-sample parametric test.
Skewness Does Not Determine Normality
A normal distribution has:
skewness = 0.
However:
skewness ≈ 0
does not prove normality.
Uniform, symmetric bimodal, and many other non-normal distributions can also have zero skewness.
Likewise, a small nonzero sample skewness can occur naturally when sampling from a normal population.
Normality assessment requires more than one third-moment statistic.
Skewness Does Not Measure Spread
A distribution can have large variance and zero skewness if it is widely dispersed but symmetric.
Another can have small variance and strong skewness if it occupies a narrow scale asymmetrically.
Therefore:
spread ≠ asymmetry.
Variance, standard deviation, IQR, and range measure aspects of dispersion.
Skewness measures directionally unbalanced shape.
Skewness Does Not Measure Tail Weight Alone
A symmetric heavy-tailed distribution can have:
skewness = 0
despite frequent extreme observations on both sides.
Skewness responds to directional imbalance rather than total tail extremity.
If equally strong upper and lower tails are present, cubic contributions can cancel.
Tail weight and asymmetry must therefore be evaluated separately.
Common Skewness Mistakes
A common mistake is saying that positive skew means most observations are on the right. In a right-skewed distribution, the tail extends toward larger values, while much of the data may actually be concentrated on the lower side.
Another mistake is assuming zero skewness proves symmetry or normality. It does neither.
Analysts also sometimes compare values from different software without checking whether one program reports unadjusted g₁ and another uses adjusted G₁.
Another error is treating mean > median as the definition of positive skewness rather than a common descriptive pattern.
Using Pearson’s coefficient and moment skewness interchangeably is also incorrect because their formulas differ.
For grouped data, midpoint-based skewness should be described as approximate.
Finally, an extreme skewness coefficient should prompt examination of raw data, because one valid or erroneous tail observation can dominate the third moment.
How to Calculate Skewness Step by Step
For a population, begin by calculating:
μ = Σxᵢ/N.
Calculate each centered deviation:
xᵢ − μ.
Find the second central moment:
μ₂ = Σ(xᵢ − μ)²/N.
Then find:
σ = √μ₂.
Calculate the third central moment:
μ₃ = Σ(xᵢ − μ)³/N.
Finally:
γ₁ = μ₃/σ³.
A positive answer indicates positive third-moment asymmetry, while a negative answer indicates negative third-moment asymmetry.
Full Worked Example
Consider:
1, 2, 3, 4, 10
Step 1: Calculate the Mean
μ = 20/5
= 4.
Step 2: Find Deviations
−3, −2, −1, 0, 6
Step 3: Square Deviations
9, 4, 1, 0, 36
Their sum is:
Therefore:
μ₂ = 50/5
= 10.
Step 4: Calculate Standard Deviation
σ = √10
≈ 3.1623.
Step 5: Cube Deviations
−27, −8, −1, 0, 216
Their sum is:
Therefore:
μ₃ = 180/5
= 36.
Step 6: Standardize the Third Moment
γ₁ = 36/(√10)³
= 36/(10√10)
≈ 1.1384.
Therefore:
Population skewness ≈ 1.14.
The distribution is positively skewed.
Verify the Direction Visually
The observations:
1, 2, 3, 4
form a compact lower cluster.
The value:
10
extends substantially farther upward.
Therefore, a positive result is consistent with the observed distribution.
A useful calculation check is always to compare the coefficient’s sign with the actual tail direction.
If the data clearly have a long upper tail but the calculation produces a large negative value, inspect the arithmetic and sign conventions.
Full Symmetric Example
Consider:
2, 3, 4, 5, 6.
Mean:
μ = 4.
Deviations:
−2, −1, 0, 1, 2.
Cubed deviations:
−8, −1, 0, 1, 8.
Their sum is:
Therefore:
μ₃ = 0
and:
γ₁ = 0.
The sample’s values are perfectly mirrored around 4.
Full Negative-Skew Example
Consider:
0, 6, 7, 8, 9.
Mean:
μ = 6.
Deviations:
−6, 0, 1, 2, 3.
Squared-deviation mean:
μ₂ = 10.
Third central moment:
μ₃ = −36.
Therefore:
γ₁ = −36/(10√10)
≈ −1.1384.
The negative coefficient reflects the distant lower-side observation.
How to Report Skewness
A clear report identifies the estimator when necessary.
For example:
“Population moment skewness was approximately 1.14, indicating positive asymmetry with greater third-moment influence in the upper tail.”
For sample data:
“The adjusted Fisher–Pearson sample skewness was 1.70.”
If the estimate appears driven by one extreme observation, mention that context.
A numerical coefficient is most informative when accompanied by a concise description of the distribution that produced it.
When Skewness Is Most Useful
Skewness is particularly useful when the analyst needs to understand whether a distribution is directionally asymmetric, whether upper and lower tails behave differently, whether the mean may be disproportionately affected by one side of the distribution, or whether a transformation might be appropriate for a strongly asymmetric variable. It also provides a compact way to compare shape across variables measured in different units because standardized moment skewness is dimensionless.
Its usefulness is greatest when combined with center, dispersion, sample size, and a graphical view of the data. A single skewness coefficient cannot reveal multiple modes, isolated gaps, the cause of an extreme tail, or whether a rare observation is valid.
Frequently Asked Questions About Skewness
What is skewness?
Skewness is a statistical measure of directional asymmetry in a distribution.
What is the population skewness formula?
γ₁ = μ₃/σ³
where μ₃ is the third central moment.
What is another form of the formula?
γ₁ = E[(X − μ)³]/σ³
What does positive skewness mean?
Positive skewness indicates greater third-moment influence in the upper or right tail.
What does negative skewness mean?
Negative skewness indicates greater third-moment influence in the lower or left tail.
What does zero skewness mean?
It means the third standardized central moment is zero.
Does zero skewness prove symmetry?
No.
Does zero skewness prove normality?
No.
Is a normal distribution skewed?
No. Its population skewness is:
What is right skewness?
Another term for positive skewness.
What is left skewness?
Another term for negative skewness.
Why does skewness use cubed deviations?
Cubing preserves the sign of deviations and gives greater influence to observations far from the mean.
Why is variance unable to show skew direction?
Variance squares deviations, so positive and negative distances contribute identically.
Can skewness be greater than 1?
Yes.
Can skewness be less than −1?
Yes.
Is skewness bounded between −1 and 1?
No.
What is sample moment skewness?
A common form is:
g₁ = m₃/m₂^(3/2).
What is adjusted sample skewness?
A commonly used adjusted Fisher–Pearson coefficient is:
G₁ = [√(n(n − 1))/(n − 2)]g₁.
Why can software report different skewness values?
Programs may use different finite-sample estimators or bias corrections.
Does skewness have units?
No. Standardized moment skewness is dimensionless.
Does converting meters to centimeters change skewness?
No.
Does adding a constant change skewness?
No.
Does multiplying all observations by a positive constant change skewness?
No.
What happens if every observation is multiplied by a negative constant?
The skewness sign reverses.
What is the general affine transformation rule?
For:
Y = aX + b,
γ₁(Y) = sign(a)γ₁(X)
when a ≠ 0.
Can an outlier affect skewness?
Yes, very strongly.
Why?
Third powers give extreme deviations disproportionate influence.
Can one upper outlier create positive skewness?
Yes.
Can one lower outlier create negative skewness?
Yes.
Is positive skewness the same as having most observations on the right?
No. Positive skew refers to the direction of the longer or more influential tail.
What is a common relationship in right-skewed unimodal data?
Often:
Mode < Median < Mean.
Is that relationship always true?
No.
What is Pearson’s first skewness coefficient?
(Mean − Mode)/Standard deviation.
What is Pearson’s second skewness coefficient?
3(Mean − Median)/Standard deviation.
Are Pearson coefficients identical to moment skewness?
No.
What is Bowley skewness?
A quartile-based robust measure:
(Q₃ + Q₁ − 2Q₂)/(Q₃ − Q₁).
Is Bowley skewness less sensitive to outliers?
Generally yes, because it uses quartiles rather than third moments.
Can a symmetric distribution have large variance?
Yes.
Can a symmetric distribution have heavy tails?
Yes.
Can it still have zero skewness?
Yes.
Does high skewness imply high variance?
No.
Does high variance imply high skewness?
No.
Can two distributions have the same range and different skewness?
Yes.
Can two distributions have the same skewness and different ranges?
Yes.
Can skewness be calculated from a frequency table?
Yes, exactly when exact values and their frequencies are known.
Can skewness be estimated from grouped data?
Yes, usually by using class midpoints and frequencies.
Is grouped-data skewness exact?
Usually not.
Can relative frequencies be used to calculate skewness?
Yes, as empirical weights when the distribution and necessary sample information are known.
Can weights change skewness?
Yes.
Why?
Weights change the distributional mass assigned to observations.
Is skewness the same as standard error?
No.
Is skewness affected by sample size?
The coefficient can be calculated at many sample sizes, but its sampling variability is generally greater in small samples.
Is there a standard error for skewness?
Yes, although the correct formula depends on assumptions and estimator convention.
What is a rough normal-theory approximation?
A commonly encountered large-sample approximation is:
√(6/n).
Does sample skewness require normal data?
No. It can be calculated for non-normal data.
Can skewness help test normality by itself?
It can provide evidence about asymmetry, but it cannot establish normality or non-normality on its own.
Can log transformation reduce positive skewness?
Often yes for suitable positive multiplicative data.
Does transformation preserve skewness?
Only positive linear transformations preserve its value exactly; nonlinear transformations can change it.
Can multimodal data have zero skewness?
Yes.
Why is that important?
Zero skewness does not imply a simple symmetric bell-shaped distribution.
What is the skewness of an exponential distribution?
Its theoretical population skewness is:
What is Bernoulli skewness?
(1 − 2p)/√[p(1 − p)]
for 0 < p < 1.
What happens when p = 0.5?
Bernoulli skewness is:
Can measurement boundaries create skewness?
Yes. Floor and ceiling effects can generate asymmetric observed distributions.
What is the biggest mistake when interpreting skewness?
Treating its sign or magnitude as a complete description of distribution shape without checking the actual data, sample size, estimator definition, and influence of extreme observations.
What is the most important rule when calculating skewness?
Use a clearly defined skewness formula, preserve the signs of cubed deviations, distinguish population from sample estimators, and interpret the coefficient as a measure of directional asymmetry rather than as a standalone test of normality or a complete description of the distribution.
