39 Measures of Skewness

39.1 What is Skewness?

Skewness is the degree of asymmetry of a frequency distribution about its central value. The mean tells us the centre; dispersion tells us the spread; skewness tells us whether the distribution leans to one side. Kurtosis, treated together with skewness in this topic, captures the peakedness of the distribution (gupta2021?; elhance2020?).

Three working ideas:

A symmetrical distribution has the same shape on both sides of the central value.
A skewed distribution has a longer tail on one side.
The direction of skew is named after the direction of the longer tail.

39.2 Symmetric vs Skewed Distributions

Three Cases of Skewness

Case	Tail	Position of Mean, Median, Mode
Symmetrical	Equal on both sides	Mean = Median = Mode
Positively skewed (right-skewed)	Longer tail on the right (high values)	Mean > Median > Mode
Negatively skewed (left-skewed)	Longer tail on the left (low values)	Mean < Median < Mode

A useful mnemonic: the mean is pulled toward the longer tail. In a positively skewed distribution, the few large values pull the mean up above the median and mode; in a negatively skewed distribution, the few small values pull the mean down below the median and mode.

flowchart LR
  S[Symmetrical<br/>Mean = Median = Mode] --- P[Positively Skewed<br/>Mean > Median > Mode<br/>Tail on the right]
  S --- N[Negatively Skewed<br/>Mean < Median < Mode<br/>Tail on the left]
  style S fill:#E8F5E9,stroke:#2E7D32
  style P fill:#FFF8E1,stroke:#F9A825
  style N fill:#FFEBEE,stroke:#C62828

39.3 Tests of Skewness

A distribution is skewed if any of the following hold:

Mean ≠ Median ≠ Mode.
The two halves on either side of the median are unequal in shape.
Quartile distances: $Q_3 - M_d \neq M_d - Q_1$.
The frequency curve drawn from the data is not symmetrical about the central vertical.

39.4 Measures of Skewness

Skewness is measured in absolute form (raw difference) or relative form (coefficient). The relative form is preferred because it is unit-free and lies in a known interval.

Four Standard Coefficients of Skewness

Coefficient	Formula	Range
Karl Pearson’s first	$Sk_p = \dfrac{\bar X - M_o}{\sigma}$	$-1$ to $+1$ generally; theoretically $-3$ to $+3$
Karl Pearson’s second (when mode is ill-defined)	$Sk_p = \dfrac{3(\bar X - M_d)}{\sigma}$	$-3$ to $+3$
Bowley’s (quartile-based)	$Sk_b = \dfrac{Q_3 + Q_1 - 2 M_d}{Q_3 - Q_1}$	$-1$ to $+1$
Kelly’s (percentile-based)	$Sk_k = \dfrac{P_{90} + P_{10} - 2 M_d}{P_{90} - P_{10}}$	$-1$ to $+1$

A fifth approach uses the moments:

\[ \beta_1 = \dfrac{\mu_3^2}{\mu_2^3}, \qquad \gamma_1 = \sqrt{\beta_1} = \dfrac{\mu_3}{\sigma^3} \]

where $\mu_2, \mu_3$ are the second and third central moments. $\gamma_1$ keeps the sign and is the standard moment-based coefficient of skewness.

39.5 Karl Pearson’s Coefficient of Skewness

The first form uses mode; the second uses median when mode is hard to identify. In a moderately skewed distribution, the empirical relation $\text{Mode} \approx 3 M_d - 2 \bar X$ is used to derive the second form from the first.

39.5.1 Worked example

A distribution has mean 50, median 45 and standard deviation 10. Karl Pearson’s second coefficient:

\[ Sk_p = \dfrac{3(50 - 45)}{10} = \dfrac{15}{10} = 1.5 \]

Since the value is positive, the distribution is positively skewed; the mean exceeds the median.

39.6 Bowley’s Coefficient of Skewness

Bowley’s measure is based on quartiles only and does not require mean, mode or standard deviation. It is therefore suitable for open-ended distributions where the extremes are not clearly defined.

The numerator compares the distance from the median to $Q_3$ against the distance from the median to $Q_1$:

If $(Q_3 - M_d) = (M_d - Q_1)$ → numerator zero → symmetric.
If $(Q_3 - M_d) > (M_d - Q_1)$ → positive skew.
If $(Q_3 - M_d) < (M_d - Q_1)$ → negative skew.

The denominator normalises by the inter-quartile range, yielding a coefficient that always lies between $-1$ and $+1$.

39.6.1 Worked example

If $Q_1 = 40$, $M_d = 50$, $Q_3 = 80$:

\[ Sk_b = \dfrac{80 + 40 - 2 \times 50}{80 - 40} = \dfrac{20}{40} = 0.5 \]

Positive skew of moderate magnitude.

39.7 Kelly’s Coefficient

Kelly extended Bowley’s idea to wider percentiles — using $P_{10}$ and $P_{90}$ — to capture asymmetry in the tails rather than only in the central 50 per cent. The interpretation is identical: positive value → positive skew; negative → negative skew; zero → symmetric.

39.8 Moment-Based Coefficients

The $\gamma_1$ coefficient is the standard in modern statistics. For a normal distribution, $\gamma_1 = 0$; positive skew gives $\gamma_1 > 0$; negative skew gives $\gamma_1 < 0$. Many statistical software packages report $\gamma_1$ along with $\gamma_2$ (kurtosis).

39.9 Kurtosis

Kurtosis measures the peakedness of a distribution — how much of the variability comes from rare, extreme deviations. It complements skewness: skewness is about asymmetry; kurtosis is about tail heaviness and peakedness.

Three Cases of Kurtosis

Case	Description	$\beta_2$	$\gamma_2$
Mesokurtic	Same as normal distribution	$\beta_2 = 3$	$\gamma_2 = 0$
Leptokurtic	More peaked, heavier tails than normal	$\beta_2 > 3$	$\gamma_2 > 0$
Platykurtic	Flatter, lighter tails than normal	$\beta_2 < 3$	$\gamma_2 < 0$

The moment formulas are:

\[ \beta_2 = \dfrac{\mu_4}{\mu_2^2}, \qquad \gamma_2 = \beta_2 - 3 \]

The convention $\gamma_2 = \beta_2 - 3$ centres the normal benchmark at zero — software output usually reports excess kurtosis (i.e. $\gamma_2$).

flowchart LR
  L[Leptokurtic<br/>β₂ > 3<br/>peaked, heavy tails] --- M[Mesokurtic<br/>β₂ = 3<br/>normal benchmark]
  M --- P[Platykurtic<br/>β₂ < 3<br/>flat, light tails]
  style L fill:#FCE4EC,stroke:#AD1457
  style M fill:#E8F5E9,stroke:#2E7D32
  style P fill:#E3F2FD,stroke:#1565C0

39.10 Why Skewness and Kurtosis Matter

Practical Uses of Skewness and Kurtosis

Use	Working content
Test the appropriateness of mean	If skewed, the median is more representative
Test the normality assumption	Many parametric tests assume normality (skew = 0, $\gamma_2 = 0$)
Identify risk in finance	Heavy tails (high $\gamma_2$) flag fat-tail risk; negative skew flags downside risk
Compare shapes of distributions	Beyond mean and SD
Diagnose quality control	Process drift produces skew or excess kurtosis

39.11 Comparison — Skewness Coefficients

Comparison of Skewness Coefficients

Measure	Inputs needed	Best for	Range
Karl Pearson 1st	Mean, Mode, SD	Unimodal data with clear mode	$-3$ to $+3$
Karl Pearson 2nd	Mean, Median, SD	When mode is ill-defined	$-3$ to $+3$
Bowley	$Q_1, Q_2, Q_3$	Open-ended classes	$-1$ to $+1$
Kelly	$P_{10}, P_{50}, P_{90}$	When tails matter	$-1$ to $+1$
Moment ($\gamma_1$)	All moments	Software computation	Real line

39.12 Exam-Pattern MCQs

Q 01

In a positively skewed distribution, the relationship among mean, median and mode is:

AMean = Median = Mode
BMean > Median > Mode
CMean < Median < Mode
DMode > Mean > Median

View solution

Correct Option: B

The few large values pull the mean highest; median is in the middle; mode is at the peak (lowest of the three).

Q 02

Match each measure of skewness with the inputs it uses:

	Measure		Inputs
(i)	Karl Pearson 1st	(a)	$Q_1, Q_2, Q_3$
(ii)	Karl Pearson 2nd	(b)	Mean, Median, SD
(iii)	Bowley	(c)	Mean, Mode, SD
(iv)	Kelly	(d)	$P_{10}, P_{50}, P_{90}$

A(i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
B(i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
C(i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
D(i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)

View solution

Correct Option: A

Q 03

A distribution has mean 60, median 55 and standard deviation 10. Karl Pearson's second coefficient of skewness is:

A0.5
B1.0
C1.5
D5.0

View solution

Correct Option: C

$Sk_p = 3(60 − 55)/10 = 1.5$.

Q 04

With $Q_1 = 20$, $M_d = 30$ and $Q_3 = 50$, Bowley's coefficient of skewness equals:

A+ 0.33
B− 0.33
C+ 0.50
D− 0.50

View solution

Correct Option: A

$Sk_b = (50 + 20 − 60)/(50 − 20) = 10/30 ≈ + 0.33$ — moderate positive skew.

Q 05

Match each kurtosis case with its description:

	Case		Description
(i)	Mesokurtic	(a)	More peaked, heavier tails than normal
(ii)	Leptokurtic	(b)	Same as normal — $\beta_2 = 3$
(iii)	Platykurtic	(c)	Flatter, lighter tails than normal

A(i)-(b), (ii)-(a), (iii)-(c)
B(i)-(a), (ii)-(b), (iii)-(c)
C(i)-(c), (ii)-(a), (iii)-(b)
D(i)-(c), (ii)-(b), (iii)-(a)

View solution

Correct Option: A

Q 06

A distribution has $\beta_2 = 4.5$. It is:

AMesokurtic
BLeptokurtic
CPlatykurtic
DSkewed but not kurtic

View solution

Correct Option: B

$\beta_2 > 3$ → leptokurtic — heavier tails and more peaked than normal.

Q 07

Arrange the following measures from most general to most restrictive in their inputs (i.e. requiring the least to most of mean, mode, etc.): (i) Bowley's coefficient (ii) Karl Pearson's first coefficient (iii) Moment coefficient $\gamma_1$ (iv) Karl Pearson's second coefficient

A(i), (iv), (ii), (iii)
B(iii), (ii), (iv), (i)
C(i), (ii), (iv), (iii)
D(iv), (iii), (i), (ii)

View solution

Correct Option: A

Bowley needs only quartiles; KP-2nd needs mean+median+SD; KP-1st needs mean+mode+SD; $\gamma_1$ needs all central moments.

Q 08

Match each statistic with what it captures:

	Statistic		Captures
(i)	Mean	(a)	Asymmetry
(ii)	Standard deviation	(b)	Centre
(iii)	Skewness	(c)	Peakedness and tail-heaviness
(iv)	Kurtosis	(d)	Spread

A(i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
B(i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
C(i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
D(i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)

View solution

Correct Option: A

Quick recall

Skewness = degree of asymmetry about the central value. Kurtosis = peakedness / tail-heaviness.
Symmetric: Mean = Median = Mode. Positive skew (right-skewed): Mean > Median > Mode. Negative skew (left-skewed): Mean < Median < Mode.
The mean is pulled toward the longer tail.
Karl Pearson 1st: $Sk_p = (\bar X - M_o) / \sigma$. Karl Pearson 2nd: $Sk_p = 3(\bar X - M_d)/\sigma$. Range: $\pm 3$.
Bowley’s: $Sk_b = (Q_3 + Q_1 - 2 M_d)/(Q_3 - Q_1)$. Range: $\pm 1$. Suitable for open-ended classes.
Kelly’s: uses $P_{10}, P_{50}, P_{90}$.
Moment: $\gamma_1 = \mu_3 / \sigma^3$.
Kurtosis: mesokurtic ($\beta_2 = 3$, normal), leptokurtic ($\beta_2 > 3$, heavy tails), platykurtic ($\beta_2 < 3$, light tails). Excess kurtosis $\gamma_2 = \beta_2 - 3$.
Mean → centre; SD → spread; Skewness → asymmetry; Kurtosis → tails / peakedness.

Coefficient	Formula	Range
Karl Pearson’s first	\(Sk_p = \dfrac{\bar X - M_o}{\sigma}\)	\(-1\) to \(+1\) generally; theoretically \(-3\) to \(+3\)
Karl Pearson’s second (when mode is ill-defined)	\(Sk_p = \dfrac{3(\bar X - M_d)}{\sigma}\)	\(-3\) to \(+3\)
Bowley’s (quartile-based)	\(Sk_b = \dfrac{Q_3 + Q_1 - 2 M_d}{Q_3 - Q_1}\)	\(-1\) to \(+1\)
Kelly’s (percentile-based)	\(Sk_k = \dfrac{P_{90} + P_{10} - 2 M_d}{P_{90} - P_{10}}\)	\(-1\) to \(+1\)