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Skewness, Kurtosis and Moments | Business Statistics MGT 202

Skewness, Kurtosis and Moments | Business Statistics MGT 202

TU BBS MGT 202 Unit 5 board exam questions on Karl Pearson and Bowley skewness, percentile kurtosis, moments and tests of normality (2078–2083).

Oct 28, 2026
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This unit covers skewness and its measures (Karl Pearson, Bowley, Kelly), the five-number summary and box-and-whisker plot, moments about the mean and about an arbitrary point, and kurtosis, followed by board exam questions listed by exam year.

Skewness

The lack of symmetry of a distribution is called skewness. It relates to the shape but not the size of a frequency curve.

A distribution which is not symmetrical is said to be skewed. So, skewness is the lack of symmetry.

Types of Skewness

TypeShape of the curveRelation of averages
Positive skewnessThe curve of the frequency distribution has a longer tail on the right.Mean > Median > Mode
Negative skewnessThe curve of the frequency distribution is elongated more on the left than the right.Mean < Median < Mode
Zero skewness (symmetrical)The tails of the curve on either side of the central value are equal.Mean = Median = Mode

Measures (Coefficients) of Skewness

Karl Pearson's coefficient of skewness

$$\begin{aligned} \text{First measure:}\quad S_{k}(P) &= \frac{\text{Mean} - \text{Mode}}{\text{Standard deviation}} = \frac{\bar{x} - M_{0}}{\sigma} \\ \text{Second measure:}\quad S_{k}(P) &= \frac{3(\bar{x} - M_{d})}{\sigma} \end{aligned}$$

Decision:

  • If \(S_k = 0\), the distribution is symmetrical (normal).
  • If \(S_k > 0\), the distribution is positively skewed.
  • If \(S_k < 0\), the distribution is negatively skewed.

Bowley's coefficient of skewness

The coefficient of skewness based on quartiles is known as Bowley's coefficient of skewness.

$$S_{k}(B) = \frac{Q_{3} + Q_{1} - 2M_{d}}{Q_{3} - Q_{1}}$$

Kelly's coefficient of skewness

Kelly's coefficient of skewness is based on percentiles or deciles.

$$\begin{aligned} S_{k}(K) &= \frac{P_{90} + P_{10} - 2P_{50}}{P_{90} - P_{10}} \\ \text{or}\quad S_{k}(K) &= \frac{D_{9} + D_{1} - 2D_{5}}{D_{9} - D_{1}} \end{aligned}$$

Five-Number Summary and Box-and-Whisker Plot

Five-Number Summary

  1. Minimum value \((x_S)\): the smallest item in the set of observations.
  2. Lower quartile \((Q_1)\): the value below which lie 25% of the set of observations.
  3. Median \((M_d)\): the value below and above which lie 50% of the observations.
  4. Upper quartile \((Q_3)\): the value above which lie 25% of the observations.
  5. Maximum value \((x_L)\): the largest item in the set of observations.

Box-and-Whisker Plot

A box-and-whisker plot displays the five-number summary: the box runs from \(Q_1\) to \(Q_3\) with the median marked inside, and the whiskers extend to the minimum and maximum values. When the right whisker is longer than the left whisker, the data have a longer tail on the right side and are positively skewed.

Moments

The arithmetic average of the various powers of the deviations of the items in a distribution from their arithmetic mean is known as the moments of the distribution.

Moments about the Mean (Central Moments)

Individual series:

$$\begin{aligned} \mu_{r} &= \frac{\sum (X-\bar{X})^{r}}{n} = \frac{\sum x^{r}}{n}, \qquad x = X - \bar{X} \\ \mu_{1} &= \frac{\sum x}{n} = 0, \quad \mu_{2} = \frac{\sum x^{2}}{n}, \quad \mu_{3} = \frac{\sum x^{3}}{n}, \quad \mu_{4} = \frac{\sum x^{4}}{n} \end{aligned}$$

where \(\bar{X}\) = arithmetic mean, \(n\) = number of terms, and \(r = 1, 2, 3, 4\) give the first four moments about the mean.

Discrete and continuous series:

$$\begin{aligned} \mu_{r} &= \frac{\sum f(X-\bar{X})^{r}}{N} = \frac{\sum f x^{r}}{N} \\ \mu_{1} &= \frac{\sum fx}{N} = 0, \quad \mu_{2} = \frac{\sum fx^{2}}{N}, \quad \mu_{3} = \frac{\sum fx^{3}}{N}, \quad \mu_{4} = \frac{\sum fx^{4}}{N} \end{aligned}$$

where \(N\) = total frequency and \(X\) = mid-value of the class (for a continuous series).

Moments about an Arbitrary Point (Raw Moments)

The calculation of the moments about the mean will be easy only when the arithmetic mean of the given series is a whole number. If the arithmetic mean is not a whole number, the calculation becomes difficult. In such cases, we find the moments about any arbitrary number \(a\). Such moments are known as raw moments and are denoted by \(\mu'_r\).

$$\begin{aligned} \text{Individual series:}\quad \mu'_{r} &= \frac{\sum (X-a)^{r}}{n} = \frac{\sum d^{r}}{n}, \qquad d = X - a \\ \text{Discrete series:}\quad \mu'_{r} &= \frac{\sum f(X-a)^{r}}{N} = \frac{\sum f d^{r}}{N} \\ \text{Continuous series:}\quad \mu'_{r} &= \frac{\sum f d'^{\,r}}{N} \times h^{r}, \qquad d' = \frac{X-a}{h} \end{aligned}$$

where \(h\) is the common factor (class width). For \(r = 1, 2, 3, 4\) these give the first four moments about \(a\).

Relation between Central Moments and Raw Moments

$$\begin{aligned} \mu_{1} &= \mu'_{1} - \mu'_{1} = 0 \\ \mu_{2} &= \mu'_{2} - \mu'^{2}_{1} \\ \mu_{3} &= \mu'_{3} - 3\mu'_{2}\mu'_{1} + 2\mu'^{3}_{1} \\ \mu_{4} &= \mu'_{4} - 4\mu'_{3}\mu'_{1} + 6\mu'_{2}\mu'^{2}_{1} - 3\mu'^{4}_{1} \end{aligned}$$

Note: \(\mu_2\) is also known as the variance \((\mu_2 = \sigma^2)\).

Kurtosis

Besides central tendency, dispersion and skewness, the fourth characteristic of a frequency distribution is kurtosis. Kurtosis indicates the peakedness of the distribution: it refers to the degree of peakedness or flatness of a distribution compared to a normal distribution.

Percentile Coefficient of Kurtosis

Kurtosis is measured by the coefficient of kurtosis. The simple measure of kurtosis based on both quartiles and percentiles is the percentile coefficient of kurtosis.

$$K = \frac{\tfrac{1}{2}(Q_{3} - Q_{1})}{P_{90} - P_{10}} = \frac{Q.D.}{P_{90} - P_{10}}$$

where \(Q_3\) = upper quartile, \(Q_1\) = lower quartile, \(P_{90}\) = 90th percentile and \(P_{10}\) = 10th percentile.

Value of KType of distribution
K = 0.263Mesokurtic (normal)
K > 0.263Leptokurtic (more peaked)
K < 0.263Platykurtic (flat topped)

Coefficients of Skewness and Kurtosis Based on Moments

The relative measure of skewness based on moments is denoted by \(\beta_1\); the alternative measure, gamma one \((\gamma_1)\), is its square root:

$$\begin{aligned} \beta_{1} &= \frac{\mu_{3}^{2}}{\mu_{2}^{3}} \\ \gamma_{1} &= \sqrt{\beta_{1}} = \frac{\mu_{3}}{\mu_{2}^{3/2}} \end{aligned}$$

where \(\mu_3\) is the third central moment and \(\mu_2\) the second central moment. If \(\mu_3 = 0\) (i.e. \(\beta_1 = 0\) or \(\gamma_1 = 0\)), the distribution is symmetrical; if \(\gamma_1 > 0\) it is positively skewed; if \(\gamma_1 < 0\) it is negatively skewed.

The relative measure of kurtosis based on moments is given by the coefficient \(\beta_2\) or \(\gamma_2\), defined by Karl Pearson:

$$\beta_{2} = \frac{\mu_{4}}{\mu_{2}^{2}}, \qquad \gamma_{2} = \beta_{2} - 3$$
\(\beta_2\)\(\gamma_2\)Type of distribution
\(\beta_2 = 3\)\(\gamma_2 = 0\)Mesokurtic
\(\beta_2 > 3\)\(\gamma_2 > 0\)Leptokurtic
\(\beta_2 < 3\)\(\gamma_2 < 0\)Platykurtic

Worked Examples

Example 1: Karl Pearson's coefficient of skewness

Calculate Karl Pearson's coefficient of skewness for the following wage distribution.

Wages (Rs.)No. of workers (f)Mid-value (X)\(d' = \frac{X-85}{10}\)fd'fd'²
40–501045−4−40160
50–601555−3−45135
60–702065−2−4080
70–802875−1−2828
80–903585000
90–100259512525
100–1101810523672
110–1201011533090
120–1308125432128
N = 169∑fd' = −30∑fd'² = 718
$$\begin{aligned} \bar{x} &= A + \frac{\sum fd'}{N} \times h = 85 + \frac{-30}{169} \times 10 = 83.22 \\ \sigma &= \sqrt{\frac{\sum fd'^{2}}{N} - \left(\frac{\sum fd'}{N}\right)^{2}} \times h = \sqrt{\frac{718}{169} - \left(\frac{-30}{169}\right)^{2}} \times 10 = 20.54 \end{aligned}$$

Since the highest frequency is 35, the modal class is 80–90, with \(L = 80\), \(f_1 = 35\), \(f_0 = 28\), \(f_2 = 25\), \(h = 10\).

$$\begin{aligned} M_{0} &= L + \frac{f_{1} - f_{0}}{2f_{1} - f_{0} - f_{2}} \times h = 80 + \frac{35 - 28}{70 - 28 - 25} \times 10 = 84.12 \\ S_{k}(P) &= \frac{\bar{x} - M_{0}}{\sigma} = \frac{83.22 - 84.12}{20.54} = -0.04 \end{aligned}$$

Since \(S_k(P) = -0.04 < 0\), the given distribution is negatively skewed.

Example 2: Mean and median from skewness and CV

Pearson's coefficient of skewness is 0.4, the coefficient of variation is 30% and the mode is 88. Find the mean and the median.

$$\begin{aligned} CV &= \frac{\sigma}{\bar{x}} \times 100 \;\Rightarrow\; \frac{30}{100} = \frac{\sigma}{\bar{x}} \;\Rightarrow\; \sigma = 0.3\bar{x} \\ S_{k}(P) &= \frac{\bar{x} - M_{0}}{\sigma} \;\Rightarrow\; 0.4 = \frac{\bar{x} - 88}{0.3\bar{x}} \\ 0.12\bar{x} &= \bar{x} - 88 \;\Rightarrow\; \bar{x} = 100, \quad \sigma = 30 \\ M_{0} &= 3M_{d} - 2\bar{x} \;\Rightarrow\; 88 = 3M_{d} - 200 \;\Rightarrow\; M_{d} = 96 \end{aligned}$$

Important Questions

2078 Exam

  1. Find Karl Pearson's coefficient of skewness when mean = 45, mode = 48 and standard deviation = 15. [2 marks]

  2. The quartile deviation of a distribution is 2 and the difference between \(P_{90}\) and \(P_{10}\) is 8. Calculate the coefficient of kurtosis. [2 marks]

  3. Calculate Karl Pearson's coefficient of skewness and interpret the result. [10 marks]

    Class0–1010–2020–3030–4040–5050–6060–7070–80
    Frequency4514172518107

  4. Test for the normality of these two distributions of wages (in Rs.) and interpret the result on the basis of the information given below: [15 marks]

    Daily wages (Rs.)No. of workers XNo. of workers Y
    20 – 301525
    30 – 403040
    40 – 504460
    50 – 606035
    60 – 703020
    70 – 801415
    80 – 9075

2079 Exam

  1. If mean = 35, mode = 40 and standard deviation = 10, find Karl Pearson's coefficient of skewness and interpret the result. [2 marks]

  2. The standard deviation of a mesokurtic distribution is 7. What must be the value of the fourth moment about the mean? [2 marks]

  3. The following data show the income distribution of families of a certain locality of a city. Calculate the coefficients of skewness and kurtosis and hence comment on the nature of the income distribution. [15 marks]

    Income (in million Rs.)Number of families
    10 – 205
    20 – 309
    30 – 4018
    40 – 5026
    50 – 6020
    60 – 7012
    70 – 807
    80 – 903

2080 Exam

  1. If mean = 25, mode = 28 and standard deviation = 5, find the coefficient of skewness. [2 marks]

  2. If the quartile coefficient of skewness is 0.6, quartile deviation is 5 and the third quartile is 28, find the median of the distribution. [2 marks]

  3. Calculate the percentile coefficient of kurtosis for the following data: [10 marks]

    Expenditure (Rs. 100)10–1920–2930–3940–4950–5960–6970–79
    No. of families35324558431710

2081 Exam

  1. Karl Pearson's coefficient of skewness is 0.5. If mean = 45 and standard deviation = 15, find the value of the mode. [2 marks]

  2. The following table shows the distribution of wages (in Rs.) of workers. Test the normality of the wage distribution. [10 marks]

    Daily wages (in Rs. '00)Number of workers
    20 – 3010
    30 – 4020
    40 – 5025
    50 – 6034
    60 – 7028
    70 – 8018
    80 – 9015

2082 Exam

  1. Find the quartile coefficient of skewness when median = 40, lower quartile = 33 and upper quartile = 45. [2 marks]

  2. Calculate the coefficient of kurtosis when first quartile = 25, third quartile = 40, \(10^{\text{th}}\) percentile = 10 and \(90^{\text{th}}\) percentile = 63. [2 marks]

  3. The following table reveals the daily wages (Rs. '00) of workers of a certain locality: [15 marks]

    Daily wages (Rs. '00)Number of workers
    20 – 3010
    30 – 4015
    40 – 5021
    50 – 6030
    60 – 7024
    70 – 8015
    80 – 905

    Calculate (a) the coefficient of skewness and interpret the result; (b) the coefficient of kurtosis and interpret the result.

2083 Exam

  1. If mean = 45, standard deviation = 10 and mode = 43.7, find Pearson's coefficient of skewness. [2 marks]

  2. A frequency distribution has Karl Pearson's coefficient of skewness 0.5, mean 5 and standard deviation 2. Find the mode of the distribution. [2 marks]

  3. The standard deviation of a symmetrical distribution is 5. What must be the value of the fourth moment about the mean in order that the distribution be mesokurtic? [2 marks]

  4. Calculate Karl Pearson's coefficient of skewness. [10 marks]

    Value0 – 55 – 1010 – 1515 – 2020 – 25
    Frequency111430123

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