**Content**

## **Skewness**

**Skewness**

The curve which is the plotted observations represents the skewness if the shape of the curve is not symmetric, of the given set. In other words the lack of symmetry in the graph of the given information represents the skewness of the given set. Depending on the tail in the right or left the skewness is known as positively skewed or negatively skewed. The distribution depending on this skewness is known as positively skewed distribution or negatively skewed distribution

The mean, mode and median shows the nature of distribution so if the nature or shape of the curve is symmetric these measure of central tendencies are equal and for the skewed distributions these measure of central tendencies varies as either mean>median>mode or mean<median<mode.

**Variance and Skewness**

Variance | Skewness |

Amount of variability can be obtained using variance | Direction of variability can be obtained using skewness |

Application of measure of variation is in Business and economics | Application of measure of Skewness is in medical and life sciences |

**Measure of Skewness**

To find the degree and the direction of the frequency distribution whether positive or negative the measure of skewness is very helpful even with the help of the graph we know the positive or negative nature of the skewness but the magnitude will not be exact in graphs hence these statistical measures gives the magnitude of lack of symmetry.

To be specific the measure of skewness must have

- Unit free so that the different distributions can be comparable if the units are same or different.
- Value of measure for symmetric distribution zero and positive or negative for positive or negative distributions accordingly.
- The value of measure should vary if we move from negative skewness to positive skewness.

There are two types of measure of skewness

**Absolute Measure of skewness****Relative Measure of skewness**

## **Absolu****te Measure of skewness **

In the symmetrical distribution the mean, mode and median are same so in absolute measure of skewness the difference of these central tendencies gives the extent of symmetry in the distribution and the nature as positive or negative skewed distribution but the absolute measure for different units is not useful while comparing two sets of information.

The Absolute skewness can be obtained using

- Skewness(S
_{k})=Mean-Median - Skewness(S
_{k})=Mean-Mode - Skewness(S
_{k})=(Q_{3}-Q_{2})-(Q_{2}-Q_{1})

## **Relative Measure of skewness**

Relative measure of skewness is used to compare the skewness in two or more distributions by eliminating the influence of variation, relative measure of skewness is known as coefficient of skewness, the following are the important relative measure of skewness.

**Karl Pearson’s Coefficient of Skewness**

This method is used most often to calculate skewness

[latex]S_k=\frac{Mean-Mode}{\sigma}[/latex]

this coefficient of skewness is positive for positive distribution, negative for negative distribution and zero for the symmetric distribution. This Karl Pearson’s coefficient usually lies between +1 and -1. If Mode is not defined then to calculate the Karl Pearson’s coefficient we use the formula as

[latex]S_k=\frac{3(Mean-Mode)}{\sigma}[/latex]

If we use this relation then Karl Pearson’s coefficient lies between +3 and -3.

**2. Bowleys’s Coefficient of Skewness|Quartile measure of skew ness**

In Bowleys’s coefficient of skewness the quartile deviations were used to find the skewness so it is also known as quartile measure of skewness

[latex]S_k=\frac{(Q_3-Q_2)-(Q_2-Q_1)}{(Q_3-Q_1)}

\\=\frac{(Q_3-2Q_2+Q_1)}{(Q_3-Q_1)}[/latex]

or we can write it as

[latex]S_k=\frac{(Q_3-M)-(M-Q_1)}{(Q_3-Q_1)}

\\=\frac{(Q_3-2M+Q_1)}{(Q_3-Q_1)}[/latex]

this value of coefficient is zero if the distribution is symmetric and the value for positive distribution is positive, for negative distribution is negative. The value of S_{k} lies between -1 and +1.

**3. Kelly’s Coefficient of Skewness**

In this measure of skewness the percentiles and deciles are used to calculate the skewness, the coefficient is

[latex]S_k=\frac{(P_{90}-P_{50})-(P_{50}-P_{10})}{(P_{90}-P_{10})}

\\=\frac{(P_{90}-2P_{50}+P_{10})}{(P_{90}-P_{10})}[/latex]

where these skewness involves the 90, 50 and 10 percentiles and using deciles we can write it as

[latex]S_k=\frac{(D_9-D_5)-(D_5-D_1)}{(D_9-D_1)}

\\=\frac{(D_9-2D_5+D_1)}{(D_9-D_1)}[/latex]

in which 9,5 and 1 deciles were used.

**4. β and γ Coefficient of Skewness| Measure of skew ness based on moments.**

Using the central moments the measure of skewness the β coefficient of skewness can be define as

[latex]\beta_1=\frac{{\mu_3}^2}{{\mu_2}^3}[/latex]

this coefficient of skewness gives value zero for the symmetric distribution but this coefficient not tells specifically for the direction either positive or negative, so this drawback can be removed by taking square root of beta as

[latex]\gamma_1=\pm \sqrt{\beta_1}=\frac{\mu_3}{{\mu_2}^{3/2}}=\frac{\mu_3}{\sigma^3}[/latex]

this value gives the positive and negative value for the positive and negative distributions respectively.

**Examples of skewness**

- Using the following information find the coefficient of skewness

Wages | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |

No. of people | 12 | 18 | 35 | 42 | 50 | 45 | 20 | 8 |

**Solution:** To find the coefficient of skewness we will use karl Pearson’s coefficient

frequency | mid-value(x) | fx | fx^{2} | |

0-10 | 12 | 5 | 60 | 300 |

10-20 | 18 | 15 | 270 | 4050 |

20-30 | 35 | 25 | 875 | 21875 |

30-40 | 42 | 35 | 1470 | 51450 |

40-50 | 50 | 45 | 2250 | 101250 |

50-60 | 45 | 55 | 2475 | 136125 |

60-70 | 20 | 65 | 1300 | 84500 |

70-80 | 8 | 75 | 600 | 45000 |

230 | 9300 | 444550 |

the karl pearson coefficient of skewness is

[latex]\begin{array}{l}

\text { Karl person’s coefficient of skewness }=J=\frac{\text { Mean }-\text { Mode }}{S . D .}\\

\begin{array}{l}

\text { Mean, } \quad \bar{x}=\frac{1}{N} \sum_{i} f_{i} x_{i}, \quad \text { Mode }=l+\frac{c\left(f_{1}-f_{0}\right)}{\left(f_{1}-f_{0}\right)+\left(f_{1}-f_{2}\right)} \\

\text { Standard deviation }=\sqrt{\frac{1}{N} \sum_{i} f_{i} x_{i}^{2}-\bar{x}^{2}}

\end{array}

\end{array}[/latex]

[latex]\begin{array}{c}

\text { Mean }=\frac{9300}{230}=40.43 \\

\text { S.D. }=\sqrt{\frac{1}{N} \sum_{i} f_{i} x_{i}^{2}-\bar{x}^{2}}=\sqrt{\frac{1}{230}(444550)-\left[\frac{9300}{230}\right]^{2}}=17.27 .

\end{array}[/latex]

the modal class is maximum frequent class 40-50 and the respective frequencies are

[latex]f_{0}=42, f_{1}=50,f_{2}=45[/latex]

thus

[latex]\text { Hence, Mode }=40+\frac{10(50-42)}{(50-42)+(50-45)}=46.15[/latex]

so the coefficient of skewness will be

[latex]=\frac{40.43-46.15}{17.27}=-0.3312[/latex]

which shows the negative skewness.

2. Find the coefficient of skewness of the frequency distributed marks of 150 students in certain examination

marks | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |

freq | 10 | 40 | 20 | 0 | 10 | 40 | 16 | 14 |

**Solution:** To calculate the coefficient of skewness we require mean, mode, median and standard deviation for the given information so for calculating these we form the following table

class interval | f | mid value x | c.f. | d’=(x-35)/10 | f*d’ | f*d’^{2} |

0-10 | 10 | 5 | 10 | -3 | -30 | 90 |

10-20 | 40 | 15 | 50 | -2 | -80 | 160 |

20-30 | 20 | 25 | 70 | -1 | -20 | 20 |

30-40 | 0 | 35 | 70 | 0 | 0 | 0 |

40-50 | 10 | 45 | 80 | 1 | 10 | 10 |

50-60 | 40 | 55 | 120 | 2 | 80 | 160 |

60-70 | 16 | 65 | 136 | 3 | 48 | 144 |

70-80 | 14 | 75 | 150 | 4 | 56 | 244 |

total=64 | total=828 |

now the measures will be

[latex]\begin{array}{l}

Median =\mathrm{L}+\frac{\left(\frac{\mathrm{N}}{2}-\mathrm{C}\right)}{\mathrm{f}} \times \mathrm{h}=40+\frac{75-70}{10} \times 10=45

\\Mean (\overline{\mathrm{x}})=\mathrm{A}+\frac{\sum_{\mathrm{i}=1}^{\mathrm{k}} \mathrm{fd}^{\prime}}{\mathrm{N}} \times \mathrm{h}=35+\frac{64}{150} \times 10=39.27

\end{array}[/latex]

and

[latex]\begin{aligned}

Standard Deviation }(\sigma) &=\mathrm{h} \times \sqrt{\frac{\sum \mathrm{fd}^{\prime 2}}{\mathrm{~N}}-\left(\frac{\sum \mathrm{fd}}{\mathrm{N}}\right)^{2}} \\ &=10 \times \sqrt{\frac{828}{150}-\left(\frac{64}{150}\right)^{2}}

\\&=10 \times \sqrt{5.33}=23.1 \end{aligned}[/latex]

hence the coefficient of skewness for the distribution is

[latex]S_k=\frac{3(Mean-Median)}{\sigma}

\\=\frac{3(39.27-45}{23.1}=-0.744[/latex]

3. Find the mean, variance and coefficient of skewness of distribution whose first four moments about 5 are 2,20,40 and 50.

**Solution:** since the first four moments are given so

[latex]\begin{array}{c}

\mu_{1}^{\prime}(5)=\frac{1}{N} \sum_{i=1}^{k} f_{i}\left(x_{i}-5\right)=2 ; \mu_{2}^{\prime}(5)=\frac{1}{N} \sum_{i=1}^{k} f_{i}\left(x_{i}-5\right)^{2}=20 ; \\

\mu_{3}^{\prime}(5)=\frac{1}{N} \sum_{i=1}^{k} f_{i}\left(x_{i}-5\right)^{3}=40 \quad \text { and } \quad \mu_{4}^{\prime}(5)=\frac{1}{N} \sum_{i=1}^{k} f_{i}\left(x_{i}-5\right)^{4}=50 . \\

\mu_{1}^{\prime}(5)=\frac{1}{N} \sum_{i=1}^{k} f_{i} x_{i}-5=2 \\

\Rightarrow \bar{x}=2+5=7

\end{array}[/latex]

so we can write it

[latex]\begin{array}{l}

\mu_{r}=\mu_{r}^{\prime}(A)-{ }^{r} C_{1} \mu_{r-1}^{\prime}(A) \mu_{1}^{\prime}(A)+{ }^{r} C_{2} \mu_{r-2}^{\prime}(A)\left[\dot{\mu}_{1}^{\prime}(A)\right]^{2}-\ldots .+(-1)^{r}\left[\mu_{1}^{\prime}(A)\right]^{r} \\

\text { Hence } \mu_{2}=\mu_{2}^{\prime}(5)-\left[\mu_{1}^{\prime}(5)\right]^{2}=20-4=16 \\

\mu_{3}=\mu_{3}^{\prime}(5)-3 \mu_{2}^{\prime}(5) \mu_{1}^{\prime}(5)+2\left[\mu_{1}^{\prime}(5)\right]^{3} \\

40-3 \times 20 \times 2+2 \times 2^{3}=-64

\end{array}[/latex]

so the coefficient of skewness is

[latex]\beta_{1}=\frac{\mu_{3}^{2}}{\mu_{2}^{3}}=\frac{(-64)^{2}}{(16)^{3}}=-1[/latex]

** ****P****ositively skewed distribution definition|Right skewed distribution meaning **

**ositively skewed distribution definition|Right skewed distribution meaning**

Any distribution in which the measure of central tendencies i.e mean, mode and median having positive values and the information in the distribution lacks the symmetry.

In other words the positively skewed distribution is the distribution in which the measure of central tendencies follows as **mean>median>mode** in the right side of the curve of the distribution.

If we sketch the information of the distribution the curve will be right tailed because of which positively skewed distribution is also known as **right skewed distribution**.

from above curve it is clear that the mode is the smallest measure in positively or right skewed distribution and the mean is the largest measure of central tendencies.

## **positively skewed distribution example|example of right skewed distribution**

- For a positively skewed or right skewed distribution if the coefficient of skewness is 0.64, find the mode and median of the distribution if mean and standard deviations are 59.2 and 13 respectively.

**Solution:** The given values are mean=59.2, s_{k}=0.64 and *σ*=13 so using the relation

[latex]S_k=\frac{mean-mode}{\sigma}

\\0.64=\frac{59.2-\text { Mode }}{13}

\\Mode =59.20-8.32=50.88

\\Mode =3 Median -2 Mean

\\50.88=3 Median -2(59.2)

\\Median =\frac{50.88+118.4}{3}=\frac{169.28}{3}=56.42[/latex]

2. Find the standard deviation of the positively skewed distribution whose coefficient of skewness is 1.28 with mean 164 and mode 100?

**Solution: **In the same way using the given information and the formula for the coefficient of positively skewed distribution

[latex]S_k=\frac{mean-mode}{\sigma}

\\1.28=\frac{164-100}{\sigma}

\\\sigma=\frac{64}{1.28}=50

[/latex]

so the standard deviation will be 50.

3. In the quarterlies deviations if the addition of first and third quarterlies is 200 with median 76 find the value of third quartile of the frequency distribution which is positively skewed with coefficient of skewness 1.2?

**S****olution: **To find the third quartile we have to use the relation of coefficient of skewness and quarterlies, since the given information is

[latex]S_k=1.2

\\Q_1+Q_3=200

\\Q_2=76[

\\S_{k}=\frac{\left(Q_{3}+Q_{1}-2 Q_{2}\right)}{\left(Q_{3}-Q_{1}\right)}

\\1.2=\frac{(200-2 \times 76)}{\left(Q_{3}-Q_{1}\right)}

\\Q_{3}-Q_{1}=\frac{48}{1.2}=40

\\Q_{3}-Q_{1}=40

[/latex]

from the given relation we have

[latex]Q_1+Q_3=200

\\Q_1=200-Q_3[/latex]

from these two equations we can write

[latex]Q_{3}-Q_{1}=40

\\ Q_{3}-(200-Q_3)=40

\\2Q_3=240

\\Q_3=120[/latex]

so the value of the third quartile is 120.

4. Find the coefficient of skewness for the following information

x | 93-97 | 98-102 | 103-107 | 108-112 | 113-117 | 118-122 | 123-127 | 128-132 |

f | 2 | 5 | 12 | 17 | 14 | 6 | 3 | 1 |

**Solution:** here we will use Bowley’s measure of skewness using quartiles

class | frequency | cumulative frequency |

92.5-97.5 | 2 | 2 |

97.5-102.5 | 5 | 7 |

102.5-107.5 | 12 | 19 |

107.5-112.5 | 17 | 36 |

112.5-117.5 | 14 | 50 |

117.5-122.5 | 6 | 56 |

122.5-127.5 | 3 | 59 |

127.5-132.5 | 1 | 60 |

N=60 |

As N^{th}/4=15^{th} observation of class is **102.5-107.5** , N^{th}/2=30^{th} observation of class is ** 107.5-112.5 **and 3N

^{th}/4=45

^{th}observation of class is

**so**

**112.5-117.5**[latex]Q_{1}=l_{1}+\frac{\left(\frac{N}{4}-m_{1}\right) c_{1}}{f_{1}}=102.5+\frac{\left(\frac{60}{4}-7\right) 5}{12}=105.83[/latex]

and

[latex]Q_{3}=l_{3}+\frac{\left(\frac{3 N}{4}-m_{3}\right) c_{3}}{f_{3}}=112.5+\frac{\left(\frac{3 \times 60}{4}-36\right) 5}{14}=115.714[/latex]

and median is

[latex]Q_{2}=l_{2}+\frac{\left(\frac{N}{2}-m_{2}\right) c_{2}}{f_{2}}=107.5+\frac{\left(\frac{60}{2}-19\right) 5}{17}=110.735[/latex]

thus

[latex]Q=\frac{Q_{3}+Q_{1}-2 M}{Q_{3}-Q_{1}}=\frac{115.714+105.83-2 \times 110.735}{115.714-105.83}=0.0075[/latex]

which is positively skewed distribution.

## **where is the mean in a positively skewed distribution**

We know that the positively skewed distribution is right skewed distribution so the curve is right tailed the meaning of this most of the information will be nearer to the tail so the mean in a positively skewed distribution is nearer to the tail and since in positively or right skewed distribution mean>median>mode so mean will be after the median.

## **Right skewed distribution mean median mode|relationship between mean median and mode in positively skewed distribution**

**Right skewed distribution mean median mode|relationship between mean median and mode in positively skewed distribution**

In the positively skewed or right skewed distribution the measure of central tendencies mean, median and mode are in the order **mean>median>mode**, as mode is the smallest one then median and the largest central tendency is the mean which for the right tailed curve is nearer to the tail of the curve for the information.

so the relationship between mean median and mode in positively skewed distribution is in the increasing order and with the help of the difference of these two central tendencies the coefficient of skewness can be calculated, so mean, median and mode gives the nature of skewness also.

## **positively skewed distribution graph|positively skewed distribution curve**

The graph either in the form of smooth curve or in the form of histogram for the discrete information, the nature is right tailed as the mean of the information gather around the tail of the curve as skewness of distribution discusses the shape of the distribution. Since the large amount of data is in left of the curve and tail of the curve onto the right is longer.

some of the graphs of positively distributed information are as follows

from the above graphs it is clear that the curve has lacking the symmetry in any aspects .

## **positively skewed score distribution**

In any distribution if the scores are in the positively skewed that is the score following the positively skewed distribution as mean>median>mode and the curve of the distribution score having right tailed curve in which score is affected by the large value.

This type of distribution is known as positively skewed score distribution. All the properties and rules for this distribution are the same from positively skewed or right skewed distribution.

** ****positive skew frequency distribution**

**positive skew frequency distribution**

In positively skewed frequency distribution on average the frequency of the information are smaller as compared to the distribution so the positive skew frequency distribution is nothing but the positively skewed or right skewed distribution where the curve is right tailed curve.

**positive vs negative skewed distribution|positively skewed distribution vs negatively skewed**

positive skewed distribution | negative skewed distribution |

In the positively skewed distribution the information is distributed as mean is the largest and mode is smallest | In the negatively skewed distribution the information is distributed as mean is the smallest and mode is largest |

the curve is right tailed | the curve is left tailed |

mean>median>mode | mean<median<mode |

## **FAQs**

**How do you know if a distribution is positively or negatively skewed**

The skewness is positive if mean>median>mode and negative if mean<median<mode,

From the distribution curve also we can judge if the curve is right tailed it is positive and if the curve is left tailed it is negative

**How do you determine positive skewness**

By calculating the measure of coefficient of skewness if positive then skewness is positive or by plotting the curve of distribution if right tailed then positive or by checking mean>median>mode

**What does a positive skew represent**

The positive skewness represent that the score of the distribution lies nearer to large values and the curve is right tailed and the mean is the largest measure

**How do you interpret a right skewed histogram**

if the histogram is right skewed then the distribution is positively skewed distribution where mean>median>mode

**In distributions that are skewed to the right what is the relationship of the mean median and mode**

The relationship is mean>median>mode

**Conclusion:**

The skewness is important concept of statistics which gives the asymmetry or lack of symmetry present in the distribution of probability depending on the positive or negative value it is classified as positively skewed distribution or negatively skewed distribution, in the above article the brief concept with examples discussed , if you require further reading go through

https://en.wikipedia.org/wiki/skewness

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