For a set of 10 observations,∑x = 452, ∑x² = 24270 and mode = 43.

1.	For a set of 10 observations,∑x = 452, ∑x² = 24270 and mode = 43.7 Find Pearson's coefficient of skewness.
Answer» $\begin{gathered}\begin{gathered}\bf \:Given-\begin{cases} &\sf{number \: of \: observations \: (n) = 10} \\ &\sf{ \sum \: x = 452}\\ &\sf{ \sum \: {x}^{2} = 24270 }\\ &\sf{<klux>MODE</klux> = 43.7} \end{cases}\end{gathered}\end{gathered}$ $\begin{gathered}\begin{gathered}\bf \: To \: Find - \begin{cases} &\sf{Coefficient \: of \: skewness \: (S_k)}\end{cases}\end{gathered}\end{gathered}$ $\begin{gathered}\Large{\bold{{\underline{Formula \: Used - }}}} \end{gathered}$ $\underline{\boxed{ \sf \: Coefficient \: of \: skewness (S_k)= \dfrac{Mean - Mode}{Standard \: Deviation}}}$ $\large\underline{\sf{Solution-}}$ We know, ↝ Mean of data is given by $\rm :\longmapsto\:Mean( \overline{x})= \dfrac{\: \displaystyle\sum_{i=1}^{n}{x _i}}{n}$ $\rm :\longmapsto\:Mean(\overline{x}) \: = \: \dfrac{\: \displaystyle\sum_{i=1}^{10}x_i}{10}$ $\rm :\longmapsto\:Mean \: ( \overline{x}) = \dfrac{452}{10} = 45.2 - - - (1)$ Now, ↝ We know that ↝ Standard deviation of given data is given by $\rm :\longmapsto\:Standard Deviation \: (\sigma) = \sqrt{\dfrac{\: \displaystyle\sum_{i=1}^{n} {(x_i)}^{2} }{n} - {\bigg(\overline{x} \bigg) }^{2} }$ $\rm :\longmapsto\:Standard Deviation \: (\sigma) = \sqrt{\dfrac{24270}{10} - {(45.2)}^{2} }$ $\rm :\longmapsto\:Standard Deviation \: (\sigma) = \sqrt{2427 - 2043.04}$ $\rm :\longmapsto\:Standard Deviation \: (\sigma) = \sqrt{383.96}$ $\rm :\longmapsto\:Standard Deviation \: (\sigma) = 19.59 \: (approimately)$ Now, ↝ we have $\begin{gathered}\begin{gathered}\bf \: \: \: \: \: \: \: \: \: \bull \: \: \: \:\begin{cases} &\sf{Mean = 45.2} \\ &\sf{Mode = 43.7}\\ &\sf{Standard \: Deviation \: = 19.59} \end{cases}\end{gathered}\end{gathered}$ So, ↝ Coefficient of Skewness is given by $\sf \: Coefficient \: of \: skewness (S_k)= \dfrac{Mean - Mode}{Standard \: Deviation}$ $\sf \: Coefficient \: of \: skewness (S_k)= \dfrac{45.2 - 43.7}{19.59}$ $\sf \: Coefficient \: of \: skewness \: (S_k) = \dfrac{1.5}{19.59}$ $\bf\implies \:Coefficient \: of \: skewness \: (S_k) = 0.077$ Additional Information The coefficient of skewness is a measure of ASYMMETRY in the distribution. A positive SKEW indicates a longer tail to the right, while a negative skew indicates a longer tail to the left. A PERFECTLY symmetric distribution, like the normal distribution, has a skew equal to zero. For small data sets this measure is unreliable. Types of Skewness 1. Positive Skewness If the given distribution is shifted to the left and with its tail on the right side, it is a positively skewed distribution. It is ALSO called the right-skewed distribution. A positively skewed distribution assumes a skewness value of more than zero. Since the skewness of the given distribution is on the right, the mean value is greater than the median and moves towards the right, and the mode occurs at the highest frequency of the distribution. 2. Negative Skewness If the given distribution is shifted to the right and with its tail on the left side, it is a negatively skewed distribution. It is also called a left-skewed distribution. The skewness value of any distribution showing a negative skew is always less than zero. The skewness of the given distribution is on the left; hence, the mean value is less than the median and moves towards the left, and the mode occurs at the highest frequency of the distribution.

For a set of 10 observations,∑x = 452, ∑x² = 24270 and mode = 43.7 Find Pearson's coefficient of skewness.

Answer»

$\begin{gathered}\begin{gathered}\bf \:Given-\begin{cases} &\sf{number \: of \: observations \: (n) = 10} \\ &\sf{ \sum \: x = 452}\\ &\sf{ \sum \: {x}^{2} = 24270 }\\ &\sf{<klux>MODE</klux> = 43.7} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: To \: Find - \begin{cases} &\sf{Coefficient \: of \: skewness \: (S_k)}\end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\Large{\bold{{\underline{Formula \: Used - }}}} \end{gathered}$

$\underline{\boxed{ \sf \: Coefficient \: of \: skewness (S_k)= \dfrac{Mean - Mode}{Standard \: Deviation}}}$

$\large\underline{\sf{Solution-}}$

We know,

↝ Mean of data is given by

$\rm :\longmapsto\:Mean( \overline{x})= \dfrac{\: \displaystyle\sum_{i=1}^{n}{x _i}}{n}$

$\rm :\longmapsto\:Mean(\overline{x}) \: = \: \dfrac{\: \displaystyle\sum_{i=1}^{10}x_i}{10}$

$\rm :\longmapsto\:Mean \: ( \overline{x}) = \dfrac{452}{10} = 45.2 - - - (1)$

Now,

↝ We know that

↝ Standard deviation of given data is given by

$\rm :\longmapsto\:Standard Deviation \: (\sigma) = \sqrt{\dfrac{\: \displaystyle\sum_{i=1}^{n} {(x_i)}^{2} }{n} - {\bigg(\overline{x} \bigg) }^{2} }$

$\rm :\longmapsto\:Standard Deviation \: (\sigma) = \sqrt{\dfrac{24270}{10} - {(45.2)}^{2} }$

$\rm :\longmapsto\:Standard Deviation \: (\sigma) = \sqrt{2427 - 2043.04}$

$\rm :\longmapsto\:Standard Deviation \: (\sigma) = \sqrt{383.96}$

$\rm :\longmapsto\:Standard Deviation \: (\sigma) = 19.59 \: (approimately)$

Now,

↝ we have

$\begin{gathered}\begin{gathered}\bf \: \: \: \: \: \: \: \: \: \bull \: \: \: \:\begin{cases} &\sf{Mean = 45.2} \\ &\sf{Mode = 43.7}\\ &\sf{Standard \: Deviation \: = 19.59} \end{cases}\end{gathered}\end{gathered}$

So,

↝ Coefficient of Skewness is given by

$\sf \: Coefficient \: of \: skewness (S_k)= \dfrac{Mean - Mode}{Standard \: Deviation}$

$\sf \: Coefficient \: of \: skewness (S_k)= \dfrac{45.2 - 43.7}{19.59}$

$\sf \: Coefficient \: of \: skewness \: (S_k) = \dfrac{1.5}{19.59}$

$\bf\implies \:Coefficient \: of \: skewness \: (S_k) = 0.077$

Additional Information

The coefficient of skewness is a measure of ASYMMETRY in the distribution.

A positive SKEW indicates a longer tail to the right, while a negative skew indicates a longer tail to the left.

A PERFECTLY symmetric distribution, like the normal distribution, has a skew equal to zero.

For small data sets this measure is unreliable.

Types of Skewness

1. Positive Skewness

If the given distribution is shifted to the left and with its tail on the right side, it is a positively skewed distribution. It is ALSO called the right-skewed distribution.

A positively skewed distribution assumes a skewness value of more than zero. Since the skewness of the given distribution is on the right, the mean value is greater than the median and moves towards the right, and the mode occurs at the highest frequency of the distribution.

2. Negative Skewness

If the given distribution is shifted to the right and with its tail on the left side, it is a negatively skewed distribution. It is also called a left-skewed distribution. The skewness value of any distribution showing a negative skew is always less than zero.

The skewness of the given distribution is on the left; hence, the mean value is less than the median and moves towards the left, and the mode occurs at the highest frequency of the distribution.