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Descriptive Statistics

Bowleys Coefficient of skewness for grouped data

  • STATISTICS
Bowley’s Coefficient of Skewness for grouped data Skewness is a measure of symmetry. The meaning of skewness is “lack of symmetry”. Skewness gives us an idea about the concentration of higher or lower data values around the central value of the data. For a symmetric distribution, the two quartiles namely $Q_1$ and $Q_3$ are equidistance from the median (i.e. $Q_2$). That is for symmetric distribution $Q_3 - Q_2 = Q_2 -Q_1$.
6 min Read 4

Bowleys Coefficient of skewness for ungrouped data

  • STATISTICS
Bowley’s Coefficient of Skewness for Ungrouped data Skewness is a measure of symmetry. The meaning of skewness is “lack of symmetry”. Skewness gives us an idea about the concentration of higher or lower data values around the central value of the data. For a symmetric distribution, the two quartiles namely $Q_1$ and $Q_3$ are equidistance from the median (i.e. $Q_2$). That is for symmetric distribution $Q_3 - Q_2 = Q_2 -Q_1$.
5 min Read 3

Central moments for grouped data

  • STATISTICS
Central moments for grouped data Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution. The mean of $X$ is denoted by $\overline{x}$ and is given by $$ \begin{eqnarray*} \overline{x}& =\frac{1}{N}\sum_{i=1}^{n}f_ix_i \end{eqnarray*} $$ Formula The first four central moments are as follows $m_1=0$ $m_2 =\frac{1}{N}\sum_{i=1}^n f_i(x_i-\overline{x})^2$ $m_3 =\frac{1}{N}\sum_{i=1}^n f_i(x_i-\overline{x})^3$ $m_4 =\frac{1}{N}\sum_{i=1}^n f_i(x_i-\overline{x})^4$ where, $N$ total number of observations $\overline{x}$ sample mean Example 1 Following tables shows a frequency distribution of daily number of car accidents at a particular cross road during a month of April.
3 min Read 1

Central moments for ungrouped data

  • STATISTICS
Central moments for ungrouped data Let $x_1, x_2,\cdots, x_n$ be $n$ observations. The mean of $X$ is denoted by $\overline{x}$ and is given by $$ \begin{eqnarray*} \overline{x}& =\frac{1}{n}\sum_{i=1}^{n}x_i \end{eqnarray*} $$ Formula The first four sample central moments are as follows $m_1=0$ (always) $m_2 =\dfrac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^2$ $m_3 =\dfrac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^3$ $m_4 =\dfrac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^4$ where, $n$ total number of observations $\overline{x}$ sample mean Example The hourly earning (in dollars) of sample of 7 workers are : 26,21,24,22,25,24,23.
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Coefficient of variation for grouped data

  • STATISTICS
Coefficient of variation for grouped data Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution. Formula Coefficient of variation is given by $CV =\dfrac{s_x}{\overline{x}}\times 100$ where, $\overline{x} =\dfrac{1}{N}\sum_{i=1}^{n}f_ix_i$ is the sample mean of $X$, $N$ total number of observations, $s_x=\sqrt{V(x)}$ is the standard deviation of $X$, $s_x^2=V(x)=\dfrac{1}{N}\sum_{i=1}^{n}f_ix_i^2 -(\overline{x})^2$ is the variance of $X$ Coefficient of variation of one data set is lower than the coefficient of variation of other data set, then the data set with lower coefficient of variation is more consistent than the other.
2 min Read 1

Coefficient of variation for ungrouped data

  • STATISTICS
Coefficient of variation for ungrouped data Coefficient of variation is an absolute measure of variation and is used for the comparison of variability between two or more frequency distribution. Let $x_i, i=1,2, \cdots , n$ be $n$ observations. Formula Coefficient of variation is given by $CV =\dfrac{s_x}{\overline{x}}\times 100$ where, $\overline{x} =\dfrac{1}{n}\sum_{i=1}^{n}x_i$ is the sample mean of $X$, $n$ total number of observations, $s_x=\sqrt{V(x)}$ is the standard deviation of $X$, $s_x^2=V(x)=\dfrac{1}{n-1}\bigg(\sum_{i=1}^{n}x_i^2 -\dfrac{\big(\sum_{i=1}^n x_i\big)^2}{n}\bigg)$ is the variance of $X$ Example 1 The following data gives the hourly wage rates (in dollars) of 10 employees of a company.
2 min Read 1

Deciles for grouped data

  • STATISTICS
Deciles for grouped data Deciles are the values of arranged data which divide whole data into ten equal parts. They are 9 in numbers namely $D_1,D_2, \cdots, D_9$. Here $D_1$ is first decile, $D_2$ is second decile, $D_3$ is third decile and so on. Decile formula for grouped data For discrete frequency distribution, the formula for $i^{th}$ decile is $D_i =\bigg(\dfrac{i(N)}{10}\bigg)^{th}$ value, $i=1,2,\cdots, 9$ where, $N$ is total number of observations.
5 min Read 4

Deciles for ungrouped data

  • STATISTICS
Deciles for ungrouped data Deciles are the values of arranged data which divide whole data into ten equal parts. They are 9 in numbers namely $D_1,D_2, \cdots, D_9$. Here $D_1$ is first decile, $D_2$ is second decile, $D_3$ is third decile and so on. Formula The formula for $i^{th}$ decile is $D_i =$ Value of $\bigg(\dfrac{i(n+1)}{10}\bigg)^{th}$ observation, $i=1,2,3,\cdots, 9$ where, $n$ is the total number of observations. Example 1 The marks obtained by a sample of 20 students in a class test are as follows:
4 min Read 3

Empirical Rule for grouped data

  • STATISTICS
Empirical rule for grouped data Empirical rule is the general rule of thumb that applies to bell shaped (symmetrical) distribution. The empirical rule can be stated as : $68$% of the data will fall within one standard deviation of the mean, $95$% of the data will fall within two standard deviations of the mean, $99.7$% of the data will fall within three standard deviations of the mean. Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution.
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Empirical Rule for ungrouped data

  • STATISTICS
Empirical Rule for ungrouped data Empirical rule is the general rule of thumb that applies to bell shaped (symmetrical) distribution. The empirical rule can be stated as : $68$% of the data will fall within one standard deviation of the mean, $95$% of the data will fall within two standard deviations of the mean, $99.7$% of the data will fall within three standard deviations of the mean. Formula Let $x_1,x_2,\cdots, x_n$ be $n$ sample observations.
2 min Read 1

Five number summary for grouped data

  • STATISTICS
Five number summary for grouped data A five number summary is a quick and easy way to determine the the center, the spread and outliers (if any) of a data set. Five number summary includes five values, namely, minimum value ($\min$), first quartile ($Q_1$), $\text{median }$ ($Q_2$), third quartile ($Q_3$), maximum value ($\max$). Formula $\min$= lower limit of the first class, $\max$= upper limit of the last class, $Q_i=l + \bigg(\dfrac{\dfrac{iN}{4} - F_<}{f}\bigg)\times h$ ; $i=1,2,\cdots,3$
5 min Read 4

Five number summary for ungrouped data

  • STATISTICS
Five number summary for ungrouped data A five number summary is a quick and easy way to determine the the center, the spread and outliers (if any) of a data set. Five number summary includes five values, namely, minimum value ($\min$), first quartile ($Q_1$), $\text{median }$ ($Q_2$), third quartile ($Q_3$), maximum value ($\max$). Formula $\min$, $Q_1$, $\text{median}$, $Q_3$ and $\max$ The five number summary of a data gives you a rough idea about the range of the data, center of the data, variation in the data and symmetry of the data.
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