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Descriptive Statistics

Geometric mean for ungrouped data

  • STATISTICS
Geometric mean for ungrouped data Let $x_i, i=1,2, \cdots , n$ be $n$ positive observations then the geometric mean of $X$ is denoted by $GM$. Formula The geometric mean is given by $GM =\big(\prod_{i=1}^n x_i\big)^{1/n}=\sqrt[n]{x_1\times x_2\times \cdots \times x_n}$ Example The marks obtained by a sample of 5 students in a class test are as follows: 20,30,21,10 and 15. Find the geometric mean of the marks. Solution The geometric mean of $X$ is
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Harmonic Mean for grouped data

  • STATISTICS
Harmonic mean for grouped data Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution. Then the harmonic mean of $X$ is denoted by $HM$ and is given by $$ \begin{aligned} HM &= \dfrac{N}{\sum_{i=1}^{n} \dfrac{f_i}{x_i}} \end{aligned} $$ where $N=\sum f$ total no. of observations Example 1 Compute harmonic mean for the following frequency distribution. x 10 15 20 25 30 f 3 12 25 10 5 Solution $x$ Freq ($f$) $f/x$ 10 3 0.
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Harmonic Mean for ungrouped data

  • STATISTICS
Harmonic mean for ungrouped data Let $x_i, i=1,2, \cdots , n$ be $n$ positive observations then the harmonic mean of $X$ is denoted by $HM$. Formula The harmonic mean is given by $HM = \dfrac{n}{\sum_{i=1}^{n} \dfrac{1}{x_i}}$ Example A bus driver travels total 30 miles. He travel first 10 miles with speed of 30 miles per hour, second 10 miles with the speed of 35 miles per hour and last 10 miles with the speed of 45 miles per hour.
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Inter Quartile Range for grouped data

  • STATISTICS
Inter Quartile Range for grouped data Inter quartile range is given by $IQR = Q_3-Q_1$ where, $Q_1$ is the first quartile $Q_3$ is the third quartile The formula for $i^{th}$ quartile is $$ \begin{aligned} Q_i=l + \bigg(\frac{\frac{iN}{4} - F_<}{f}\bigg)\times h; \quad i=1,2,3 \end{aligned} $$ where, $l :$ the lower limit of the $i^{th}$ quartile class $N=\sum f :$ total number of observations $f :$ frequency of the $i^{th}$ quartile class $F_< :$ cumulative frequency of the class previous to $i^{th}$ quartile class $h :$ the class width Example 1 A class teacher has the following data about the number of absences of 35 students of a class.
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Inter Quartile Range for ungrouped data

  • STATISTICS
Inter Quartile Range for ungrouped data Inter quartile range is given by $IQR = Q_3-Q_1$ where $Q_1$ is the first quartile $Q_3$ is the third quartile The formula for $i^{th}$ quartile is $Q_i =$ Value of $\bigg(\dfrac{i(n+1)}{4}\bigg)^{th}$ observation, $i=1,2,3$ where $n$ is the total number of observations. Example 1 A random sample of 15 patients yielded the following data on the length of stay (in days) in the hospital. 5, 6, 9, 10, 15, 10, 14, 12, 10, 13, 13, 9, 8, 10, 12.
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Karl Pearson coefficient of skewness for grouped data

  • STATISTICS
Karl Pearson coefficient of skewness for grouped data Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution. Karl Pearson coefficient of skewness formula The Karl Pearson’s coefficient skewness for grouped data is given by $S_k =\dfrac{Mean-Mode)}{sd}=\dfrac{\overline{x}-\text{Mode}}{s_x}$ OR $S_k =\dfrac{3(Mean-Median)}{sd}=\dfrac{\overline{x}-M}{s_x}$ where, $\overline{x}$ is the sample mean, $M$ is the median, $s_x$ is the sample standard deviation. Sample mean The sample mean $\overline{x}$ is given by $$ \begin{eqnarray*} \overline{x}& =\frac{1}{N}\sum_{i=1}^{n}f_ix_i \end{eqnarray*} $$
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Karl Pearson coefficient of skewness for ungrouped data

  • STATISTICS
Karl Pearson coefficient of skewness for ungrouped data Let $x_i, i=1,2, \cdots , n$ be $n$ observations. Formula The Karl Pearson’s coefficient skewness is given by $S_K =\dfrac{3(Mean-Median)}{sd}=\dfrac{3(\overline{x}-M)}{s_x}$ where, $\overline{x}$ is the sample mean, $M$ is the median, $s_x$ is the sample standard deviation. Sample mean The sample mean $\overline{x}$ is given by $$ \begin{eqnarray*} \overline{x}& =\frac{1}{n}\sum_{i=1}^{n}x_i \end{eqnarray*} $$ Sample Median Arrange the data in ascending order of magnitude. Median of $X$ is given by
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Kelly Coefficient of skewness for grouped data

  • STATISTICS
Kelly’s coefficient of skewness for grouped data For a symmetric distribution, the first decile namely $D_1$ and nineth decile $D_9$ are equidistance from the median i.e. $D_5$. Thus, $D_9 - D_5 = D_5 -D_1$. Kelly’s coefficient of skewness is based on deciles $D_{1}$, $1^{st}$ decile, $D_{5}$, $5^{th}$ decile, and $D_{9}$, $9^{th}$ decile). Only 20% of the observations are excluded from the measure. Formula $S_k =\dfrac{D_9+D_1 -2*Median}{D_9-D_1}$ OR $S_k =\dfrac{P_{90}+P_{10} -2*Median}{P_{90}-P_{10}}$
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Kelly Coefficient of skewness for ungrouped data

  • STATISTICS
Kelly’s Coefficient of Skewness for ungrouped data For a symmetric distribution, the first decile namely $D_1$ and nineth decile $D_9$ are equidistance from the median i.e. $D_5$. Thus, $D_9 - D_5 = D_5 -D_1$. The coefficient of skewness based on deciles is defined as $S_k = \dfrac{D_9+D_1 - 2D_5}{D_9 -D_1}$ OR $S_k =\dfrac{P_{90}+P_{10} -2*Median}{P_{90}-P_{10}}$ where, $D_1 (=P_{10})$ is the first decile (or tenth percentile) of the data $Median=D_5 (=P_{50})$ is the median of the data $D_9 (=P_{90})$ is the ninth decile (or nineteenth percentile) of the data.
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Mean Absolute Deviation for grouped data

  • STATISTICS
Mean absolute deviation 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 mean absolute deviation about mean is given by $MAD =\dfrac{1}{N}\sum_{i=1}^{n}f_i|x_i -\overline{x}|$ where, $N$ total number of observations $\overline{x}$ sample mean Example 1 A librarian keeps the records about the amount of time spent (in minutes) in a library by college students.
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Mean Absolute Deviation for ungrouped data

  • STATISTICS
Mean absolute deviation for ungrouped data Let $x_i, i=1,2, \cdots , n$ 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: $MAD =\dfrac{1}{n}\sum_{i=1}^{n}|x_i -\overline{x}|$ where, $n$ total number of observations $\overline{x}$ sample mean Example 1 The age (in years) of 6 randomly selected students from a class are : 22,25,24,23,24,20. Compute mean absolute deviation about mean. Solution $x_i$ $(x_i-xb)$ $|x_i-xb|$ 22 -1 1 25 2 2 24 1 1 23 0 0 24 1 1 20 -3 3 Total 138 8 The sample mean of $X$ is
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Mean median and mode for grouped data

  • STATISTICS
Mean, median and mode for grouped data Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution. Formula Sample mean The mean of $X$ is denoted by $\overline{x}$ and is given by $\overline{x} =\dfrac{1}{N}\sum_{i=1}^{n}f_ix_i$ In case of continuous frequency distribution, $x_i$’s are the mid-values of the respective classes. Sample median The median is given by $\text{Median } = l + \bigg(\dfrac{\frac{N}{2} - F_<}{f}\bigg)\times h$ where, $N$, total number of observations $l$, the lower limit of the median class $f$, frequency of the median class $F_<$, cumulative frequency of the pre median class $h$, the class width Sample mode The mode of the distribution is given by
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