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Descriptive Statistics

Mean median and mode for ungrouped data

  • STATISTICS
Mean, median and mode for ungrouped data Let $x_i, i=1,2, \cdots , n$ be $n$ observations. Formula The mean of $X$ is denoted by $\overline{x}$ and is given by $\overline{x} =\dfrac{1}{n}\sum_{i=1}^{n}x_i$ Arrange the data in ascending order of magnitude. Median of $X$ is given by Median $$ \begin{equation*} M= \left\{ \begin{array}{ll} \text{value of }\big(\frac{n+1}{2}\big)^{th}\text{ observation}, & \hbox{if $n$ is odd;} \\ \text{average of }\big(\frac{n}{2}\big)^{th}\text{ and }\big(\frac{n}{2}+1\big)^{th} \text{ observation}, & \hbox{if $n$ is even.
2 min Read 1

Moment coefficient of kurtosis for grouped data

  • STATISTICS
Moment coefficient of kurtosis 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 moment coefficient of kurtosis $\beta_2$ is defined as $\beta_2=\dfrac{m_4}{m_2^2}$ The moment coefficient of kurtosis $\gamma_2$ is defined as $\gamma_2=\beta_2-3$ where $N$ total number of observations $\overline{x}$ sample mean $m_2 =\frac{1}{N}\sum_{i=1}^n f_i(x_i-\overline{x})^2$ is second central moment $m_4 =\frac{1}{N}\sum_{i=1}^n f_i(x_i-\overline{x})^4$ is fourth central moment 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 2

Moment coefficient of kurtosis for ungrouped data

  • STATISTICS
Moment coefficient of kurtosis 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 moment coefficient of kurtosis $\beta_2$ is defined as $\beta_2=\dfrac{m_4}{m_2^2}$ The moment coefficient of kurtosis $\gamma_2$ is defined as $\gamma_2=\beta_2-3$ where $n$ total number of observations $\overline{x}$ sample mean $m_2 =\dfrac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^2$ is second sample central moment $m_4 =\dfrac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^4$ is fourth sample central moment Example The hourly earning (in dollars) of sample of 7 workers are :
2 min Read 1

Moment coefficient of Skewness for grouped data

  • STATISTICS
Moment coefficient of skewness 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 moment coefficient of skewness $\beta_1$ is defined as $\beta_1=\dfrac{m_3^2}{m_2^3}$ The moment coefficient of skewness $\gamma_1$ is defined as $\gamma_1=\sqrt{\beta_1}=\dfrac{m_3}{m_2^{3/2}}$ where $n$ total number of observations $\overline{x}$ sample mean $m_2 =\frac{1}{N}\sum_{i=1}^n f_i(x_i-\overline{x})^2$ is second central moment $m_3 =\frac{1}{N}\sum_{i=1}^n f_i(x_i-\overline{x})^3$ is third central moment 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 2

Moment coefficient of skewness for ungrouped data

  • STATISTICS
Moment coefficient of skewness 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 moment coefficient of skewness $\beta_1$ is defined as $\beta_1=\dfrac{m_3^2}{m_2^3}$ The moment coefficient of skewness $\gamma_1$ is defined as $\gamma_1=\sqrt{\beta_1}=\dfrac{m_3}{m_2^{3/2}}$ where $n$ total number of observations $\overline{x}$ sample mean $m_2 =\dfrac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^2$ is second sample central moment $m_3 =\dfrac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^3$ is third sample central moment Example The hourly earning (in dollars) of sample of 7 workers are :
2 min Read 1

Octiles for grouped data

  • STATISTICS
Octiles for grouped data Octiles are the values of arranged data which divide whole data into eight equal parts. They are 7 in numbers namely $O_1,O_2, \cdots, O_7$. Here $O_1$ is first octile, $O_2$ is second octile, $O_3$ is third octile and so on. Formula For discrete frequency distribution, the formula for $i^{th}$ octile is $O_i =\bigg(\dfrac{i(N)}{8}\bigg)^{th}$ value, $i=1,2,\cdots, 7$ where, $N$ is total number of observations. For continuous frequency distribution, the formula for $i^{th}$ octile is
4 min Read 3

Octiles for ungrouped data

  • STATISTICS
Octiles for ungrouped data Octiles are the values of arranged data which divide whole data into eight equal parts. They are 7 in numbers namely $O_1,O_2, \cdots, O_7$. Here $O_1$ is first octile, $O_2$ is second octile, $O_3$ is third octile and so on. The formula for $i^{th}$ octile is $O_i =$ Value of $\bigg(\dfrac{i(n+1)}{8}\bigg)^{th}$ observation, $i=1,2,3,\cdots, 7$ where, $n$ is the total number of observations. Example Diastolic blood pressure (in mmHg) of a sample of 18 patients admitted to the hospitals are as follows:
2 min Read 1

Outliers for ungrouped data

  • STATISTICS
Outliers for ungrouped data An outlier is any observation that is $1.5(IQR)$ away from the first quartile ($Q_1$) or third quartile ($Q_3$). Formula $x$ is an outlier if $x$ is below $Q_1 -1.5\times IQR$ or above $Q_3+1.5\times IQR$, where, $Q_1$ is the first quartile, $Q_3$ is the third quartile, $IQR = Q_3-Q_1$ is an inter-quartile range. $Q_i =$ Value of $\bigg(\dfrac{i(n+1)}{4}\bigg)^{th}$ observation, $i=1,2,3$ where $n$ is the total number of observations.
3 min Read 2

Percentiles for grouped data

  • STATISTICS
Percentiles for grouped data Percentiles are the values which divide whole distriution into hundred equal parts. They are 99 in numbers namely $P_1, P_2, \cdots, P_{99}$. Here $P_1$ is first percentile, $P_2$ is second percentile and so on. Formula For discrete frequency distribution, the formula for $i^{th}$ percentile is $P_i =\bigg(\dfrac{i(N)}{100}\bigg)^{th}$ value, $i=1,2,\cdots, 99$ where, $N$ is total number of observations. For continuous frequency distribution, the formula for $i^{th}$ percentile is
4 min Read 3

Percentiles for ungrouped data

  • STATISTICS
Percentiles for ungrouped data Percentiles are the values of arranged data which divide whole data into hundred equal parts. They are 9 in numbers namely $P_1,P_2, \cdots, P_{99}$. Here $P_1$ is first percentile, $P_2$ is second percentile, $P_3$ is third percentile and so on. Formula The formula for $i^{th}$ percentile is $P_i =$ Value of $\bigg(\dfrac{i(n+1)}{100}\bigg)^{th}$ observation, $i=1,2,3,\cdots,99$ where $n$ is the total number of observations. Example 1 The test score of a sample of 20 students in a class are as follows:
4 min Read 3

Quartile Deviation for ungrouped data

  • STATISTICS
Quartile Deviation for ungrouped data Quartile Deviation is given by $QD = \dfrac{Q_3-Q_1}{2}$ Coefficient of Quartile Deviation is given by Coefficient of $QD = \dfrac{Q_3-Q_1}{Q_3+Q_1}$ 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.
3 min Read 2

Quartiles for grouped data

  • STATISTICS
Quartile for grouped data Quartiles are the values which divide whole distriution into four equal parts. They are 3 in numbers namely $Q_1$, $Q_2$ and $Q_3$. Here $Q_1$ is first quartile, $Q_2$ is second quartile and $Q_3$ is third quartile. Quartile formula for grouped data For discrete frequency distribution, the formula for $i^{th}$ quartile is $Q_i =\bigg(\dfrac{i(N)}{4}\bigg)^{th}$ value, $i=1,2,3$ where, $N$ is total number of observations. For continuous frequency distribution, the formula for $i^{th}$ quartile is
5 min Read 4