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Descriptive Statistics

Quartiles for ungrouped data

  • STATISTICS
Quartiles for ungrouped data Quartiles are the values of arranged data which divide whole data 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. 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.
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Summary Statistics for grouped data

  • STATISTICS
Summary statistic for grouped data Summary statistic summarize and provide information about the sample data. It includes the minimum value of the data, first quartile ($Q_1$), median (i.e., $Q_2$), mean ($\overline{x}$), third quartile ($Q_3$) and the minimum value of the data. Summary statistic includes minimum value ($\min$), first quartile ($Q_1$), $\text{median }$ ($Q_2$), sample mean ($\overline{x}$), third quartile ($Q_3$), maximum value ($\max$). Formula $\min$, $Q_1$, $\text{median}$, $\overline{x}$, $Q_3$ and $\max$
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Summary Statistics for ungrouped data

  • STATISTICS
Summary statistic for ungrouped data Summary statistic summarize and provide information about the sample data. It includes the minimum value of the data, first quartile ($Q_1$), median (i.e., $Q_2$), mean ($\overline{x}$), third quartile ($Q_3$) and the minimum value of the data. Summary statistic includes minimum value ($\min$), first quartile ($Q_1$), $\text{median }$ ($Q_2$), sample mean ($\overline{x}$), third quartile ($Q_3$), maximum value ($\max$). Formula $\min$, $Q_1$, $\text{median}$, $\overline{x}$, $Q_3$ and $\max$
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Variance and standard deviation for grouped data

  • STATISTICS
Variance and standard deviation for grouped data Let $(x_i,f_i), i=1,2, \cdots , n$ be the observed frequency distribution. Formula Sample variance The sample variance of $X$ is denoted by $s_x^2$ and is given by $s_x^2 =\dfrac{1}{N-1}\sum_{i=1}^{n}f_i(x_i -\overline{x})^2$ OR $s_x^2 =\dfrac{1}{N-1}\bigg(\sum_{i=1}^{n}f_ix_i^2-\frac{\big(\sum_{i=1}^n f_ix_i\big)^2}{N}\bigg)$ where, $N=\sum_{i=1}^n f_i$ is the total number of observations, $\overline{x}$ is the sample mean. Sample standard deviation The sample standard deviation of $X$ is defined as the positive square root of sample variance.
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Variance and standard deviation for ungrouped data

  • STATISTICS
Variance and standard deviation for ungrouped data Let $x_i, i=1,2, \cdots , n$ be $n$ observations. Formula Sample variance of $X$ is denoted by $s_{x}^2$ and is given by $s_x^2 =\dfrac{1}{n-1}\sum_{i=1}^{n}(x_i -\overline{x})^2=\dfrac{1}{n-1}\bigg(\sum_{i=1}^{n}x_i^2-\dfrac{\big(\sum_{i=1}^n x_i\big)^2}{n}\bigg)$ where, $\overline{x}=\dfrac{1}{n}\sum_{i=1}^{n}x_i$ is the sample mean The sample standard deviation of $X$ is defined as the positive square root of the sample variance. The sample standard deviation of $X$ is given by $s_x =\sqrt{s_x^2}$ Example 1 The age (in years) of 6 randomly selected students from a class are
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Geometric mean for grouped data

  • STATISTICS
Geometric mean for grouped data Let $(x_i,f_i), i=1,2, \cdots , n$ be the given frequency distribution then the geometric mean of $X$ is denoted by $GM$. Formula The geometric mean of grouped data is given by $GM=\bigg(\prod_{i=1}^n x_i^{f_i}\bigg)^{1/N}$ OR $\log (GM) =\frac{1}{N}\sum_{i=1}^n f_i\log(x_i)$ where, $N=\sum_i f_i$ total number of observations Example Compute geometric mean for the following frequency distribution. x 5-8 9-12 13-16 17-20 21-24 f 2 13 21 14 5 Solution Class Interval Class Boundries mid-value ($x_i$) Freq ($f_i$) $log(x_i)$ $f_i*log(x_i)$ 5-8 4.
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Quartile Deviation for grouped data

  • STATISTICS
Quartile Deviation for grouped data Quartile deviation is half the difference between the thrid quartile $Q_3$ and the first quartile $Q_1$ of a frequency distribution. It is also known as Semi interquartile range. Quartile Deviation is given by $QD = \dfrac{Q_3-Q_1}{2}$ where, $Q_1$ is the first quartile $Q_3$ is the third quartile Quartile Deviation Formula for Grouped Data The formula for $i^{th}$ quartile is $$ \begin{aligned} Q_i=l + \bigg(\frac{\frac{iN}{4} - F_<}{f}\bigg)\times h; \quad i=1,2,3 \end{aligned} $$
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