# Stastistics

Curriculum

Dispersion

• Quartile Deviation (Grouped data only)
• Mean Deviation (Grouped data only)
• Standard Deviation (Grouped data only)
• Variation (Grouped data only)

Quartile Deviation:
• The difference between the upper quartile and the lower quartile is called inter-quartile range. The semi-inter quartile range of the data is known as Quartile deviation( Q. D. )
• Here, $Q_3 - Q_1$ is known as inter-quartile range, $\frac{Q_3-Q_1}{2}$ is known as semi-inter quartile rangle and $\frac{Q_3-Q_1}{Q_3+Q_1}$ is known as the coefficient of quartile defiation.
• For continuous or grouped data
Position of $Q_1 = \left( \frac{N}{4} \right)^{th}$ term
Position of $Q_3 = \left( \frac{3N}{4} \right)^{th}$ item
• Value of $\displaystyle Q_1 = L+ \frac{\frac{N}{4}-cf}{f} \times h$
• Value of $\displaystyle Q_3 = L+ \frac{\frac{3N}{4}-cf}{f} \times h$
Where,
$L=$lower limit of quartile class
$c.f.=$ c.f of preceeding class
$f =$ frequency of quartile class
$h =$ class - interval
Mean Deviation:
• The average of the absolute values of the deviation of each item from mean, median or mode is known as a mean deviation. It is also known as average deviation. It is denoted by M.D.
Calculation of Mean Deviation
• M.D. from mean $\displaystyle \frac{ \Sigma f|x-\overline{x}|} {N}$
• M.D. from mean $\displaystyle \frac{ \Sigma f|x-median|} {N}$
Where $x$ indicates the mid-value of each class interval.
• Coefficient of M.D. from mean $\displaystyle \frac{M.D.}{Mean}$
• Coefficient of M.D. from mean $\displaystyle \frac{M.D.}{Median}$
Standard Deviation
• Standard deviation is the positive square root of the arithmetic mean of the squares of deviations of given data taken from mean. It is also known as " Root mean square deviation". It is denoted by Greek Letter $\sigma$ ( read as sigma ). It is considered as the best measure of dispersion.
Calculation of Standard Deviation
• Actual mean method:
Standard deviation$\displaystyle ( \sigma ) = \sqrt{\frac{\Sigma f(x-\overline{x})}{N}}$
Where $x$ is mid-value of each class-interval.
• Direct Method:
Standard deviation$\displaystyle ( \sigma ) = \sqrt{\frac{\Sigma fx^2}{N}-\left(\frac{\Sigma fx}{N} \right)^2}$
Where $x$ is mid-value of each class-interval.
• Assumed mean method ( or short - cut method )
Standard deviation$\displaystyle ( \sigma ) = \sqrt{\frac{\Sigma fd^2}{N}-\left(\frac{\Sigma fd}{N} \right)^2}$
Where $d=x-A, A= \text{ assumed mean }$, $x= \text{ mid-value of each class-interval}$
• Step deviation method:
When class-interval is very large then step deviation method is used to find the standard deviation.
Standard deviation$\displaystyle ( \sigma ) = \sqrt{\frac{\Sigma fd'^2}{N}-\left(\frac{\Sigma fd'}{N} \right)^2} \times h$
Where,
$\displaystyle d'=\frac{x-A}{h}$
$A= \text{ assumed mean }$
$x= \text{ mid-value of each class-interval}$
$h = class - height$
Formula to calculate the coefficient of standard deviation
• Coefficient of S.D. $= \frac{S.D.}{Mean} \text{ or, } \frac{\sigma}{Mean }$
• If the coefficient of S.D. is multiplied by 100, then it is known as coefficient of variation.
Thus, coefficient of variation (C.V.) $= \frac{\sigma}{ Mean } \times 100\%$
• The square of S.D. is called variance.
i.e. $\text{ Variance } = \sigma^2$
1. Find the Quartile Deviation and its coefficient from the following frequency table.

$\begin{bmatrix} CI&0\le x < 10 & 10\le x < 20& 20\le x <30 & 30 \le x <40 & 40 \le x <50 \\f & 5 & 2&9&2&2 \end{bmatrix}$

2. Find the mean deviation from median and its coefficient from the following frequency table.
$\begin{bmatrix} CI&10\le x < 20 & 20\le x < 30& 30\le x <40 & 40 \le x <50 & 50 \le x <60 \\f & 2 & 3&5&4&1 \end{bmatrix}$
3. Compute mean deviation mean and it's coefficient from  the following data.

$\begin{bmatrix}\text{Marks} & 0 -10& 0-20&0-30&0-40&0-50 \\ \text{No of students} &9&15&19&31&40 \end{bmatrix}$

4. Compute standard deviation and it's coefficient from the following data.

$\begin{bmatrix}\text{Marks} & 0 -50& 10-50&20-50&30-50&40-50 \\ \text{No of students} &15&14&10&5&2 \end{bmatrix}$

5. Compute coefficient of variation from the following data.

$\begin{bmatrix}\text{Mid Value } & 5 & 15 & 25 & 35 & 45 \\ \text{No of students} &4&6&10&7&2 \end{bmatrix}$