- The semi-interquartile range of the data is known as Quartile Deviation(Q.D.)
- Here ,inter-quartile range \(=Q_3-Q_1 \)
- Semi- interquartile range \(=\frac{Q_3-Q_1}{2} \)
- Coefficient of quartile deviation \(=\frac{Q_3-Q_1}{Q_3+Q_1} \)

For continuous or grouped data.

- \(\text{Position of }Q_1=\left(\frac{N}{4}\right)^{th}\text{ item} \)
- \( Q_1 =L+\frac{\frac{N}{4}-c.f.}{f} \times h \)
- \(\text{Position of }Q_3=\left(\frac{3N}{4}\right)^{th}\text{ item} \)
- \(Q_3=L+\frac{\frac{3N}{4}-c.f.}{f}\times h \)

Where,

\( L= \) lower limit of quartile class

\(c.f.= \) c.f. of the preceding class

\( f= \) frequency of quartile class

\(h= \) class -height or, class - size or class intervalalign

# 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 For continuous series- M.D. from mean \( = \frac{\Sigma f|x - \bar{x} |}{N} \)
- M.D. from median \( = \frac{\Sigma f|x - m_d |}{N} \)
- Coefficient of M.D. from mean \( = \frac{M.D.}{Mean} \)
- Coefficient of M.D. from median \( = \frac{M.D.}{Median} \)
Where, \begin{align} \text{ Mean }( \bar{x} ) &= \frac{\Sigma fx }{N} \\ \text{ Median }(m_d) & = L + \frac{\frac{N}{2} - c.f. }{f} \times h \\ x & = \text{ mid-value of class interval } \end{align}

**Standard deviation**is the positive square root of the arithmetic mean of the square of deviations of given data taken from mean. It is also known as

**"Root mean square deviation"**. It is denoted by Greek letter (read as sigma). It is considered as the best measure of dispersion because:

- It's value is based on all the observations.
- Deviation of each term is taken from the central value.
- All algebraic sign are also considered

**Calculation of Standard Deviation**

**Actual mean method:**Standard deviation(σ)\( = \sqrt{\frac{\Sigma f\left(x-\overline{x}\right)^2}{N}} \), where \( x \) is the mid-value of each class-interval.

**Direct method:**

**Standard deviation(σ) \( = \sqrt{\frac{\Sigma fx^2}{N}-\left(\frac{\Sigma fx}{N}\right)^2} \), where \( x \) is the mid-value of each class-interval.**

**Assumed mean method:**

**Standard deviation(σ)\( =\sqrt{\frac{\Sigma fd^2}{N}-\left(\frac{\Sigma fd}{N}\right)^2} \), where, \( d = x - A \)**

\( A = \) Assumed mean

\( x = \) mid-value of class interval.

**Step deviation method:**When the class-interval is very large then step deviation method is used to find the standard. Standard deviation(σ)\( =\sqrt{\frac{\Sigma fd'^2}{N}-\left(\frac{\Sigma fd'}{N}\right)^2}\times h \) Where\(d'=\frac{x-A}{h},\text{ }x=\text{mid- value of class -interval}\)

\(A=\text{Assumed mean}\)

\(h=\text{class - size}\)

**Coefficient of variation (C.V.)**The relative measure of standard deviation is known as the coefficient of standard deviation and is defined by
Coefficient of standard deviation \( = \frac{ \text{Standard Deviation} } {\text{mean} }=\frac{\sigma}{\overline{x}} \)
If the coefficient of standard deviation is multiplied by 100, then it is known as coefficient of variation. Coefficient of variation is denoted by C.V. and is calculated as:

\( C.V. = \frac{\sigma}{\overline{x} } \times 100\% \)
Greater the coefficient of variation, greater will be the variation and less will be the consistency or uniformity. Less the C.V., greater will be the consistency or uniformity. For the consistency or uniformity of distribution, we use the C.V. So, C.V. is used to compare given distributions.

**Variance:**The square of standard deviation(σ) is called variation. It is given by\( \text{Varaince} = (\sigma)^2 \)

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