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.

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

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