NCERT Solutions for Class 11 Statistics Chapter 6 Measures Of Dispersion

Ans : The study of the averages is only one sided distribution story. In order to understand the frequency distribution fully, it is essential to study the variability of the observations. The average measures center of the data whereas the quantum of the variation is measured by the measures of dispersion like range, quartile deviation, mean deviation and Standard Deviation. For example, if a country has very high income group people and very low income group people, then we can say that the country has large Income disparity. 

Ans : Standard Deviation is the best measure of dispersion as it satisfies the most essentials of the good measure of dispersion. 
The following points make Standard Deviation the best measure of dispersion: 
1. Most of the statistical theory is based on Standard Deviation. It helps to make comparison between variability of two or  more sets of data. Also, Standard Deviation helps in testing the significance of random samples and in regression and  correlation analysis. 
2. It is based on the values of all the observations. In other words, Standard Deviation makes use of every item in a particular distribution. 
3. Standard Deviation has a precise value and is a well-defined and definite measure of dispersion. That is, it is rigidly defined.
4. It is independent of the origin. 
5. It is widely used measure of dispersion as all data distribution is nearer to the normal distribution. 
6. It enables algebraic treatment. It has correct mathematical processes in comparison to range, quartile deviation and mean deviation. 

Ans : Yes, is true that some measures of dispersion depend upon the spread of values, whereas some calculate the variation of values from the central value. The spread of values is determined by the absolute measures of dispersion like Range, Quartile Mean Deviation, and Standard Deviation. These measures express dispersion in terms of original unit of the series and it cannot be used for the comparison of statistical data having different units. On the other hand, the relative measures of the dispersion calculate the variability of the values from a central value. The relative measure includes coefficient of Range, Mean Deviation and Variation. It is used when the comparison has to be made between two statistical sets. These measures are free from any units. 

Ans : Absolute Value of Dispersion 
Range = L - S = 45000-18,000 = Rs27,OOO 
Relative Value of Dispersion
\(\begin{array}{c}{\text { Coefficient of Range }=\frac{L-S}{L+S}} \\ {=\frac{45000-18000}{45000+18000}} \\ {=0.428}\end{array}\) 

Ans : (i) (a) Wheat Highest value of distribution (H) = 25 
Lowest value of distribution (L) = 9
\(\begin{array}{l}{\text { Range }=\mathrm{H}-\mathrm{L}} \\ {=25-9} \\ {=16}\end{array}\)


(b) Highest value of distribution (H) = 34 
Lowest value of distribution (L) = 12 
\(\begin{array}{l}{\text { Range }=\mathrm{H}-\mathrm{L}} \\ {=34-12} \\ {=22}\end{array}\)


(ii) (a) Wheat 
Arranging the production of wheat in increasing order 
9, 10, 10, 12, 15, 16, 18, 19, 21, 25
\(\begin{array}{l}{Q_{1}=\frac{N+1}{4} \text { th item }} \\ {=\frac{10+1}{4} \text { th item }} \\ {=\frac{11}{4} \text { th item }} \\ {=2.75^{\text { th }} \text { item }}\end{array}\)
\(\begin{array}{l}{=\text { Size of } 2 \text { th item }+0.75\left(\text { size of } 3^{\text { rd }} \text { item }-\text { size of } 2^{\text { nd }} \text { item }\right)} \\ {=10+0.75(10-10)} \\ {=10+0.75 \times 0} \\ {=10}\end{array}\)
\(\begin{array}{l}{Q_{3}=\frac{3(N+1)}{4} \text { th item }} \\ {=\frac{3(10+1)}{4} \text { th item }} \\ {=\frac{33}{4} \text { th item }} \\ {=8.25 \mathrm{th}} \\ {=\text { Size of } 8^{\text { th }} \text { item }+0.25 \text { (size of } 9^{\text { th }} \text { item }-\text { size of } 8^{\text { th }} \text { item) }}\end{array}\)
\(\begin{array}{l}{=19+0.25(21-19)} \\ {=19+0.25 \times 2} \\ {=19+0.50=19.50}\end{array}\)
\(\begin{aligned} \text { Quartile Deviation } &=\frac{Q_{3}-Q_{1}}{2} \\ &=\frac{19.50-10}{2} \\ &=\frac{9.50}{2} \\ &=4.75 \end{aligned}\)


(b) Rice 
Arranging the data of production of rice 
12, 12, 12, 15, 18, 18, 22, 23, 29, 34


\(\begin{array}{l}{Q_{1}=\frac{N+1}{4} \text { th item }} \\ {=\left(\frac{10+1}{4}\right) \text { th item }} \\ {=2.75 \text { th item }} \\ {=\text { size of } 2^{\text { nd }} \text { item }+0.75 \text { (size of } 3^{\text { rd }} \text { item }-\text { size of } 2^{\text { nd }} \text { item) }}\end{array}\)
\(\begin{aligned} &=12+0.75(12-12) \\ &=12+0.75 \times 0 \\ &=12 \end{aligned}\)
\(\begin{array}{l}{Q_{3}=\frac{3(N+1)}{4} \text { th item }} \\ {=\frac{3(10+1)}{4} \text { th item }} \\ {=\frac{33}{4} \text { th item }}\end{array}\)
\(\begin{array}{l}{=8.25 \text { th item }} \\ {=\text { Size of } 8^{\text { th }} \text { item }+0.25 \text { (size of 9th item - size of gth item) }} \\ {=23+0.25(29-23)} \\ {=23+0.25 \times 6}\end{array}\)
\(\begin{aligned} &=23+1.5 \\ &=24.5 \end{aligned}\)
\(\begin{aligned} & \text { Quartile Deviation }=\frac{Q_{3}-Q_{1}}{2} \\ &=\frac{24.5-12}{2} \\ &=\frac{12.50}{2} \\ &=6.25 \end{aligned}\)


(iii) (a) Wheat

\(\begin{array}{l}{\text { Mean }=\frac{\sum x}{N}=\frac{155}{10}=15.5} \\ {\text { Mean Deviation from Mean }} \\ {M D(\overline{X})=\frac{\sum|d|+(\overline{x}-A \overline{x})}{n}} \\ {=\frac{43+(15.5-15)(5-5)}{10}}\end{array}\)
\(=\frac{43}{10}=4.3\)


(b) Rice



(iv) (a) Wheat



\(\begin{array}{l}{\text { Median }=\text { size of }\left(\frac{N+1}{2}\right) \text { th item }} \\ {=\text { size of }\left(\frac{10+1}{2}\right) \text { th item }} \\ {=\text { size of }(5.5) \text { th item }=\frac{\text { size of } 5^{\text { th }} \text { item }+\text { size of } 6^{\text { th }} \text { item }}{2}=\frac{15+16}{2}=15.5}\end{array}\)
\(\begin{array}{l}{\mathrm{MD}_{\mathrm{Na}_{\mathrm{Na} \operatorname{dan}}=}=\frac{\sum|d|+(\mathrm{Median}-\mathrm{A})\left(\sum f_{\mathrm{B}}-\sum f_{\mathrm{A}}\right)}{n}} \\ {=\frac{57+(18-18)(6-4)}{10}} \\ {=\frac{57}{10}=5.7}\end{array}\)


(b) Rice

\(\begin{array}{l}{\text { since } n \text { is even }} \\ {\text { Therefore, Median }=\text { size of }\left(\frac{N+1}{2}\right) \text { th item }} \\ {=\text { size of }\left(\frac{10+1}{2}\right) \text { th item }} \\ {\Rightarrow \text { size of }(5.5) \text { th item }=\frac{\text { size of } 5 \text { th item }+\text { size of } 6 \text { th item }}{2}=\frac{36}{2}=18}\end{array}\)
\(\begin{array}{l}{M D \text { Median }=\frac{\Sigma|d|+(\text { Median }-A)\left(\sum f_{B}-\Sigma f_{A}\right)}{n}} \\ {=\frac{57+(18-18)(6-4)}{10}} \\ {=\frac{57}{10}=5.7}\end{array}\)


(v) (a) Wheat

\(\begin{array}{l}{\sigma=\sqrt{\frac{\sum d^{2}}{n}-\left(\frac{\sum d}{n}\right)^{2}}} \\ {=\sqrt{\frac{257}{10}-\left(\frac{5}{10}\right)^{2}}} \\ {=\sqrt{\frac{257}{100}}{\frac{2570-25}{100}}} \\ {-5.04}\end{array}\)


(b) Rice

\(\begin{aligned} \sigma &=\sqrt{\frac{\sum d^{2}}{n}-\left(\frac{\sum d}{n}\right)^{2}} \\ &=\sqrt{\frac{535}{10}-\left(\frac{15}{10}\right)^{2}} \\ &=\sqrt{\frac{535}{10}-\frac{225}{100}} \\ &=\sqrt{51.25} \\ &=7.16 \end{aligned}\)


(vi) (a) Wheat
\(\begin{array}{l}{C V=\frac{\sigma}{\overline{X}} \times 100} \\ {=\frac{5.04}{15.5} \times 100} \\ {=32.51}\end{array}\)


(b) Rice
\(\begin{array}{l}{C V=\frac{\sigma}{\overline{X}} \times 100} \\ {=\frac{7 \cdot 16}{19.5} \times 100} \\ {=36.71}\end{array}\)


(vi) Rice crop has greater variation as the coefficient of variation is higher for rice as compared to that or wheat 


(vii) Rice crop has higher Range, Quartile Deviation, Mean Deviation amount mean, Mean Deviation about median Standard Deviation and Coefficient of Variation.