Question 1: Find the standard deviation for the following distribution:

$\displaystyle \begin{array} {|c | c| c| c| c | c | c | c |} \hline x: & 4.5 & 14.5 & 24.5 & 34.5 & 44.5 & 54.5 & 64.5 \\ \hline f: & 1 & 5 & 12 & 22 & 17 & 9 & 4 \\ \hline \end{array}$

Median value of $x$ is 34.5

Calculation of Variance and Standard Deviation

$\displaystyle \begin{array} {|c | c| c| c| c | c | c | } \hline x_i & f_i & d_i & u_i & f_iu_i & {u_i}^2 & f_i {u_i}^2 \\ & & = x_i - 34.5 & = \frac{x_i - 34.5}{10} & & & \\ \hline 4.5 & 1 & -30 & -3 & -3 & 9 & 9 \\ \hline 14.5 & 5 & -20 & -2 & -10 & 4 & 20 \\ \hline 24.5 & 12 & -10 & -1& -12 & 1 & 12 \\ \hline 34.5 & 22 & 0 & 0 & 0 & 0 & 0 \\ \hline 44.5 & 17 & 10 & 1 & 17 & 1 & 17 \\ \hline 54.5 & 9 & 20 & 2 & 18 & 4 & 36 \\ \hline 64.5 & 4 & 30 & 3 & 12 & 9 & 36 \\ \hline & N = \sum f_i = 70 & & & \sum f_i u_i = 22 & & \sum f_i {u_i}^2 = 130 \\ \hline \end{array}$

$\displaystyle \text{Here } N = 70, \hspace{1.0cm} \sum f_iu_i = 22, \hspace{1.0cm} \sum f_i {u_i}^2 = 130, \hspace{1.0cm} h = 10$

$\displaystyle Var(X) = h^2 \Bigg[ \Bigg( \frac{1}{N} \sum f_i {u_i}^2 \Bigg) - \Bigg( \frac{1}{N} \sum f_iu_i \Bigg)^2 \Bigg]$

$\displaystyle \Rightarrow Var(X) = 10^2 \Bigg[ \Bigg( \frac{1}{70} \times 130 \Bigg) - \Bigg( \frac{1}{70} \times 22 \Bigg)^2 \Bigg] \\ \\ \\ { \hspace{2.0cm} = 100 \Bigg[ \frac{13}{7} - \frac{121}{1225} \Bigg] = 100( 1.857 - 0.098) = 175.822}$

$\displaystyle \text{Hence } S.D. = \sqrt{Var(X)} = \sqrt{175.822} = 13.259$

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Question 2: Table below shows the frequency $f$ with which $x$ alpha particles were radiated from a diskette.

$\displaystyle \begin{array} {|c | c| c| c| c | c | c | c |c | c| c| c| c | c | } \hline x: & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline f: & 51 & 203 & 383 & 525 & 532 & 408 & 273 & 139 & 43 & 27 & 10 & 4 & 2 \\ \hline \end{array}$

Calculate the mean and variance.

$\displaystyle \text{Mean } \overline{X} = \frac{\sum f_ix_i}{\sum f_i} = \frac{10078}{2600} = 3.88$

Calculation of Variance and Standard Deviation

$\displaystyle \begin{array} {|c | c| c| c| c | c | c| } \hline x_i & f_i & f_i x_i & d_i = x_i - \overline{X} & {d_i}^2 & f_id_i & f_i {d_i}^2 \\ \hline 0 & 51 & 0 & -3.88 & 15.05 & -197.88 & 767.55 \\ \hline 1 & 203 & 203 & -2.88 & 8.29 & -584.64 & 1682.87 \\ \hline 2 & 383 & 766 &-1.88 & 3.53 & -720.04 & 1351.99 \\ \hline 3 & 525 & 1575 & -0.88 & 0.77 & -462 & 404.25 \\ \hline 4 & 532 & 2128 & 0.12 & 0.014 & 63.84 & 7.448 \\ \hline 5 & 408 & 2040 & 1.12 & 1.25 & 456.96 & 510 \\ \hline 6 & 273 & 1638 & 2.12 & 4.49 & 578.76 & 1225.77 \\ \hline 7 & 139 & 973 & 3.12 & 9.73 & 433.68 & 1352.47 \\ \hline 8 & 43 & 344 & 4.12 & 16.97 & 177.16 & 729.71 \\ \hline 9 & 27 & 243 & 5.12 & 26.21 & 138.24 & 707.67 \\ \hline 10 & 10 & 100 & 6.12 & 37.45 & 61.2 & 374.5 \\ \hline 11 & 4 & 44 & 7.12 & 50.69 & 28.48 & 202.76 \\ \hline 12 & 2 & 24 & 8.12 & 65.93 & 16.24 & 131.86 \\ \hline & N = \sum f_i & \sum f_ix_i & & & \sum f_i d_i & \sum f_i {d_i}^2 \\ & = 2600 & = 10078 & & & = -10 & = 9448.848 \\ \hline \end{array}$

$\displaystyle \text{Here } N = 2600, \hspace{1.0cm} \sum f_id_i = -10, \hspace{1.0cm} \sum f_i {d_i}^2 = 9448.848$

$\displaystyle Var(X) = \Bigg( \frac{1}{N} \sum f_i {d_i}^2 \Bigg) - \Bigg( \frac{1}{N} \sum f_id_i \Bigg)^2$

$\displaystyle \Rightarrow Var(X) = \frac{9448.848}{2600} - \Bigg( \frac{-10}{2600} \Bigg)^2 = 3.6341 - 0.000015 = 3.63415$

$\displaystyle \text{Hence } S.D. = \sqrt{Var(X)} = \sqrt{3.63415} = 1.9063$

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Question 3: Find the mean, and standard deviation for the following data:

$\displaystyle \text{(i) } \begin{array} {|l | c| c| c| c | c | c| } \hline \text{Year render}: & 10 & 20 & 30 & 40 & 50 & 60 \\ \hline \text{Number of persons ( cumulative)}: & 15 & 32 & 51 & 78 & 97 & 109 \\ \hline \end{array}$

$\displaystyle \text{(ii) } \begin{array} {|c | c| c| c| c | c | c | c |c | c| c| c| c | c | c| c| } \hline \text{Marks} : & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ \hline \text{Frequency}: & 1 & 6 & 6 & 8 & 8 & 2 & 2 & 3 & 0 & 2 & 1 & 0 & 0 & 0 & 1 \\ \hline \end{array}$

(i)

Computation of mean and standard deviation

$\displaystyle \begin{array} {|l | c| c| c| c | c | } \hline x_i & \text{Cumulative Frequency} & f_i & x_i f_i & {x_i}^2 f_i \\ \hline 10 & 15 & 15 & 150 & 1500 \\ \hline 20 & 32 & 17 & 340 & 6800 \\ \hline 30 & 51 & 19 & 570 & 17100 \\ \hline 40 & 78 & 27 & 1080 & 43200 \\ \hline 50 & 97 & 19 & 950 & 47500 \\ \hline 60 & 109 & 12 & 720 & 43200 \\ \hline & & \sum f_i = 109 & \sum f_i x_i = 3810 & \sum f_i {x_i}^2 = 159300 \\ \hline \end{array}$

$\displaystyle \text{Mean } = \frac{\sum f_i x_i }{N} = \frac{3810}{109} = 34.95$

$\displaystyle Var(X) = \Bigg( \frac{1}{N} \sum f_i {x_i}^2 \Bigg) - \Bigg( \frac{1}{N} \sum f_ix_i \Bigg)^2 \\ \\ \\ { \hspace{2.0cm} = \frac{159300}{109} - \Bigg( \frac{3810}{109} \Bigg)^2 = 1461.46789 - 1221.7910 = 239.67}$

$\displaystyle \text{Hence } S.D. = \sqrt{Var(X)} = \sqrt{239.67} = 15.4815$

$\\$

(ii)

Computation of mean and standard deviation

$\displaystyle \begin{array} {|l | c| c| c| c | c | } \hline x_i & f_i & x_i f_i & {x_i}^2 f_i \\ \hline 2 & 1 & 2 & 4 \\ \hline 3 & 6 & 18 & 54 \\ \hline 4 & 6 & 24 & 96 \\ \hline 5 & 8 & 40 & 200 \\ \hline 6 & 8 & 48 & 288 \\ \hline 7 & 2 & 14 & 98 \\ \hline 8 & 2 & 16 & 128 \\ \hline 9 & 3 & 27 & 243 \\ \hline 10 & 0 & 0 & 0 \\ \hline 11 & 2 & 22 & 242 \\ \hline 12 & 1 & 12 & 144 \\ \hline 13 & 0 & 0 & 0 \\ \hline 14 & 0 & 0 & 0 \\ \hline 15 & 0 & 0 & 0 \\ \hline 16 & 1 & 16 & 256 \\ \hline & N = \sum f_i = 40 & \sum f_i x_i = 239 & \sum f_i {x_i}^2 = 1753 \\ \hline \end{array}$

$\displaystyle \text{Mean } = \frac{\sum f_i x_i }{N} = \frac{239}{40} = 5.975$

$\displaystyle Var(X) = \Bigg( \frac{1}{N} \sum f_i {x_i}^2 \Bigg) - \Bigg( \frac{1}{N} \sum f_ix_i \Bigg)^2 \\ \\ \\ { \hspace{2.0cm} = \frac{17530}{409} - \Bigg( \frac{239}{40} \Bigg)^2 = 43.825 - 5.975 = 8.12 }$

$\displaystyle \text{Hence } S.D. = \sqrt{Var(X)} = \sqrt{8.12} = 2.85$

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Question 4: Find the standard deviation for the following data:

$\displaystyle \text{(i) } \begin{array} {|c | c| c| c| c | c | } \hline x: & 3 & 8 & 13 & 18 & 23 \\ \hline f: & 7 & 10 & 15 & 10 & 6 \\ \hline \end{array}$

$\displaystyle \text{(ii) } \begin{array} {|c | c| c| c| c | c | c| } \hline x: & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline f: & 4 & 9 & 16 & 14 & 11 & 6 \\ \hline \end{array}$

(i)

$\displaystyle \text{Mean } \overline{X} = \frac{\sum f_ix_i}{\sum f_i} = \frac{614}{48} = 12.79$

Calculation of Variance and Standard Deviation

$\displaystyle \begin{array} {|c | c| c| c| c | c | c| } \hline x_i & f_i & f_i x_i & d_i = x_i - \overline{X} & {d_i}^2 & f_id_i & f_i {d_i}^2 \\ \hline 3 & 7 & 21 & -9.79 & 95.84 & - 68.53 & 670.88 \\ \hline 8 & 10 & 80 & -4.79 & 22.94 & -47.9 & 229.4 \\ \hline 13 & 15 &195 & 0.21 & 0.04 & 0.6 & 0.6 \\ \hline 18 & 10 & 180 & 5.21 & 27.14 & 52.1 & 271.4 \\ \hline 23 & 6 & 138 & 10.21 & 104.24 & 61.26 & 625.44 \\ \hline & N = \sum f_i & \sum f_ix_i & & & \sum f_i d_i & \sum f_i {d_i}^2 \\ & = 48 & = 614 & & & = -2.47 & = 2797.32 \\ \hline \end{array}$

$\displaystyle \text{Here } N = 48, \hspace{1.0cm} \sum f_id_i = -2.47, \hspace{1.0cm} \sum f_i {d_i}^2 = 1797.32$

$\displaystyle Var(X) = \Bigg( \frac{1}{N} \sum f_i {d_i}^2 \Bigg) - \Bigg( \frac{1}{N} \sum f_id_i \Bigg)^2$

$\displaystyle \Rightarrow Var(X) = \frac{1797.32}{48} - \Bigg( \frac{-2.47}{48} \Bigg)^2 =37.44417 - 0.00265 = 37.4415 = 37.44$

$\displaystyle \text{Hence } S.D. = \sqrt{Var(X)} = \sqrt{37.4415} = 6.1189 = 6.12$

(ii)

$\displaystyle \text{Mean } \overline{X} = \frac{\sum f_ix_i}{\sum f_i} = \frac{277}{60} = 4.62$

Calculation of Variance and Standard Deviation

$\displaystyle \begin{array} {|c | c| c| c| c | c | c| } \hline x_i & f_i & f_i x_i & d_i = x_i - \overline{X} & {d_i}^2 & f_id_i & f_i {d_i}^2 \\ \hline 2 & 4 & 8 & -2.62 & 6.86 & -10.48 & 27.44 \\ \hline 3 & 9 & 27 & -1.62 & 2.62 & -14.58 & 23.58 \\ \hline 4 & 16 & 64 & -0.62 & 0.38 & -9.92 & 6.08 \\ \hline 5 & 14 & 70 & 0.38 & 0.14 & 5.32 & 1.96 \\ \hline 6 & 11 & 66 & 1.38 & 1.90 & 15.18 & 20.90 \\ \hline 7 & 6 & 42 & 2.38 & 5.66 & 14.28 & 33.96 \\ \hline & N = \sum f_i & \sum f_ix_i & & & \sum f_i d_i & \sum f_i {d_i}^2 \\ & = 60 & = 277 & & & = -0.2 & = 113.92 \\ \hline \end{array}$

$\displaystyle \text{Here } N = 60, \hspace{1.0cm} \sum f_id_i = -0.2, \hspace{1.0cm} \sum f_i {d_i}^2 = 113.92$

$\displaystyle Var(X) = \Bigg( \frac{1}{N} \sum f_i {d_i}^2 \Bigg) - \Bigg( \frac{1}{N} \sum f_id_i \Bigg)^2$

$\displaystyle \Rightarrow Var(X) = \frac{113.92}{60} - \Bigg( \frac{-0.2}{60} \Bigg)^2 =1.8987 - 0.0000011 = 1.8987$

$\displaystyle \text{Hence } S.D. = \sqrt{Var(X)} = \sqrt{1.8987} = 1.3779 = 1.38$