Question

1. Using the table below, find the standard deviation and the variance of the scores of 30 students in Statistics exam.

Class Interval Class Frequency (f)

47 - 51 4

52 – 56 3

57 – 61 3

62 – 66 4

67 – 71 5

72 – 76 3

77 – 81 3

82 – 86 1

87 - 91 4

2. Determine the standard deviation and the variance of the scores obtained by the students in Mathematics using the table below.

Class Interval Class Frequency (f)

29 – 33 5

34 – 38 5

39 – 43 4

44 – 48 4

49 – 53 2

54 – 58 4

59 – 63 6

64 - 68 2

Answer 1

1.

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} & Classes & M_i & f_i & M_i\cdot f_i & M_i^2\cdot f_i \\ \hline & 47-51 & 49 & 4 & 196 & 9604 \\ \hdashline & 52-56 & 54 & 3 & 162 & 8748 \\ \hdashline & 57-61 & 49 & 3 & 177 & 10443 \\ \hdashline & 62-66 & 64 & 4 & 256 & 16384 \\ \hdashline & 67-71 & 69 & 5 & 345 & 23805 \\ \hdashline & 72-76 & 74 & 3 & 222 & 16428 \\ \hdashline & 77-81 & 79 & 3 & 237 & 18723 \\ \hdashline & 82-86 & 84 & 1 & 84 & 7056 \\ \hdashline & 87-91 & 89 & 4 & 356 & 31684 \\ \hdashline Sum= & & & 30 & 2035 & 142875 \\ \hdashline \end{array}$$

$$\sigma^2=\dfrac{1}{N}\bigg(\sum _iM_i^2\cdot f_i-\dfrac{1}{N}\big(\sum_iM_i\cdot f_i\big)^2\bigg)$$

$$=\dfrac{1}{30}(142875-\dfrac{1}{30}(2035)^2)$$

$$\approx161.1389$$

$$\sigma=\sqrt{\sigma^2}=\sqrt{161.38888889}\approx12.6940$$

2.

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} & Classes & M_i & f_i & M_i\cdot f_i & M_i^2\cdot f_i \\ \hline & 29-33 & 31 & 5 & 155 & 4805 \\ \hdashline & 34-38 & 36 & 5 & 180 & 6480 \\ \hdashline & 39-43 & 41 & 4 & 164 & 6724\\ \hdashline &44-48 & 46 & 4 & 184 & 8464 \\ \hdashline & 49-53 & 51 & 2 & 102 & 5202 \\ \hdashline & 54-58 & 56 & 4 & 224 & 16428 \\ \hdashline & 59-63 & 61 & 6 & 366 & 22326 \\ \hdashline & 64-68 & 66 & 2 & 132 & 8712 \\ \hdashline & 87-91 & 89 & 4 & 356 & 31684 \\ \hdashline Sum= & & & 32 & 1507 & 75257 \\ \hdashline \end{array}$$

$$\sigma^2=\dfrac{1}{N}\bigg(\sum _iM_i^2\cdot f_i-\dfrac{1}{N}\big(\sum_iM_i\cdot f_i\big)^2\bigg)$$

$$=\dfrac{1}{32}(75257-\dfrac{1}{32}(1507)^2)$$

$$=133.9599609375\approx133.9600$$

$$\sigma=\sqrt{\sigma^2}=\sqrt{133.9599609375}\approx11.5741$$