Answer to Question #287549 in Statistics and Probability for Ace

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.

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

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

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

$$=133.9599609375\approx133.9600$$

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