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Statistics

Long Answer Type

511. Find the mean and variance for the following frequency distribution:

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512. Find the mean and variance for the following distribution:

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513. Find the mean, variance and standard deviation using short-cut method.

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514. Calculate the mean, variance and standard deviation for the following distribution:

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515. Calculate the mean and S.D. for the following distribution:

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516. The following table gives the distribution of the income of 100 families in a village. Calculate the variance and standard deviation.

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517.

The runs scored by two batsmen in 10 innings are: A: 65, 58, 68, 44, 48, 45, 60, 62, 60, 50
B: 48, 80, 58, 44, 52.65, 73, 56, 64.50.
Which of the batsmen is more consistent and which one is more variable?

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518.

The marks obtained (out of 100) by two students in 10 qualifying tests were: A: 48, 53, 58, 41, 54, 52, 54, 49, 51, 50
B: 11, 98, 60, 94, 48, 52,17, 90, 20, 20
Who is more consistent in performance and who is more variable?

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519.
The mean and standard deviation of marks obtained by 50 students of a class in three subjects, Mathematics, Physics and Chemistry are given below:

Subject Mathematics Physics Chemistry
Mean 42 32 40.9

Standard
deviation 12 15 20
which of the three subjects shows the highest variability in marks and which shows the lowest?

The means are all different.
∴ we will compare co-efficients of variations.
Mathematics:
Mean, $top enclose straight X subscript 1 end enclose space equals space 42 comma space space space space space straight S. straight D. comma space space straight sigma subscript 1 space equals space 12$

 $left parenthesis straight C. straight V. right parenthesis subscript straight M space equals space fraction numerator straight sigma over denominator top enclose straight X subscript 1 end enclose end fraction cross times 100 % space equals space 12 over 42 cross times 100 % space equals space 200 over 7 %$

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Physics:
Mean, $space space top enclose straight X subscript 2 end enclose space equals space 32 comma space space straight S. straight D. comma space straight sigma subscript 2 space equals space 15$

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$rightwards double arrow$ $space space space space space left parenthesis straight C. straight V. right parenthesis subscript straight P space equals space 46.9 %$ ...(ii)

Chemistry
 Mean, $top enclose straight X subscript 3 end enclose space equals space 40.9 comma space straight S. straight D. comma space straight sigma subscript 3 equals 20$

$left parenthesis straight C. straight V. right parenthesis subscript straight C space equals space fraction numerator straight sigma subscript 3 over denominator top enclose straight X subscript 3 end enclose end fraction cross times 100 % space equals space fraction numerator 20 over denominator 40.9 end fraction cross times 100 % space equals space 20000 over 409 % space equals space 48.9 %$

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From (i), (ii) and (iii), we conclude that Chemistry with highest C.V. = 48.9% shows the highest variability and Mathematics with C.V. = 28.6% shows the lowest variability.

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520.

An analysis of monthly wages paid to workers in two firms A and B, belonging to the same industry, gives the following results:
Firm A Firm B
No. of wage earners 586 648
Mean of monthly wages Rs 5253 Rs 5253
Variance of the distribution of wages 100 121

(i) Which firm A or B pays larger amount as monthly wages?
(ii) Which firm A or B shows greater variability in individual wages?

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