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Question -

The diameters of circles (in mm) drawn in a design are given below:

Diameters

33 – 36

37 – 40

41 – 44

45 – 48

49 – 52

No. of circles

15

17

21

22

25
Calculate the standard deviation and mean diameter of the circles.
[Hint first make the data continuous by making the classes as 32.5-36.5, 36.5-40.5, 40.5-44.5, 44.5 – 48.5, 48.5 – 52.5 and then proceed.]



Answer -

Let the assumed mean,A = 42.5 and h = 4

Let us make the tableof the given data and append other columns after calculations.

Height (class)

Number of children

(Frequency fi)

Midpoint

Xi

Yi = (xi – A)/h

Yi2

fiyi

fiyi2

32.5 – 36.5

15

34.5

-2

4

-30

60

36.5 – 40.5

17

38.5

-1

1

-17

17

40.5 – 44.5

21

42.5

0

0

0

0

44.5 – 48.5

22

46.5

1

1

22

22

48.5 – 52.5

25

50.5

2

4

50

100

N = 100

25

199

Mean,

Where, A = 42.5, h = 4

So, x̅ = 42.5 +(25/100) × 4

= 42.5 + 1

= 43.5

Then, Variance,

σ2 =(42/1002)[100(199) – 252]

= (1/625) [19900 –625]

= 19275/625

= 771/25

= 30.84

Hence, standarddeviation = σ = √30.84

= 5.553

∴ Mean = 43.5,variance = 30.84 and Standard Deviation = 5.553.

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