The diameters of circles (in mm) drawn in a ...

NCERT Class XI Mathematics - Statistics - Solutions

Question : 22 of 34

Marks: +1, -0

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.]

Solution:

Diameters	Modified Classes	Mid values $x_i$	$f_i$	$u_i$ = $\frac{x_i-42.5}{4}$	$f_i u_i$	$f_i u_i^2$
33 - 36	32.5 - 36.5	34.5	15	- 2	- 30	60
37 - 40	36.5 - 40.5	38.5	17	- 1	- 17	17
41 - 44	40.5 - 44.5	42.5	21	0	0	0
45 - 48	44.5 - 48.5	46.5	22	1	22	22
49 - 52	48.5 - 52.5	50.5	25	2	50	100
			100		25	199

Let assumed mean (A) = 42.5

Mean

\left(\overset{-}{x}\right)

= A +

\frac{\Sigma f_i u_i}{N}

× h = 42.5 +

\frac{25}{100}

× 4 = 42.5 + 1 = 43.5

Standard deviation (σ) =

\frac{h}{N} \sqrt{N \Sigma f_i u_i^2 - (\Sigma f_i u_i)^2}

\frac{4}{100} \sqrt{100 \times 199 - (25)^2}

\frac{1}{25} \sqrt{19275}

\frac{1}{25}

× 138.83 = 5.55

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