The diameters of circles (in mm) drawn in a d | vedclass

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(A) First,we make the class intervals continuous by adjusting the boundaries:						Class Interval			Frequency $(f_i)$			Mid-point $(x_i)$			$y_i = \fr

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$5.55$

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(A) First,we make the class intervals continuous by adjusting the boundaries:						Class Interval			Frequency $(f_i)$			Mid-point $(x_i)$			$y_i = \frac{x_i - 42.5}{4}$			$f_i y_i$			$f_i y_i^2$							$32.5-36.5$			$15$			$34.5$			$-2$			$-30$			$60$							$36.5-40.5$			$17$			$38.5$			$-1$			$-17$			$17$							$40.5-44.5$			$21$			$42.5$			$0$			$0$			$0$							$44.5-48.5$			$22$			$46.5$			$1$			$22$			$22$							$48.5-52.5$			$25$			$50.5$			$2$			$50$			$100$							Total			$N = 100$			-			-			$\sum f_i y_i = 25$			$\sum f_i y_i^2 = 199$			Mean $\bar{x} = A + \frac{\sum f_i y_i}{N} 	imes h = 42.5 + \frac{25}{100} 	imes 4 = 42.5 + 1 = 43.5 	ext{ mm}$.Variance $\sigma^2 = h^2 \left[ \frac{\sum f_i y_i^2}{N} - \left( \frac{\sum f_i y_i}{N} ight)^2 ight] = 4^2 \left[ \frac{199}{100} - \left( \frac{25}{100} ight)^2 ight] = 16 \left[ 1.99 - 0.0625 ight] = 16 	imes 1.9275 = 30.84$.Standard Deviation $\sigma = \sqrt{30.84} \approx 5.55 	ext{ mm}$.

Diameters	$33-36$	$37-40$	$41-44$	$45-48$	$49-52$
No. of circles	$15$	$17$	$21$	$22$	$25$

$x_i$	$5$	$7$	$9$	$11$
$f_i$	$3$	$2$	$1$	$2$

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