Name: Arithmetic Mean of Grouped and Ungrouped Data
Uploaded: 2019-07-19
Duration: 8 min 59 s
Description: The arithmetic mean is used to measure the central tendency of data. Watch this video and learn to find the arithmetic mean of grouped and ungrouped data wit...

Question

Find the median, lower and upper quartiles, $8 t h$ decile, $56 t h$ percentile and the mode of the following distribution of $245$ workers.

Daily savings (in ₹)	Number of workers
$1 - 2.99$	$6$
$3 - 4.99$	$53$
$5 - 6.99$	$85$
$7 - 8.99$	$56$
$9 - 10.99$	$21$
$11 - 12.99$	$16$
$13 - 14.99$	$4$
$15 - 16.99$	$4$

Accepted Answer

Daily savings (in ₹) Class interval Number of workers Cumulative frequency

$1 - 2.99$ $1 - 2.995$ $6$ $6$

$3 - 4.99$ $2.995 - 4.995$ $53$ $59$

$5 - 6.99$ $4.995 - 6.995$ $21$ $144$

$7 - 8.99$ $6.995 - 8.995$ $56$ $200$

$9 - 10.99$ $8.995 - 10.995$ $21$ $221$

$11 - 12.99$ $10.995 - 12.995$ $3 - 4.99$ $237$

$53$ $12.995 - 14.995$ $4$ $241$

$15 - 16.99$ $14.995 - 16.995$ $4$ $245$

$N = 245$ , $\frac{N}{2} = \frac{245}{2} = 122.5$

Thus, cumulative frequency near to $\frac{N}{2}$ is $144$ , so the median class is $(4.995 - 6.995)$ .

We use the formula, $M = l + \frac{\frac{N}{2} - c}{f} \times h$ where,

$l =$ lower limit of the median class

$N =$ size of the sample or population

$c =$ cumulative frequency of the class just before the median class

$f =$ frequency of the median class

$h =$ width of the median class

$l = 4.995, N = 245, c = 59, f = 85, h = 2$

$\Rightarrow M = 4.995 + \frac{\frac{245}{2} - 59}{85} \times 2 = 4.995 + \frac{122.5 - 59}{85} \times 2 = 4.995 + \frac{63.5}{85} \times 2$

$= 4.995 + 0.747 \times 2 = 4.995 + 1.494 = 6.49$

Hence, the median is $6.49$ .

The first or lower Quartile $Q_{1} = l + \frac{\frac{N}{4} - c}{f} \times h$

$\Rightarrow \frac{N}{4} = \frac{245}{4} = 61.25$ , so first Quartile class is $(2.995 - 4.995)$ .

$l = 2.995, N = 245, c = 6, f = 53, h = 2$

$Q_{1} = 2.995 + \frac{\frac{245}{4} - 6}{53} \times 2 = 2.995 + \frac{61.25 - 6}{53} \times 2$

$= 2.995 + \frac{55.25}{53} \times 2 = 2.995 + 2.085 = 5.08$

Third or upper Quartile $Q_{3} = l + \frac{\frac{3}{4} N - c}{f} \times h$

$\Rightarrow \frac{3}{4} N = \frac{3 \times 245}{4} = 183.75$

Third Quartile class is $(6.995 - 8.995)$ .

$Q_{3} = 6.995 + \frac{\frac{3}{4} \times 245 - 144}{56} \times 2 = 6.995 + \frac{183.75 - 144}{56} \times 2$

$Q_{3} = 6.995 + \frac{39.75}{56} \times 2 = 6.995 + 1.4196 = 8.41$

In same way we have to find $8 t h$ decile or the $80^{t h}$ percentile of the data given by formula,

$D_{8} = P_{80} = l + \frac{\frac{80}{100} N - c}{f} \times h$

$\Rightarrow \frac{80}{100} \times 245 = 196$ , so class is $(6.995 - 8.995)$

$D_{8} = 6.995 + \frac{\frac{80}{100} \times 245 - 144}{56} \times 2 = 6.995 + \frac{196 - 144}{56} \times 2$

$= 6.995 + \frac{52}{56} \times 2 = 6.995 + 1.9259 = 8.92$

$P_{56} = l + \frac{\frac{56}{100} N - c}{f} \times h$

$\Rightarrow \frac{56}{100} \times 245 = 137.2$ , so class is $(4.995 - 6.995)$

$P_{56} = 4.995 + \frac{137.2 - 59}{85} \times 2 = 4.995 + \frac{78.2}{85} \times 2$

$P_{56} = 4.995 + \frac{78.2}{85} \times 2 = 4.995 + 1.84 = 6.84$

From the table we can state that $(4.995 - 6.995)$ is the modal class as it has the highest frequency.

For finding the mode, we use the formula

Mode $= l + \frac{f_{m} - f_{1}}{2 f_{m} - f_{1} - f_{2}} \times h$

$\Rightarrow 4.995 + \frac{85 - 53}{2 \times 85 - 53 - 56} \times 2 = 4.995 + \frac{32}{61} \times 2$

$= 4.995 + 1.0491 = 6.04$

Hence, mode is $6.04$ .

Months	No. of T.V. sets sold
January	$14$
February	$17$
March	$16$
April	$12$
May	$7$
June	$(2.995 - 4.995)$
July	$8$
August	$9$
September	$6$
October	$14$
November	$15$
December	$18$

Weight of children in $k g$	Number of children
$2.5$	$12$
$3.0$	$175$
$3.5$	$156$
$3.8$	$42$
$4.0$	$15$
$4.2$	$5$

Number of heads per toss $(x)$	Number of tosses $(f)$
$0$	$38$
$1$	$144$
$2$	$342$
$3$	$287$
$4$	$164$
$5$	$25$

Number of students absent	Number of days
$0$	$2$
$1$	$7$
$2$	$6$
$D_{8} = 6.995 + \frac{\frac{80}{100} \times 245 - 144}{56} \times 2 = 6.995 + \frac{196 - 144}{56} \times 2$	$8$
$4$	$30$
$5$	$25$
$6$	$7$
$7$	$12$
$8$	$8$
$9$	$5$
	Total $= 110$

Number of Teachers	Number of Higher Secondary Schools
$6 - 10$	$955$
$11 - 15$	$1067$
$16 - 20$	$1663$
$21 - 25$	$1492$
$26 - 30$	$1220$
$31 - 35$	$1129$
$36 - 40$	$745$
$41 - 45$	$637$
$46 - 50$	$442$

Important Questions on Statistics

Learn and Prepare for any exam you want !

Expenditure on water (in ₹)	Number of families
$30 - 40$	$12$
$40 - 50$	$18$
$50 - 60$	$20$
$60 - 70$	$15$
$70 - 80$	$12$
$80 - 90$	$11$
$90 - 100$	$6$
$100 - 110$	$4$
$110 - 120$	$2$

Find the median, lower and upper quartiles, 8th decile, 56th percentile and the mode of the following distribution of 245 workers. Daily savings (in ₹) Number of workers 1-2.99 6 3-4.99 53 5-6.99 85 7-8.99 56 9-10.99 21 11-12.99 16 13-14.99 4 15-16.99 4

Important Questions on Statistics

Learn and Prepare for any exam you want !