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, $4 t h$ decile and $56 t h$ percentile of the following distribution of marks:

Marks	Number of students
$0 - 4$	$10$
$4 - 8$	$12$
$8 - 12$	$18$
$12 - 14$	$7$
$14 - 18$	$5$
$18 - 20$	$8$
$20 - 25$	$4$
$25$ and above	$6$

Accepted Answer

We have to find the median, lower and upper quartiles, $4 t h$ decile and $50 t h$ percentile of the following distribution of marks:

Marks Number of students Cumulative frequency

$0 - 4$ $10$ $10$

$4 - 8$ $12$ $22$

$8 - 12$ $18$ $40$

$12 - 14$ $7$ $18 - 20$

$14 - 18$ $20 - 25$ $52$

$18 - 20$ $8$ $60$

$20 - 25$ $4$ $64$

$25$ and above $6$ $70$

Here, $N$ is $70$ so $\frac{N}{2} = \frac{70}{2} = 35$

Median class $= (8 - 12)$

Median $= M = 8 +$ a fraction of the class width $4$

We assume that the frequency $8 - 12$ of the median class is uniformly distributed over the width i.e, $4$ of the class.

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

$l =$ lower limit of the median class $= 8$

$N =$ size of the sample or population $8$

$c =$ cumulative frequency of the class just before the median class $= 22$

$f =$ frequency of the median class $= 18$

$h =$ width of the median class $= 4$

$\Rightarrow M = 8 + \frac{\frac{70}{2} - 22}{18} \times 4 = 8 + \frac{35 - 22}{18} \times 4$

$\Rightarrow M = 8 + \frac{13}{9} \times 2 = 10.89$

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

First Quartile class is $(4 - 8)$

$Q_{1} = 4 + \frac{\frac{70}{4} - 10}{12} \times 4 = 4 + \frac{17.5 - 10}{12} \times 4 = 4 + 0.625 \times 4 = 6.5$

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

Third Quartile class is $(12 - 14)$ .

$Q_{3} = 14 + \frac{\frac{3}{4} \times 70 - 47}{5} \times 4 = 14 + \frac{52.5 - 47}{5} \times 4$

$Q_{3} = 14 + \frac{5.5}{5} \times 4 = 14 + 4.4 = 18.4$

In same way we have to find $4 t h$ decile or the $40^{t h}$ percentile of the data given by formula,

$D_{4} = P_{40} = l + \frac{\frac{40}{100} N - c}{f} \times h$

$\Rightarrow \frac{40}{100} \times 70 = 28$ , so class is $(8 - 12)$

$D_{4} = 8 + \frac{\frac{40}{100} \times 70 - 22}{18} \times 4 = 8 + \frac{28 - 22}{18} \times 4$

$= 8 + \frac{6}{18} \times 4 = 8 + \frac{4}{3} = 9.33$

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

$\Rightarrow \frac{56}{100} \times 70 = 39.2$ , so class is $(8 - 12)$

$P_{56} = 8 + \frac{39.2 - 22}{18} \times 4 = 8 + \frac{17.2}{18} \times 4$

$P_{56} = 8 + 0.96 \times 4 = 11.82$

Months	No. of T.V. sets sold
January	$14$
February	$17$
March	$16$
April	$12$
May	$7$
June	$6$
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 Teachers	Number of Higher Secondary Schools
$6 - 10$	$955$
$11 - 15$	$1067$
$16 - 20$	$1663$
$21 - 25$	$1492$
$26 - 30$	$1220$
$31 - 35$	$1129$
$36 - 40$	$745$
$41 - 45$	$637$
$46 - 50$	$442$

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

Marks below	Number of students
$10$	$15$
$20$	$35$
$30$	$60$
$40$	$84$
$50$	$96$
$60$	$127$
$70$	$198$
$80$	$250$

Find the median, lower and upper quartiles, $4 t h$ decile and $56 t h$ percentile of the following distribution of marks:

Marks Number of students

$0 - 4$ $10$

$4 - 8$ $12$

$8 - 12$ $18$

$12 - 14$ $7$

$14 - 18$ $5$

$18 - 20$ $8$

$20 - 25$ $4$

$25$ and above $6$

Important Questions on Statistics

Learn and Prepare for any exam you want !

Number of students absent	Number of days
$0$	$2$
$1$	$7$
$\Rightarrow \frac{40}{100} \times 70 = 28$	$6$
$3$	$8$
$4$	$30$
$5$	$25$
$6$	$7$
$7$	$12$
$8$	$= 8 + \frac{6}{18} \times 4 = 8 + \frac{4}{3} = 9.33$
$9$	$5$
	Total $= 110$

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

Find the median, lower and upper quartiles, 4th decile and 56th percentile of the following distribution of marks: Marks Number of students 0-4 10 4-8 12 8-12 18 12-14 7 14-18 5 18-20 8 20-25 4 25 and above 6

Important Questions on Statistics

Learn and Prepare for any exam you want !

Find the median, lower and upper quartiles, $4 t h$ decile and $56 t h$ percentile of the following distribution of marks:

Marks Number of students

$0 - 4$ $10$

$4 - 8$ $12$

$8 - 12$ $18$

$12 - 14$ $7$

$14 - 18$ $5$

$18 - 20$ $8$

$20 - 25$ $4$

$25$ and above $6$