Question

a) Determine the missing frequencies of the following distribution given that the median

is 33.5 and themode is 34.0.

Class limits 0-9 10-19 20-29 30-39 40-49 50-59 60-69 Total

Freq. 4 16 f3 f4 f5 6 4 230

b) Compute the arithmetic mean.

c) Compute the value below which 25% of the observations lie.

d) Compute the value above which 25% of the observations lie

Answer 1

a) missing frequencies

Cumulative Frequency total = 30+f3+f4+f5=230

f3+f4+f5=200...........(i)

median class is 29.5 - 39.5

$33.5=29.5+[\frac{\frac{230}{2}-(20+f3)}{f4}]*10$

$4=[\frac{115-20-f3}{f4}]*10$

$0.4=\frac{95-f3}{f4}$

$0.4f4+f3=95$ ..........(ii)

modal class is 29.5 - 39.5

$34=29.5+{\frac{f4-f3}{(f4-f3)+(f4-f5)}}*10$

$4.5=[\frac{f4-f3}{f4-f3+f4-f5}]*10$

$0.45=\frac{f4-f3}{2f4-f3-f5}$

$0.45(2f4-f3-f5)=f4-f3$

$-0.1f4+0.55f3-0.45f5=0$ .....................(iii)

Solving equations (i), (ii) and (iii) yields

$f3=55$

$f4=100$

$f5=45$

b) Arithmetic mean

$=\frac{(4*4.75)+(16*14.5)+(55*24.5)+(100*34.5)+45*44.5)+(6*54.5)+(4*64.5)}{230}=\frac{7636}{230}=33.2$

c) value below 25%

$\frac{25}{100}*230=57.5$

d) value above 25%

$100-57.5=42.5$