Question

Given the following grouped frequency distribution

Data	Frequency
30 - 34	15
35 - 39	20
40 - 44	15
45 - 49	10
50 - 54	5
55 - 59	4
60 - 64	3
65 - 69	1
70 - 74	1

Find the mean (give you answer to one decimal) =

Find the standard deviation (give your answer to two decimals) =

What is the shape of this distribution?

Answer 1

Solution:
First, we will calculate the midpoints of the classes which can be calculated as
Midpoints = (Upper limit + Lower limit)/2

Data	Midpoints
30-34	32
35-39	37
40-44	42
45-49	47
50-54	52
55-59	57
60-64	62
65-69	67
70-74	72

Mean of grouped data can be calculated as
Mean = (Midpoints*Frequency)/Frequency = ((32*15) + (37*20) + (42*15) + (47*10) + (52*5) + (57*4) + (62*3) + (67*1) + (72*1))/(15+20+15+10+5+4+3+1+1) = 3133/74 = 42.3

Data	Midpoints	Frequency	Midpoints * Frequency
30-34	32	15	480
35-39	37	20	740
40-44	42	15	630
45-49	47	10	470
50-54	52	5	260
55-59	57	4	228
60-64	62	3	186
65-69	67	1	67
70-74	72	1	72

standard deviation can be calculated as
Standard deviation = sqrt(((frequency*(Midpoints - Mean)^2)/(n-1))

Data	Midpoints	Frequency	Midpoints * Frequency	Midpoints-mean	(Midpoints-mean)^2	(Midpoints - mean)^2 * f
30-34	32	15	480	-10.3	106.09	1591.35
35-39	37	20	740	-5.3	28.09	561.8
40-44	42	15	630	-0.3	0.09	1.35
45-49	47	10	470	4.7	22.09	220.9
50-54	52	5	260	9.7	94.09	470.45
55-59	57	4	228	14.7	216.09	864.36
60-64	62	3	186	19.7	388.09	1164.27
65-69	67	1	67	24.7	610.09	610.09
70-74	72	1	72	29.7	882.09	882.09

Standard deviation = sqrt((1591.35+561.8+1.35+220.9+470.45+864.36+1164.27+610.09+882.09)/(74-1)) = sqrt(6366.66/73) = 9.34
Now when create bar graph for this group data than

Also from the data and bar graph, we can say that this distribution is right-skewed distribution.