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Question

An analysis for more efficiency in a factory , indicating the distribution of ages of workers was as follows:
Age (in years) 16-19 20-29 30-39 40-49 50-59 60-64
Frequency 15 46 49 32 28 14
(a) Calculate the mean and median of the above data.
(b) Draw a histogram and indicate mode therein.

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Solution

(a)

Age

Frequency (f)

Midpoint

(m)

fm

16 – 19

15

17.5

262.5

20 – 29

46

24.5

1127

30 – 39

49

34.5

1690.5

40 – 49

32

44.5

1424

50 – 59

28

54.5

1526

60 – 64

14

62

868

f=184

fm=6898

Mean

X¯=ΣfmΣf =6898184 =37.48

Hence, the mean of the given distribution is 37.48

Median

Note that the given distribution is in the form of inclusive class intervals. For the calculation of median, first the class intervals must be converted into exclusive form using the following formula.
Value of Adjustment = Lower limit of one class - Upper limit of the preceeding class2
Value of lower limit of one class Value of upper limit of the preceeding class2
The value of adjustment as calculated is then added to the upper limit of each class and subtracted from the lower limit of each class. In this manner, we get the following distribution.
Age

Frequency
(f)

Cumulative
Frequency

(c.f)

15.5 – 19.5
19.5 – 29.5
L129.5 – 39.5
39.5 – 49.5
49.5 – 59.5
59.5 – 64.5
15
46
49 f1
32
28
14
15
61→c.f
110
142
170
184

Median class is given by the
Size of
N2th item = 1842th item = 92th item.
This corresponds to the class interval of (29.5 39.5), so this is the median class.
Median=L1+N2-c.ff×iso, Median=29.5+1842-6149×10or, Median=29.5+3149×10 Median= 29.5+6.32 =35.826

Hence, the median is 35.826

(b)
Here the data is in the form of unequal class interval. So, we will first make appropriate adjustment in the frequencies to make the class intervals equal.
Age

Frequency
(f)

Adjusted
Frequency

15.5 – 19.5

19.5 – 29.5
29.5 – 39.5
39.5 – 49.5
49.5 – 59.5
59.5 – 64.5
15

46
49
32
28
14

15×104=37.5
-
-
-
-
14×105=28



From the above graph, it can be seen that mode is 30.5. However, mean can not be determined graphically.

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