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Measures of Central Tendency — Economics STD 11 Commerce — Question

CBSE BoardEnglish MediumSTD 11 CommerceEconomicsMeasures of Central Tendency6 Marks
Question
If the arithmetic mean of the data given below is 28, find (a) the missing frequency, and (b) the median of the series:
Profit per retail shop (in ₹)
0-10
10-20
20-30
30-40
40-50
50-60
Number of retail shops
12
18
27
-
17
6
✓

Answer

Let us take the missing frequency as x.
Profit per retail shop (in ₹) (X)
Number of retail shops (f)
Mid values (m)
fm
0-10
10-20
20-30
30-40
40-50
50-60
12
18
27
x
17
6
5
15
25
35
45
55
60
270
675
35x
765
330
 
$\sum\text{f} = 80 +\text{x}$
 
$\sum\text{Fm} = 2100 + 35\text{x}$
$\text{Mean}=\frac{\sum\text{fm}}{\sum\text{f}}$
$\text{Mean}=28$
Substituting the values in the formula we get,$28=\frac{2100\ +\ 35\text{x}}{80\ +\ \text{x}}$
$28\text{x}(80+\text{x})=2100+35\text{x}$
$2240+28\text{x}=2100+35\text{x}$
$7\text{x}=140\text{x}$
$\text{x}=20$
Thus, the missing value frequency is 20.
Class Interval
Frequency (f)
Cumulative frequency (CF)
0-10
12
12
10-20
18
30
20-30
27
57
30-40
x
77
40-50
17
94
50-60
6
100
Total
$\sum\text{f}=100$
  
Formula of median is as follow: Median$=\text{L}+\frac{\frac{\text{N}}{2}-\text{Cf}}{\text{f}}\times\text{h}$
By substituting the value in the formula we get,$\text{Median}=20+\frac{(50-30)}{27}\times(30-20)$
$\text{Median}=20+\frac{(20)}{27}\times(10)$
$\text{Median}=20+\frac{200}{27}$
$\text{Median}=20+7.41$
$\text{Median}=27.41$
Thus the median value of the series is 27.41

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