Calculate Median of the following series:
|
Wages Rate (in Rs.) |
No. of Workers |
|
Less than 10 Less than 20 Less than 30 Less than 40 Less than 50 Less than 60 Less than 70 Less than 80 |
15 35 60 84 96 127 198 250 |
Here, we have been given cumulative frequency of "less than". This frequency has been converted into simple frequency as under for calculating median.
Calculation of Median:
|
Wages (Rs.) |
Cumulative frequency |
Frequency |
|
0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 |
15 35 60 84 96 127 198 250 |
15-0= 15 35-15 = 20 60-35 = 25 84 - 60= 24 96 - 84 = 12 127-96 = 31 198-127 = 71 250-198 = 52 |
Calculate the median of the following distribution of salaries of a employees of a firm.
|
Income (in Rs.) |
No. of Persons |
|
400-500 500-600 600-700 700-800 800-900 900-1000 1000-1100 |
25 69 107 170 201 142 64 |
|
ncome (Rs.) |
No. of Persons (f) |
c.f |
|
400-500 500-600 600-700 700-800 800-900 900-1000 1000-1100 |
25 69 107 170 201 142 64 |
25 94 201 371 572 714 778 |
11 = Lower limit of median group.
c.f. = Cumulative frequency of the class preceding the median class.
f = Frequency of the median group,
c = The class interval of the median group
Calculation of Median :
Median lies in the group 800 - 900. Applying the formula we get,
Hence Median Income = 800.89.
Calculate median of the following:
|
Marks |
No. of Students |
|
46-50 41-45 36-40 31-35 26-30 21-25 16-20 11-15 |
5 11 22 35 26 13 10 7 |
In the question the inclusive series has been given in the descending order. For calculating, it would be converted into an exclusives series in the ascending order.
Calculation of Median
|
Class Interval |
No. of Students |
Cumulative frequency (c.f.) |
|
10.5-15.5 15.5-20.5 20.5-25.5 25.5-30.5 30.5-35.5 35.5-40.5 40.5-45.5 45.5-50.5 |
7 10 13 26 35 22 11 5 |
7 17 30 50 91 113 124 129 |
|
N=129 |
Hence Median = 31.7 marks.
Prove with an example that the weighted arithmetic mean will be less than the simple mean, when items of small values are given greater weights and items of big values are given less weight.
In order to prove the statement given in the example, we take the following table :
|
Type |
Weight (Rs.) X |
Workers W |
WX |
|
Man Woman Children |
8 6 4 |
50 20 10 |
400 120 40 |
|
Σ x=18 |
Σ W=80 |
ΣWX=560 |
Compare the arithmetic mean, median and mode as measure of Central Tendency. Describe situations where one is more suitable than the others.
As compared to Mean and Median, Mode is less suitable. Mean is simple to calculate, its value is definite, it can be given algebraic treatment and is not affected by fluctuations of sampling. Median is even more simple to calculate but is affected by fluctuations and cannot be given algebraic treatment. Mode is the most popular item of a series but it is not suitable for most elementary studies because it is not based on all the observations of the series and is unrepresentative. i
In case of arithmetic mean it is the numerical magnitude of the deviations that balances. In case of median it is the number of values greater than the Median which balances against the number of values less than the Median. The median is always between the arithmetic mean and the mode.
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