Some measures of dispersion depend upon the spread of values whereas some are estimated on the basis of the variation of values from a central value. Do you agree?
Yes, some measures of dispersion such as range, quartile deviation and interquartile range depend upon the spread of values; and some measures like mean deviation and standard deviation are estimated on the basis of a variation of values from a central value. Mean deviation, standard deviation takes into account all the values of the data through variation between them and a central value; whereas range, Q.D and interquartile range takes into account only some positional values. Other than these, Lorenz curve is also used to represent variation graphically.
The sum of 10 values is 100 and the sum of their squares is 1090. Find out the coefficient of variation.
If in the previous question, each worker is given a hike of 10% in wages. How are the Mean and Standard deviation values affected?
If each worker is given a hike of 10%
Then, total sum of wages (∑X) 10000 increases by 10%.
New total sum of wages = 10000 + 1000 = 11000
New mean = Rs. 220
New standard deviation:
With 10% increase in wage,
New Standard Deviation= Old Standard Deviation + 10% of Old Standard Deviation
= 40 + (40*10%)
= 40+4 = 44
With the hike of 10% in wages, mean increases to 220 and standard deviation increases to 44.
Average daily wage of 50 workers of a factory was ₹200 with a standard Deviation of ₹40. Each worker is given a raise of ₹20. What is the new average daily wage and standard deviation? Have the wages become more or less uniform?
Average daily wage (Mean) = ₹ 200
Standard deviation = ₹ 40
Since, wage of each worker is increased by ₹ 20 this is a situation of change in origin. With change in origin value of Mean changes but it has no impact on value of Standard Deviation.
i.e New Standard Deviation = Old Standard Deviation = ₹ 40
Sum of total wages of workers = 40 × 200 = ₹ 10000
Conclusion,
The coefficient of variation has declined with increase in wage rate; this means that wages have become more uniform than earlier.
In a town, 25% of the persons earned more than ₹ 45000 whereas 75% earned more than ₹ 18000. Calculate the absolute and relative values of dispersion.
A measure of dispersion is a good supplement to the central value in understanding a frequency distribution. Comment.
Measure of dispersion is better supplement to the central value in understanding a frequency distribution because it provides much clearer interpretation of the distribution. Measures of central tendency do not highlight variability present in the data; whereas measures of dispersion provide clear interpretation about the variability. They also explain the spread of individual values around the central value and also about the scatter and expansion of data in the series.
Calculate the Mean deviation using mean and Standard deviation for the following distribution.
Classes 
Frequencies 
20–40 
3 
40–80 
6 
80–100 
20 
100–120 
12 
120–140 
9 

50 
First we calculate all the relevant elements for mean deviation from mean and standard deviation
Classes 
f 
m 
fm 
d = m – 94.8 
d 
fd 
fd 
fd^{2} 
20–40 
3 
30 
90 
–64.8 
64.8 
194.4 
–194.4 
12597.12 
40–80 
6 
60 
360 
–34.8 
34.8 
208.8 
–208.8 
7266.24 
80–100 
20 
90 
1800 
–4.8 
4.8 
96 
–96 
460.8 
100–120 
12 
110 
1320 
15.2 
15.2 
182.4 
182.4 
2772.48 
120–140 
9 
130 
1170 
35.2 
35.2 
316.8 
316.8 
11151.36 

50 

∑fm = 4740 


∑fd = 998.4 

∑fd^{2 }= 34248 
A batsman is to be selected for a cricket team. The choice is between X and Y on the basis of their scores in five previous tests which are:
X: 25 85 40 80 120
Y: 50 70 65 45 80
Which bats man should be selected if we want.
To select batsmen first we need to calculate mean and standard deviation.
Batsmen X 
d 
d^{2} 
Batsmen Y 
d 
d^{2} 
25 
–45 
2025 
50 
–12 
144 
85 
15 
225 
70 
8 
64 
40 
–30 
900 
65 
3 
9 
80 
10 
100 
45 
–17 
289 
120 
50 
2500 
80 
18 
324 
∑x = 350 

∑ d^{2 }= 5750 
∑y = 310 

∑ d^{2 }= 830 
Now, calculate coefficient of variation to check for reliability of the batsmen
Coefficient of variation for Batsmen X = 33.91/70 × 100 = 48.44%
Coefficient of variation for Batsmen Y = 12.88/62 × 100 = 20.77%
Which measure of dispersion is the best and how?
Standard deviation is the best measure of dispersion. It is based on all values of the distribution, thus a change in value of even one variable affects it. It is most widely used measure of dispersion:
To check the quality of two brands of light bulbs, their life in burning hours was estimated as under for 100 bulbs of each brand.
Life (in hrs) 
No. of bulbs 


Brand A 
Brand B 

0–50 
15 
2 

50–100 
20 
8 

100–150 
18 
60 

150–200 
25 
25 

200–250 
22 
5 


100 
100 

To check for the quality of two brands of light bulbs first we calculate the mean and standard deviation of both the brand’s.
In the previous question, calculate the relative measures of variation and indicate the value which, in your opinion, is more reliable.
The yield of wheat and rice per acre for 10 districts of state is as under:
District 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
Wheat 
12 
10 
15 
19 
21 
16 
18 
9 
25 
10 
Rice 
22 
29 
12 
23 
18 
15 
12 
34 
18 
12 
Calculate for each crop,
1. Range
2. Q.D.
3. Mean deviation about Mean
4. Mean deviation about Median
5. Standard Deviation
6. Which crop has greater variation?
7. Compare the values of different measures of for each crop.
First, we arrange the values in an ascending order.
Wheat
Wheat 
d(mean) 
D(median) 
d 
d^{2} 
9 
6.5 
6.5 
–6.5 
42.25 
10 
5.5 
5.5 
–5.5 
30.25 
10 
5.5 
5.5 
–5.5 
30.25 
12 
3.5 
3.5 
–3.5 
12.25 
15 
0.5 
0.5 
–0.5 
0.25 
16 
0.5 
0.5 
0.5 
0.25 
18 
2.5 
2.5 
2.5 
6.25 
19 
3.5 
3.5 
3.5 
12.25 
21 
5.5 
5.5 
5.5 
30.25 
25 
9.5 
9.5 
9.5 
90.25 
∑X = 155 
∑d = 43 
∑D = 43 

∑d^{2} =254.50 
Rice
Rice 
d(mean) 
D(median) 
d 
d^{2} 
12 
7.5 
6 
–7.5 
56.25 
12 
7.5 
6 
–7.5 
56.25 
12 
7.5 
6 
–7.5 
56.25 
15 
4.5 
3 
–4.5 
20.25 
18 
1.5 
0 
–1.5 
2.25 
18 
1.5 
0 
–1.5 
2.25 
22 
2.5 
4 
2.5 
6.25 
23 
3.5 
5 
3.5 
12.25 
29 
9.5 
11 
9.5 
90.25 
34 
14.5 
16 
14.5 
210.25 
∑X = 195 
∑d = 57.5 
∑D = 57 

∑d^{2} = 512.50 
1. Range
Range for wheat = L – S
Largest value = 25 Smallest value = 9
= 25 – 9 = 16
Range for rice = L – S
Largest value = 34 Smallest value = 12
= 34 – 12 = 22
2. Quartile Deviation
3. Mean Deviation about mean
Mean for wheat = ∑X/N = 155/10 = 15.5
Mean for rice = 195/10 = 19.5
Mean Deviation from Mean = ∑d/n
Substituting values from above tables in the formula
Mean Deviation from Mean for Wheat = 43/10 = 4.3
Mean Deviation from Mean for Rice = 57.5/10 = 5.75
4. Mean Deviation about median
Position of Median = N+1^{th}/2 item = 11/2 item = 5.5 item
Median= 5^{th} + 6^{th} /2 item
Median for wheat = 15 +16 / 2 = 15.5
Median for rice = 18 +18 / 2 = 18
Mean Deviation from Median= ∑D/n
here, D = X– median
Substituting values from above tables in the formula
Mean Deviation for wheat = 43/10 = 4.3
Mean Deviation for rice = 57/10 = 5.7
5. Standard Deviation
6. After carefully comparing the values for different measure of dispersion for each crop we reach to the conclusion that Rice has greater variation than Wheat.
7. Comparing values of different measures:
Wheat 
Rice 

Range 
16 
22 
Quartile Deviation 
4.5 
5.5 
Mean Deviation about mean 
4.3 
5.75 
Mean deviation about median 
4.3 
5.7 
Standard Deviation 
5.04 
7.15 
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