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.
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 (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.
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.
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 
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%
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:
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.
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 
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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