Yes, some measures of dispersion such as range, quartile deviation and inter-quartile 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 inter-quartile 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| |
f|d| |
fd |
fd2 |
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 |
|
|
∑f|d| = 998.4 |
|
∑fd2 = 34248 |
To select batsmen first we need to calculate mean and standard deviation.
Batsmen X |
d |
d2 |
Batsmen Y |
d |
d2 |
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 |
|
∑ d2 = 5750 |
∑y = 310 |
|
∑ d2 = 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 |
d2 |
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 |
|
∑d2 =254.50 |
Rice
Rice |
|d|(mean) |
|D|(median) |
d |
d2 |
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 |
|
∑d2 = 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+1th/2 item = 11/2 item = 5.5 item
Median= 5th + 6th /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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