NCERT Solutions For Class 11 Economics Chapter 11 Measures of Dispersion

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.

First we calculate all the relevant elements for mean deviation from mean and standard deviation

To select batsmen first we need to calculate mean and standard deviation.

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%

1. Batsmen X has a greater average (70 runs) as compared to Batsmen Y (62 runs), so he shall be selected if we want a high run getter.
2. But, if we want reliable batsmen for the team then Batsmen Y shall be selected as he has a lower coefficient of variation (20.77) as compared to batsmen X (48.44). This means that batsmen Y are very consistent with his performance.

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:

1. It is independent of origin but not of scale.
2. It can be used if the samples are combined in various ways.
3. It is very useful in advanced statistical works and further algebraic treatment is also possible.
4. The square of deviations from mean is least in case of standard deviation.
5. It is the best method to compare variability between two or more distributions.
6. It plays a central role in normal distribution which in turn is very useful.

To check for the quality of two brands of light bulbs first we calculate the mean and standard deviation of both the brand’s.

1. Brand B gives a higher life as average life time of bulb from brand B(136.5 hrs) is greater than Brand A(134.5 hrs).
2. Brand B is more dependable because variation in Brand B (27.34%) is lesser than Brand A (51.15%). This means that the average lifetime of samples of bulb from Brand B only differ by 27.34%, but average lifetime of samples of bulb from Brand B differ by 51.15%.

First, we arrange the values in an ascending order.

Wheat

Rice

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: