NCERT Solution for Class 11 Statistics Chapter 5: Correlation

(i) Coefficient of Range
(a) Wheat

The coefficient of variation is more reliable than all other measures.

 

Dispersion is the extent to which values in a distribution differ from the average of the distribution. Knowledge of only average is insufficient as it does not reflect the quantum of variation in values.

Measures of dispersion enhance the understanding of a distribution considerably by providing information about how much the actual value of items in a series deviate from the central value, e.g., per capita income gives only the average income but a measure of dispersion can tell you about income inequalities, thereby improving the understanding of the relative living standards of different sections of the society. Through the value of dispersion one can better understand the distribution.

Thus a measure of dispersion is a good supplement to the central value in understanding a frequency distribution.

 

Standard Deviation is considered to be the best measure of dispersion and is therefore the most widely used measure of dispersion.

• It is based on all values and thus provides information about the complete series. Because of this reason, a change in even one value affects the value of standard deviation.

• It is independent of origin but not of scale.

• It is useful in advanced statistical calculations like comparison of variability in two data sets.

• It can be used in testing of hypothesis.

• It is capable of further algebraic treatment.

 

Yes, it is true that some measures of dispersion depend upon the spread of values, whereas some calculate the variation of values from the central value. Range and Quartile Deviation measure the dispersion by calculating the spread within which the value lies. Mean Deviation and Standard Deviation calculate the extent to which the values differ from the average or the central value.

 

25% of the persons earned more than ₹ 45,000. This implies that upper quartile Q3 = 45,000 75% earned more than 18,000. This implies that lower quartile Q1 =18,000
Absolute Measure of Dispersion = Q3 – Q1 = 45,000 – 18,000 = 27,000
Relative Measure of Dispersion
Co-efficient of Quartile Deviation

Calculate for each crop,
(i) Range
(ii) QD
(iii) Mean Deviation about Mean
(iv) Mean Deviation about Median
(v) Standard Deviation
(vi) Which crop has greater variation?
(vii) Compare the value of different measures for each crop.
Answer:
(i) Range
(a) Wheat Highest value of distribution (H) = 25
Lowest value of distribution (L) = 9
Range = H – L = 25 – 9 = 16
(b) Rice Highest value of distribution (H) = 34
Lowest value of distribution (L)=12
Range = H – L = 34 – 12 = 22
(ii) Quartile Deviation
(a) Wheat Arranging the production of wheat in increasing order 9, 10, 10, 12, 15, 16, 18, 19, 21, 25
Q1 = N+1/4th item = 10+1/4th item = 11/4

th item
= 2.75th item
= Size of 2nd item + 0.75 (size of 3rd item – size of 2nd item)
= 10 + 0.75(10 – 10)
= 10 + 0.75 × 0
= 10
Q3 = 3(N+1)/4th item = 3(10+1)/4th item= 33/4th item 

= 8.25th
= Size of 8th item + 0.25 (size of 9th item – size of 8th item)
= 19 + 0.25(21 – 19)
= 19 + 0.25 × 2
= 19 + 0.50 = 19.50
Quartile Deviation = 

Q3- Q1 = 4.75
(b) Rice Arranging the data of production of rice
12, 12, 12, 15, 18, 18, 22, 23, 29, 34 item
Q1 = N+1/4th item = 10+1/4th item
= 2.75 th item
= Size of 2nd item + 0.75 (size of 3rd item – size of 2nd item)
= 12 + 0.75(12 – 12) = 12 + 0.75 × 0
= 12

= 8.25th item
= Size of 8th item + 0.25 (size of 9th item – size of 8th item)
= 23 + 0.25(29 – 23)
= 23 + 0.25 × 6
= 23 + 1.5
= 24.5
Quartile Deviation = 6.25

(iii) Mean Deviation about Mean
(a) Wheat

(b) Rice

(iv) Mean Deviation about Median
(a) Wheat

(b) Rice

(v) Standard Deviation
(a) Wheat

(b) Rice

(vi) Coefficient of Variation
(a) Wheat
= 32.51
(b) Rice
= 36.71
Rice crop has greater variation as the coefficient of variation is higher for rice as compared to that of wheat.
(vii) Rice crop has higher Range, Quartile Deviation, Mean Deviation about Mean, Mean Deviation about Median, Standard Deviation and Coefficient of Variation.

 

Batsman X

(i) Average of Batsman X is higher than that of Batsman Y, so he should be selected if we want a high scorer.
(ii) The Batsman Y is more reliable than Batsman X. This is because the coefficient of variation of Batsman X is higher than that of Batsman Y.

 

For Brand A

For Brand B

(i) The average life of bulb of Brand B is comparatively higher than that of Brand A.
(ii) The bulbs of Brand B are more dependable as CV of Brand B is lesser than CV of Brand A.

 

N = 50

x

¯

¯

¯

 = 200
σ = 40
Average wage = 

 Total wages 

 Number of workers 

200 = 

 Total wages 

50

So, total wages = 200 × 50 = ₹ 10,000
Now, increase in wage rate = ₹20
Total raise = 50 × 20= ₹ 1,000
Total wage after raise = ₹ 10,000 + 1,000 = ₹ 11,000
New average wage = 

 New total wages 

 Number of workers 

=

11,000

50

 = ₹220
Thus, Mean increases by the amount of increase in wage of each worker as the absolute increase was equal for all.
Standard Deviation will remain the same that is ₹40 as Standard Deviation is independent of origin and hence addition of equal amount in all the values will not cause any change in the Standard Deviation.
Uniformity of wages can be seen by coefficient of variation.
Previously, the coefficient of variation was
CV = 

σ

X

¯

¯

¯

¯

¯

×100

 = (40/200) × 100 = 20
The new coefficient of variation after wage increase is given by
CV = σ×100

 = (40/220) × 100 = 18.18
This shows that wages have become more uniform now as the new CV is lower.

 

Average wage = ₹ 200
Hike in wages = 10%
Since arithmetic mean is not independent of scale, the mean will also increase by 10%.
= 10/100×200

= ₹ 20
Hence, the new Mean will be
200 + 20 = ₹ 220
Standard Deviation is also not independent of scale, hence, the Standard Deviation will also increase by 10% Initial Standard Deviation = ₹ 40
So, New Standard Deviation = ₹40 +10% of 40
= ₹ (40 + 4)
= ₹ 44