Measure of Dispersion Exercise: 14.2 Class 11 Basic Mathematics Solution [NEB UPDATED]

Measure of Dispersion Exercise: 14.2

Exercise 14.2

1. What do you mean by skewness? How does it differ from dispersion? Describe the various measures of skewness.
Solution:

Part-1:

Skewness is a measure of the asymmetry of a distribution.

$Skewness{\rm{ }} = {\rm{ }}\frac{{\left( {mean{\rm{ }} - {\rm{ }}mode} \right)}}{{standard{\rm{ }}deviation}}{\rm{ }}$

Part-2:
Dispersion is a measure of how spread out a distribution is, while skewness is a measure of the asymmetry of a distribution.

Part-3:

The various measures of skewness are:

Absolute measure of skewnwss

Relative Measure of Skewness

2. What are relative measures and absolute measure of skewness? Describe the positive and negative skewness with figures.
Solution:

Part-1:

Absolute measure of skewnwss

(i) Karl Pearson’s measure of skewness = mean – mode or mean – median

(ii) Bowley’s measure of skewness = Q₃ + Q₁ – 2M_d

Relative Measure of skewness

(i) Karl Pearson’s coefficient of skewness

S_k(P) = (Mean – Mode)/S.D

Part-2:

Positive skewness: When Mean > Median > Mode

Negaive skewness: When Mean < Median < Mode.

3.The values of mean, median and mode of three distributions are given:

Estimate whether the distribution are symmetrical or skewed.

i) Mode = 75 and mean = 78.6

Solution:

Here, Mean > Mode

So, it is positively skewed.

ii) Median = 71.4, Mean = 71.4 and Mode = 71.4

Solution:

Here, Mean = Median = Mode,

So, it is symmetrical distribution.

iii) Median = 12.04 and Mean = 11.28
Solution:

Here, Median > Mean,

So, it is negatively skewed.

4. a) For a group of 50 items; Sigma x 7 ^ 2 = underline 600, Sigma*x = 150 and M_{o} = 1.75; find Pearsonian

coefficient of skewness.

b) If*Sigma*fx =110, Sigma fx^ 2 =I650 , N = 10 and M_{o} = 12.45 find the skewness based on mean,

mode and standard deviation.

cf For a distribution, ifn =20, Sigma x=120, Sigma x^ 2 =94 xi and M_{d} = 7.5 , find the coefficient of skewness.

5.

overline a = a/1 a) A frequency distribution gives the following results -(p9)* 6/107 + 1007 = 1

i )C.V.=5\%

ii) s.d.= sigma = 2

iii) Karl Pearson's coefficient of skewness = 0.5.

Find the mean and the mode of the distribution.

G 2 ^ * = (Gt)/2 = |f_{1}|/2 * G

b) The median, mode and coefficient of skewness for Icertain distribution are respectively 17.4, 15.3 and 0.35. Calculate mean and the coefficient of variation. In a distribution, if sum of the difference of two quartiles is 25, their sum is 83 and median is 44. Find the coefficient of skewness.

In a certain distribution, if median = 45, mode = 39 and standard deviation I = 15 , find

the coefficient of variation and the coefficient of skewness.

6. Consider the following distribution:

Distribution A

Distribution B

Arithmetic mean:

100

90

90

Median:

Standard deviation:

400-10

80

10

a) Give as much information as you can from two distributions.

7.

b) Is the distribution A same as the distribution B regarding the degree of variation and skewness?

Calculate the coefficient of skewness based on mean, mode and the standard deviation from the following data: