**Class 11 statistics Chapter 6 Measures of dispersion**

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**NCERT Notes for Class 11 statistics Chapter 6 Measures of dispersion**

**Class 11 statistics Chapter 6 Measures of dispersion**

Dispersion: The word Dispersion means deviation or difference. In statistics, dispersion refers to deviation of various items of the series from its Central value. It is also called averages of second order. Dispersion is two types. They are the following.

- Absolute Measures of Dispersion: It is expressed in the same statistical unit in which the original data are given.
- Relative Measures of Dispersion: It is the ratio of absolute dispersion to an appropriate average. It is independent of the unit .

Measures of Dispersion: The following are the important measures of Dispersion.

- Range
^{[1]} - Quartile deviation
- Mean deviation
- Standard deviation
- Lorenz curve

**Range:It** is defined as the difference between the largest and the smallest value of the series.

Range=L-S

Here L= largest value, S=smallest value

coefficient of Range = 𝐿−𝑆 / 𝐿+𝑆 |

Merits and Demerits of Range Merits:

- Easy to calculate
- Easy to understand Demerits:
- It is affected by extreme values
- It is not based on all observations
- It cannot be used in case of open-end distribution.

**Quartile Deviation**: It is the average of the difference between the upper quartile (Q3) and the lower quartile (Q1).

**Merits of quartile deviation **

- Easy to calculate
- Easy to understand
- Not affected by extreme values
- More reliable than range

**Demerits of quartile deviation **

- It is not based on all observations
- It ignores the first 25% observations as well as the last 25%
- Not capable of further algebraic treatment
- It doesn’t measure variation of items
^{2}from the average.

**MEAN DEVIATION:** Mean deviation of a series is the arithmetic average of the deviations of various items from a measure of Central tendency.

* MERITS OF MEAN DEVIATION: *

- Easy to calculate
- Easy to understand
- Rigidly defined
- Based on all items
- Not very much affected by extreme values

** DEMERITS OF MEAN DEVIATION**:

- Very much affected by sampling fluctuations
- Not capable of further algebraic treatment
- Ignoring signs reduces its application in scientific problems.

** STANDARD DEVIATION:** It is defined as the square root of the arithmetic average of the squares of deviations taken from the arithmetic average of a series.

SD in individual series =

SD in discrete and continuous series =

CO efficient of variation=

## MERITS AND LIMITATIONS OF STANDARD DEVIATION

- It is the best measure of Dispersion
- It is capable of further algebraic treatment
- It is less affected by sampling fluctuations
- It is based on every item of the distribution
- The value is always definite

### LIMITATIONS

- It is difficult to compute
- It gives more importance to extreme items
- It can not be used for purposes of comparison

**INTERPRETATIONS **

- If the value of Coefficient of Variation is less, it means it is more consistent.
- A series with more Coefficient of Variation is regarded as less consistent or less stable than a series with less coefficient of variation.

**LORENZ CURVE **

**It is the graphical representation of Dispersion Developed by Dr. Max O. Lorenz **

**LORENZ CURVE STEPS **

- Find Class Mid Point
- Cumulate the Class Mid Points
- Cumulate the frequencies
^{3} - Take the grand total of class mid points and grand total of frequencies as 100
- Then convert all the other cumulative class mid points and cumulative frequencies into their respective percentages
- Mark cumulative percentages of frequencies on the x axis and cumulative class mid points on the y axis 7) Each axis should have values from 0 – 100.
- Draw a line from the origin to the point whose cordinate is 100, 100.
- This Line is called the LINE OF EQUAL DISTRIBUTION
- Then plot the cumulative values and cumulative frequencies.

**MERITS AND LIMITATIONS MERITS**

- It is most commonly used to show the various kinds of inequality.
- The curve uses the information expressed in a cumulative manner to indicate the degree of variability.
- It is especially useful in comparing the variability of two or more distributions.
- Since it gives a picture, it is easy to understand.

**LIMITATIONS**

- It does not give any numerical value of the measure of dispersion
- It merely gives a picture of the extent
^{4}to which a series is pulled away from an equal distribution

**Join those points to get a curve which is called LORENZ CURVE**

INTERPRETATION (with reference to the given diagram)

Any curve similar to ‘OAC’ closer to the line of Equality ‘OC’ indicates high degree of equality or limited degree of inequality. The farther the curve OC, the greater is the variability present in the distribution. The farthest curve from the line ‘OC’ has the highest Dispersion