CBSE Class 11 Economics Chapter 6: Measures of Dispersion NCERT Solutions

Measures of central tendency, like the mean or median, provide a single value that represents the center of a dataset. However, they do not tell us how spread out the data is. Measures of dispersion, such as range, quartile deviation, mean deviation, and standard deviation, are essential because they quantify this spread or variability. For instance, two countries might have the same average income, but one could have extreme wealth disparity (high dispersion) while the other has more evenly distributed income (low dispersion). Therefore, dispersion measures offer a more complete and nuanced understanding of a frequency distribution by complementing the information provided by central values.

Standard Deviation is widely considered the best measure of dispersion because it satisfies most of the essential criteria for a good measure. Here's why:

1. Foundation of Statistical Theory: Much of statistical theory, including hypothesis testing, correlation, and regression analysis, is built upon the concept of Standard Deviation. It is also highly effective for comparing the variability between two or more datasets.
2. Uses All Observations: Unlike measures like the range, Standard Deviation incorporates the value of every single observation in the distribution, leading to a more accurate representation of the overall spread.
3. Precise and Rigidly Defined: It has a unique, well-defined mathematical value, making it unambiguous and reliable.
4. Independent of Origin: Shifting the data by a constant value (e.g., adding 10 to every score) does not change the Standard Deviation, which is a desirable property.
5. Suitability for Normal Distribution: It is particularly useful when data distributions approximate a normal distribution, a common scenario in many fields.
6. Enables Algebraic Treatment: Standard Deviation allows for advanced mathematical operations and manipulations, which are not as easily performed with other measures like range or quartile deviation.

Due to these advantages, Standard Deviation is the most preferred and versatile measure of dispersion in statistical analysis.

Yes, I agree with this statement. Measures of dispersion can be broadly categorized based on how they quantify variability:

Measures Based on Spread: These measures focus on the overall range or span of the data values. The most common example is the Range, which is simply the difference between the highest and lowest values in a dataset. Other measures like Quartile Deviation also consider the spread by looking at the middle 50% of the data.

Measures Based on Variation from a Central Value: These measures calculate the deviation of individual data points from a measure of central tendency, typically the mean. Examples include Mean Deviation (average of absolute deviations from the mean) and Standard Deviation (square root of the average of squared deviations from the mean). These methods assess how much individual values typically differ from the average value.

Furthermore, it's important to distinguish between absolute and relative measures. Absolute measures (like Range, Standard Deviation) are expressed in the same units as the original data, making them useful for understanding the spread within a single dataset but difficult for comparing datasets with different units. Relative measures (like the Coefficient of Variation) are unit-free ratios, making them ideal for comparing the dispersion of datasets with different units or means.

The problem provides information about the distribution of earnings in a town:

• 25% of persons earned more than Rs 45,000. This means Rs 45,000 is the 75th percentile (P75 or Q3).
• 75% of persons earned more than Rs 18,000. This means Rs 18,000 is the 25th percentile (P25 or Q1).

We can use these values to calculate measures of dispersion.

1. Absolute Value of Dispersion:

We can calculate the Interquartile Range (IQR), which is the difference between the third quartile (Q3) and the first quartile (Q1).

Q3 = Rs 45,000

Q1 = Rs 18,000

Interquartile Range (IQR) = Q3 - Q1

$$IQR\; =\; 45,000\; -\; 18,000\; =\; 27,000$$

The absolute value of dispersion, represented by the Interquartile Range, is Rs 27,000.

We can also calculate the Quartile Deviation (QD):

Quartile Deviation (QD) = $\backslash frac\{Q3\; -\; Q1\}\{2\}$

$$QD\; =\; \backslash frac\{45,000\; -\; 18,000\}\{2\}\; =\; \backslash frac\{27,000\}\{2\}\; =\; 13,500$$

The Quartile Deviation is Rs 13,500.

2. Relative Value of Dispersion:

The relative measure corresponding to Quartile Deviation is the Coefficient of Quartile Deviation (CQD).

Coefficient of Quartile Deviation (CQD) = $\backslash frac\{Q3\; -\; Q1\}\{Q3\; +\; Q1\}$

$$CQD\; =\; \backslash frac\{45,000\; -\; 18,000\}\{45,000\; +\; 18,000\}\; =\; \backslash frac\{27,000\}\{63,000\}$$

$$CQD\; =\; \backslash frac\{27\}\{63\}\; =\; \backslash frac\{3\}\{7\}\; \backslash approx\; 0.4286$$

The relative value of dispersion (Coefficient of Quartile Deviation) is approximately 0.4286.