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

Solution: It is absolutely correct that a measure of dispersion significantly enhances our understanding of a frequency distribution when used alongside a measure of central tendency (like the mean or median). While a central value gives us a typical or representative value of the data, it doesn't tell us how spread out the data points are. For instance, two different datasets might have the same average, but one could have all its values clustered closely around the average, while the other has values widely scattered. Measures of dispersion, such as the range, quartile deviation, mean deviation, and standard deviation, quantify this spread. They help us understand the variability, consistency, or diversity within the data. For example, in economics, comparing the average income of two countries might be misleading if one country has a very wide income gap (high dispersion) while the other has more evenly distributed incomes (low dispersion), even if their averages are similar.

Solution: Among the various measures of dispersion, the **Standard Deviation (SD)** is generally considered the best and most widely used. This is due to several key advantages it possesses:

1. Foundation of Statistical Theory: Much of advanced statistical theory, including hypothesis testing, correlation, and regression analysis, is built upon the concept of standard deviation. It is crucial for comparing the variability between two or more datasets and for inferring population characteristics from samples.
2. Uses All Observations: Unlike measures like the range (which only uses the minimum and maximum values), the standard deviation is calculated using every single value in the dataset. This makes it a more representative measure of the overall spread.
3. Precise and Rigidly Defined: The standard deviation has a unique, unambiguous mathematical definition, ensuring consistency in its calculation and interpretation.
4. Independent of Origin: Adding or subtracting a constant value from every data point does not change the standard deviation, which is a desirable property.
5. Widely Applicable: It is particularly useful when data distributions tend to be normal or bell-shaped, which is common in many natural and social phenomena.
6. Enables Algebraic Treatment: Standard deviation allows for more complex mathematical manipulations and derivations compared to range, quartile deviation, or mean deviation, making it more versatile in statistical analysis.

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

Measures based on the spread of values: These measures focus on the overall range or span covered by the data. The most prominent example is the **Range**, which is simply the difference between the highest and lowest values in a dataset. It directly measures the total spread of the data.

Measures based on variation from a central value: These measures calculate how much individual data points deviate from a central point, typically the mean. Examples include **Mean Deviation** and **Standard Deviation**. They assess the average distance of data points from the center, providing insight into the consistency or dispersion around that central value.

Furthermore, dispersion measures can be classified as **absolute** or **relative**. Absolute measures (like Range, Quartile Deviation, Mean Deviation, Standard Deviation) are expressed in the same units as the original data. Relative measures (like the Coefficient of Range, Coefficient of Variation) are ratios or percentages, making them unit-free and suitable for comparing datasets with different units or scales.

Solution:

Absolute Value of Dispersion:

The problem provides information about quartiles. We are given:

• 25% of persons earned more than Rs 45,000. This means that 75% earned less than Rs 45,000. Therefore, Rs 45,000 is the 75th percentile, which corresponds to the third quartile (Q3). So, $Q3\; =\; 45,000$.
• 75% of persons earned more than Rs 18,000. This means that 25% earned less than Rs 18,000. Therefore, Rs 18,000 is the 25th percentile, which corresponds to the first quartile (Q1). So, $Q1\; =\; 18,000$.

Using these quartiles, we can calculate the **Quartile Deviation (QD)**, which is an absolute measure of dispersion:

$QD\; =\; \backslash frac\{Q3\; -\; Q1\}\{2\}$

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

$QD\; =\; \backslash frac\{27,000\}\{2\}$

$QD\; =\; 13,500$

Thus, the absolute value of dispersion, measured by Quartile Deviation, is Rs 13,500.

Relative Value of Dispersion:

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

$CQD\; =\; \backslash frac\{Q3\; -\; Q1\}\{Q3\; +\; Q1\}$

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

$CQD\; =\; \backslash frac\{27,000\}\{63,000\}$

$CQD\; =\; \backslash frac\{27\}\{63\}$

$CQD\; =\; \backslash frac\{3\}\{7\}$

$CQD\; \backslash approx\; 0.476$

Thus, the relative value of dispersion is approximately 0.476 (or 47.6%).