1. Give the definition of Statistics due to Croxton and Cowden, and note down the importance of Statistics in the field of Economics. 2. Find Mean, Variance and Coefficient of vari... 1. Give the definition of Statistics due to Croxton and Cowden, and note down the importance of Statistics in the field of Economics. 2. Find Mean, Variance and Coefficient of variation for first n natural numbers. 3. The first three moments of a distribution about 2 are 1, 22 and 10. Find its mean, standard deviation and the third central moment. Read more

Steps to Solve

1. Definition of Statistics

   According to Croxton and Cowden, statistics can be defined as "the science of collecting, organizing, presenting, analyzing, and interpreting data."

   Importance in Economics Statistics is crucial in economics as it helps economists analyze data to make informed decisions, forecast economic trends, and evaluate policies.

2. Finding Mean, Variance, and Coefficient of Variation for First n Natural Numbers

   • Mean Calculation
     The mean (average) of the first $n$ natural numbers is calculated using the formula:
     $$ \text{Mean} = \frac{1 + 2 + 3 + \ldots + n}{n} = \frac{n(n + 1)/2}{n} = \frac{n + 1}{2} $$

   • Variance Calculation
     The variance measures how spread out the numbers are. It can be calculated using:
     $$ \text{Variance} = \frac{1^2 + 2^2 + 3^2 + \ldots + n^2}{n} - \left(\frac{n + 1}{2}\right)^2 $$
     The sum of squares for the first $n$ natural numbers is given by:
     $$ S = \frac{n(n + 1)(2n + 1)}{6} $$
     Thus, variance becomes:
     $$ \text{Variance} = \frac{S}{n} - \left(\frac{n + 1}{2}\right)^2 $$

   • Coefficient of Variation
     The coefficient of variation (CV) is given by:
     $$ \text{CV} = \frac{\sigma}{\mu} \times 100 $$
     where $\sigma$ is the standard deviation and $\mu$ is the mean.

3. Finding Mean, Standard Deviation, and Third Central Moment

   • Mean Calculation
     The mean is given by the first moment about 2, which is:
     $$ M = 2 + 1 = 3 $$

   • Standard Deviation Calculation
     The standard deviation ($\sigma$) is the square root of the variance. Variance can be derived from the second moment:
     $$ \text{Variance} = 22 - (3 - 2)^2 = 22 - 1 = 21 $$
     Thus,
     $$ \sigma = \sqrt{21} $$

   • Third Central Moment
     The third central moment is given and can be calculated using moments about the mean:
     It is the third moment about the mean, given by:
     $$ M_3 = \text{central moments} $$
     Since we want the third central moment, we use:
     $$ M_3 = 10 $$