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.
Understand the Problem
The questions are asking for definitions and importance of statistical concepts, statistical calculations involving mean, variance, and coefficients, as well as deriving moments from a distribution. The approach involves providing definitions, performing statistical calculations, and deriving specific values based on given moments.
Answer
1. Statistics is the science of collection, classification, and interpretation of numerical data. 2. Mean = $\frac{n+1}{2}$, Variance = $\frac{(n(n+1)(2n+1))}{6n} - \left(\frac{n+1}{2}\right)^2$, Coefficient of Variation = $\frac{\sqrt{Variance}}{Mean} \times 100$. 3. Mean = $3$, Standard Deviation = $\sqrt{22}$, Third Central Moment = $10$.
Answer for screen readers
- Statistics is the science of collection, classification, and interpretation of numerical data.
- Mean = $\frac{n+1}{2}$, Variance = $\frac{(n(n+1)(2n+1))}{6n} - \left(\frac{n+1}{2}\right)^2$, Coefficient of Variation = $\frac{\sqrt{Variance}}{Mean} \times 100$
- Mean = $3$, Standard Deviation = $\sqrt{22}$, Third Central Moment = $10$.
Steps to Solve
- Definition of Statistics (Croxton and Cowden)
Statistics is defined by Croxton and Cowden as the science of collection, classification, and interpretation of numerical data. It involves methods for gathering data, summarizing it, and drawing conclusions from it.
- Importance of Statistics in Economics
Statistics plays a crucial role in economics as it helps in:
- Data analysis and interpretation for economic policies.
- Forecasting trends, such as GDP growth and inflation rates.
- Making informed decisions based on empirical evidence.
- Finding Mean, Variance, and Coefficient of Variation for First n Natural Numbers
- Calculate Mean:
The formula for the mean of the first $n$ natural numbers is:
$$ \text{Mean} = \frac{n(n+1)}{2n} = \frac{n+1}{2} $$
- Calculate Variance:
The variance can be calculated using:
$$ \text{Variance} = \frac{(n(n+1)(2n+1))}{6n} - \left(\frac{n+1}{2}\right)^2 $$
- Calculate Coefficient of Variation:
The coefficient of variation (CV) is given by:
$$ \text{CV} = \frac{\text{Standard Deviation}}{\text{Mean}} \times 100 $$
- Finding Mean, Standard Deviation, and Third Central Moment
- Mean:
The first moment about the origin is $1$. To find the mean:
$$ \text{Mean} = \text{First Moment} + 2 = 1 + 2 = 3 $$
- Standard Deviation:
Using the second moment (variance), we find:
$$ \text{Variance} = 22 $$
So the standard deviation is:
$$ \text{Standard Deviation} = \sqrt{22} $$
- Third Central Moment:
The third central moment can be derived from the third moment about a point using the formula:
$$ \text{Third Central Moment} = \text{Third Moment about 2} - 3 \cdot \text{Mean} \cdot \text{Variance} + 2 \cdot \text{Mean}^3 $$
Plugging in the values will provide the third central moment.
- Statistics is the science of collection, classification, and interpretation of numerical data.
- Mean = $\frac{n+1}{2}$, Variance = $\frac{(n(n+1)(2n+1))}{6n} - \left(\frac{n+1}{2}\right)^2$, Coefficient of Variation = $\frac{\sqrt{Variance}}{Mean} \times 100$
- Mean = $3$, Standard Deviation = $\sqrt{22}$, Third Central Moment = $10$.
More Information
Statistics is essential for making data-driven decisions in economics, allowing for the effective analysis of trends and evaluation of policies. In calculating statistics for natural numbers, we gain insights into fundamental properties like mean and variation.
Tips
- Confusing the formulas for population variance and sample variance.
- Forgetting to convert the moment about a point to central moments correctly.
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