(2)

The standard deviation of a distribution is given as 4. To make the distribution mesokurtic, with a kurtosis value of 3, what should be the value of the 4th central moment that will ensure that the distribution is mesokurtic

a 768

b 3

C 786

d Higher than 786.

To determine the value of the 4th central moment (\(\mu_4\)) that will ensure the distribution is mesokurtic (with a kurtosis value of 3), given a standard deviation (\(\sigma\)) of 4, we use the formula for kurtosis:

\[ \text{kurtosis} = \frac{\mu_4}{\sigma^4} \]

For a mesokurtic distribution, the kurtosis is 3. Given that the standard deviation (\(\sigma\)) is 4, we substitute these values into the formula:

\[ 3 = \frac{\mu_4}{4^4} \]

First, calculate \(4^4\):

\[ 4^4 = 4 \times 4 \times 4 \times 4 = 256 \]

Now, solve for \(\mu_4\):

\[ 3 = \frac{\mu_4}{256} \]

\[ \mu_4 = 3 \times 256 \]

\[ \mu_4 = 768 \]

Therefore, the 4th central moment that will ensure the distribution is mesokurtic is:

\[ a) 768 \]

Would you like more details or have any questions? Here are some questions you might want to ask:

1. What is the significance of kurtosis in a distribution?
2. How do you calculate the standard deviation of a dataset?
3. What are the different types of kurtosis (mesokurtic, leptokurtic, platykurtic)?
4. Can you explain the relationship between central moments and kurtosis?
5. What other moments are important in statistical analysis?

**Tip:** In statistics, the kurtosis of a distribution measures the "tailedness" of the distribution. A mesokurtic distribution has a kurtosis of 3, indicating it has similar tail behavior to the normal distribution.

Learn how to calculate the 4th central moment (μ_4) to achieve mesokurtic distribution with a kurtosis value of 3, given the standard deviation (σ) is 4. Detailed step-by-step solution provided.

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