Understanding Standard Deviation and Variance in Statistics

The question asks us to find the variance of a distribution given its standard deviation. Standard deviation and variance are both important measures of dispersion in statistics, telling us how spread out the data points are from the mean.

The relationship between standard deviation and variance is straightforward. The variance is simply the square of the standard deviation.

The formula relating variance and standard deviation is:

\(\text{Variance} = (\text{Standard Deviation})^2\)

This can also be written using symbols:

\(\sigma^2 = \sigma^2\)

Where:

• \(\sigma^2\) represents the variance
• \(\sigma\) represents the standard deviation

In this question, we are given that the standard deviation of the distribution is 9.

Given:

Standard Deviation (\(\sigma\)) = 9

We need to find the variance (\(\sigma^2\)).

Calculating the Variance

Using the formula, we can calculate the variance by squaring the given standard deviation:

\(\text{Variance} = (\text{Standard Deviation})^2\)

\(\text{Variance} = (9)^2\)

\(\text{Variance} = 9 \times 9\)

\(\text{Variance} = 81\)

So, the value of the variance is 81.

Now, let's look at the given options to find the one that matches our calculated variance:

• A. 18
• B. 27
• C. 81
• D. 36

Our calculated variance is 81, which corresponds to option C.

Revision Table: Standard Deviation vs. Variance

Additional Information on Measures of Dispersion

Standard deviation and variance are part of a group of statistical measures called measures of dispersion or measures of variability. These measures help us understand the spread or variability of a dataset.

Other common measures of dispersion include:

• Range: The difference between the maximum and minimum values in a dataset.
• Interquartile Range (IQR): The range of the middle 50% of the data, calculated as the difference between the third quartile (Q3) and the first quartile (Q1).

Understanding dispersion is crucial in statistics as it complements measures of central tendency (like mean, median, mode) by providing a complete picture of the dataset's characteristics.