Understanding Variance and Standard Deviation

This question asks us to find the variance of a population given its standard deviation. Standard deviation and variance are both measures of the spread or dispersion of a dataset. They tell us how much individual data points typically deviate from the mean of the dataset.

Relationship Between Variance and Standard Deviation

The relationship between variance and standard deviation is quite simple and direct. The variance is the square of the standard deviation. Conversely, the standard deviation is the square root of the variance.

The formula relating them is:

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

or

Standard Deviation $= \sqrt{\text{Variance}}$

Calculating the Variance

We are given that the standard deviation of the population is 5.

Using the formula, we can calculate the variance:

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

Variance $= (5)^2$

Variance $= 5 \times 5$

Variance $= 25$

Therefore, the variance of the population is 25.

Comparing with Options

Let's look at the given options:

• A. 10
• B. 15
• C. 25
• D. 12.5

Our calculated variance is 25, which matches option C.

Calculation steps:

1. Identify the given measure: Standard Deviation = 5.
2. Recall the relationship: Variance = (Standard Deviation)$^2$.
3. Substitute the value: Variance = $(5)^2$.
4. Calculate the square: Variance = 25.

Revision Table: Key Concepts in Statistics

Additional Information: Why Use Both Standard Deviation and Variance?

Both variance and standard deviation measure dispersion, but they are used in different contexts.

• Variance: The calculation of variance involves squaring the differences from the mean. This makes larger deviations contribute more significantly to the variance. Variance is often used in statistical tests and theoretical calculations (like ANOVA) because its mathematical properties are convenient. However, variance is in squared units, which can make it difficult to interpret in the context of the original data.
• Standard Deviation: By taking the square root of the variance, the standard deviation brings the measure of spread back into the original units of the data. This makes it easier to understand how much the data points typically vary from the mean. For example, if the data is in meters, the standard deviation will also be in meters. This makes standard deviation a more commonly reported measure of dispersion in descriptive statistics.

Understanding the simple squared relationship between variance and standard deviation is fundamental in statistics.