If the variance of 3+x, 6, 4, x and 7-x is 4 and the mean is 5, find the standard deviation

Analysis

To solve this problem, we need to understand the relationship between variance and standard deviation. In statistics, the standard deviation is simply the square root of the variance.

Concept

• Variance ($\sigma^2$): A measure of how spread out a set of numbers is from their mean.
• Standard Deviation ($\sigma$): The square root of the variance ($\sigma = \sqrt{\text{Variance}}$).

The question provides the variance directly as 4. Even though the question gives us the data set and the mean, we do not actually need to perform complex calculations on the variables $x$ because the definition of standard deviation is independent of the specific values once the variance is known.

Step-by-Step Solution

1. Identify the given information: • Variance ($\sigma^2$) = 4 • Mean = 5 (Note: This information is extra and not required to find the standard deviation since the variance is already provided).

2. Apply the formula: • Standard Deviation ($\sigma$) = $\sqrt{\text{Variance}}$ • $\sigma = \sqrt{4}$

3. Calculate: • $\sigma = 2$

Verification

The extra information provided in the question (the data set and the mean) is often included in JAMB questions to test if a student can identify the necessary information versus "distractor" information.

Exam Tip

Always check if the question asks for the variance or the standard deviation. If you are given the variance and asked for the standard deviation, just take the square root. If you are given the standard deviation and asked for the variance, just square it. Don't waste time calculating the mean or the variance of the variables if the variance is already explicitly stated!