Question

Which of the following values best represents the proportion of data within 2 standard deviations in a normal distribution?

• A. 0.90
• B. 0.68
• C. 0.95
• D. 0.93

Hint:

Remember the 68–95–99.7 rule: 1$\sigma$ → 68% \quad 2$\sigma$ → 95% \quad 3$\sigma$ → 99.7%.

Answer 1

The Correct Option is C

To solve this problem, we need to understand the properties of a normal distribution, specifically regarding the proportions of data that lie within certain standard deviations from the mean. 

A normal distribution is a probability distribution that is symmetric about the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. The standard deviation is a measure of the amount of variation or dispersion present in a dataset.

One of the key properties of a normal distribution is the empirical rule, also known as the 68-95-99.7 rule, which applies to all normal distributions:

1. Approximately 68% of the data falls within one standard deviation (\(\mu \pm \sigma\)) of the mean.
2. Approximately 95% of the data falls within two standard deviations (\(\mu \pm 2\sigma\)) of the mean.
3. Approximately 99.7% of the data falls within three standard deviations (\(\mu \pm 3\sigma\)) of the mean.

Given the options, the proportion of data that falls within two standard deviations of the mean is approximately 0.95, as per the empirical rule. Let's rule out the other options:

• 0.90: This option is incorrect as it does not align with the empirical rule for two standard deviations (which is 0.95).
• 0.68: This corresponds to one standard deviation, not two, so this is incorrect.
• 0.93: Although this value is close, the empirical rule specifies 0.95 as the correct proportion for two standard deviations.

Thus, the correct answer is indeed 0.95, which aligns with the empirical rule for data within two standard deviations in a normal distribution.

Standard Deviations	Proportion of Data
1 standard deviation (\(\mu \pm \sigma\))	68%
2 standard deviations (\(\mu \pm 2\sigma\))	95%
3 standard deviations (\(\mu \pm 3\sigma\))	99.7%