Question

fill in each blank so that the resulting statement is true.
in a normal distribution, approximately____% of the data items fall within 1 standard deviation of the mean, approximately__% of the data items fall within 2 standard deviations of the mean, and approximately____% of the data items fall within 3 standard deviations of the mean.
in a normal distribution, approximately% of the data items fall within 1 standard deviation of the mean, approximately% of the data items fall within 2 standard deviations of the mean, and approximately% of the data items fall within 3 standard deviations of the mean

Explanation:

In a normal distribution, the empirical rule (also known as the 68 - 95 - 99.7 rule) is a key concept. This rule provides approximate percentages of data within a certain number of standard deviations from the mean.

• For 1 standard deviation: By the empirical rule, approximately 68% of the data lies within \( \mu\pm\sigma\) (where \(\mu\) is the mean and \(\sigma\) is the standard deviation).
• For 2 standard deviations: Approximately 95% of the data lies within \( \mu\pm2\sigma\).
• For 3 standard deviations: Approximately 99.7% of the data lies within \( \mu\pm3\sigma\).

Answer:

In a normal distribution, approximately \(68\%\) of the data items fall within 1 standard deviation of the mean, approximately \(95\%\) of the data items fall within 2 standard deviations of the mean, and approximately \(99.7\%\) of the data items fall within 3 standard deviations of the mean.