Question

Rough Rule of Thumb for the Standard Deviation According to the \(95 \%\) rule, the largest value in a sample from a distribution that is approximately symmetric and bell-shaped should be between 2 and 3 standard deviations above the mean, while the smallest value should be between 2 and 3 standard

Short answer

For a distribution that is approximately symmetric and bell-shaped, according to the \(95\%\) rule, the largest value should be between \(µ + 2σ\) and \(µ + 3σ\), while the smallest value should be between \(µ - 2σ\) and \(µ - 3σ\), where 'µ' is the mean and '\(σ\)' is the standard deviation.

Step-by-step solution

Understanding the 95% Rule for Standard Deviation

According to the \(95\%\) rule, which is also known as the Empirical Rule or the Normal Distribution Rule, approximately \(95\%\) of the data values in a symmetric and bell-shaped distribution (also known as a normal distribution) will fall within 2 standard deviations of the mean.

Apply the 95% Rule to Determine the Range for the Largest Value

Since the largest value in the sample should be between 2 and 3 standard deviations above the mean, we find this range by adding 2 and 3 times the standard deviation to the mean. If 'µ' is the mean and '\(σ\)' is the standard deviation, then the largest value should fall in the range \(µ + 2σ\) to \(µ + 3σ\).

Apply the 95% Rule to Determine the Range for the Smallest Value

Similarly, since the smallest value in the sample should be between 2 and 3 standard deviations below the mean, we find this range by subtracting 2 and 3 times the standard deviation from the mean. Again, if 'µ' is the mean and '\(σ\)' is the standard deviation, then the smallest value should fall in the range \(µ - 2σ\) to \(µ - 3σ\).