Standard Deviation Rule

Question

The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 55 ounces and a standard deviation of 3 ounces.

Use the Standard Deviation Rule, also known as the Empirical Rule.

Suggestion: sketch the distribution in order to answer these questions.

a) 95% of the widget weights lie between _____ oz and _____ oz

b) What percentage of the widget weights lie between 52 and 61 ounces? ___________%

c) What percentage of the widget weights lie below? ___________%

Answer 1

The empirical rule states that about 68% of the distribution lies between -1 and 1 standard deviations from the mean and that about 95% of the distribution lies between -2 and 2 standard deviations.

Label a bell curve from SD = -2 to SD = 2:

49, 52, 55 (mean), 58, 61

a) Since 49 is -2 SD (2 SD below the mean) and 61 is 2 SD (2 SD above the mean), 95% of the distribution (widget weights) lie between 49 and 61.

b) Since 68% of the distribution lies between -1 SD and 1 SD, we can use the symmetry of the distribution to show that half of 68% = 34% of the distribution lies between -1 SD and 0 SD. So 34% of the distribution lies between 52 and 55. Similarly, since 95% of the distribution lies between -2 SD and 2 SD, half of 95% = 47.5% will lie between 0 and 2 SD. So 47.5% of the distribution lies between 55 and 61. Adding 34% and 47.5% we find that 81.5% of the widget weights lie between 52 and 61.