6.5. Empirical Rule http://www.ck12.org
of the mean. This is referred to as theEmpirical Rule, which is also known as the 68-95-99.7 Rule. To accommodate
the percentages given by the Empirical Rule, there are defined values in each of the regions to the left and to the
right of the mean.
These percentages are used to answer real-world problems when both the mean and the standard deviation of a data
set are known. Also keep in mind that since 99.7% of the data in a normal distribution is within 3 standard deviations
of the mean, 1− 99 .7%= 0 .3% of the data does not fall within 3 standard deviations of the mean. This means that
0 .3%
2 =^0 .15% of the data is beyond 3 standard deviations on either end of the normal distribution curve. This is not
shown in the figure above.
Example A
The lifetimes of a certain type of light bulb are normally distributed. The mean life is 400 hours, and the standard
deviation is 75 hours. For a group of 5,000 light bulbs, how many are expected to last each of the following times?
a) between 325 hours and 475 hours
b) more than 250 hours
c) less than 250 hours
a) 68% of the light bulbs are expected to last between 325 hours and 475 hours. This means that 5, 000 × 0. 68 = 3 , 400
light bulbs are expected to last between 325 and 475 hours.
b) 95%+ 2 .35%+ 0 .15%= 97 .5% of the light bulbs are expected to last more than 250 hours. This means that
5 , 000 × 0. 975 = 4 ,875 of the light bulbs are expected to last more than 250 hours.
c) Only 2.35%+ 0 .15%= 2 .5% of the light bulbs are expected to last less than 250 hours. This means that 5, 000 ×
0. 025 =125 of the light bulbs are expected to last less than 250 hours.
Example B
A bag of chips has a mean mass of 70 g, with a standard deviation of 3 g. Assuming a normal distribution, create a
normal curve, including all necessary values.
a) If 1,250 bags of chips are processed each day, how many bags will have a mass between 67 g and 73 g?
b) What percentage of the bags of chips will have a mass greater than 64 g?