Using the Normal Distribution applet, you explored the area under the normal curve. The area under the normal curve within a specified number of standard deviations of the mean will always be the same.

These three areas are sometimes called the 68.3 - 95.4 - 99.7 rule. The areas of different regions, using this rule, can be shown as follows.

The area of a region under the normal curve corresponds to the probability that a data value from a set of normally distributed data will lie in that region.

Example 1

Determine the area under a normal curve between two standard deviations below the mean and one standard deviation above the mean.

Begin by sketching a diagram to represent the scenario.

0.136 + 0.341 + 0.341 = 0.818

The area between two standard deviations below the mean and one standard deviation above the mean is 0.818.

Example 2

A pea soup company is canning their product and the amount of soup in each can is normally distributed. If the mean amount of soup per can is 800 mL with a standard deviation of 2 mL, determine the percentage of cans that will have between 796 and 806 mL of soup.

Begin by drawing a diagram to represent the problem. The mean is 800 so it is at the centre of the normal curve and the standard deviation is 2 so count by 2's to show standard deviation increments on either side of the mean.

Look up the area for each of the included regions.

0.136 + 0.341 + 0.341 + 0.136 + 0.021 = 0.975

So, 97.5% of soup cans will hold between 796 and 806 mL of soup.