The Normal Curve

1XX1X 2X2X3X 3X

It’s a frequency distribution that often occurs when there is a large number of values in a data set.

The graph is a symmetric, bell-shaped curve known as the normal curve.

Most of the data occurs around the mean.

A small portion of the data occurs in the tails.

Properties of a Normal Distribution•The maximum point of the curve is at the mean.

•The curve extends indefinitely far to the left and right and approaches the x-axis.

•With a large standard deviation the curve will be flat.

•With a small standard deviation the curve will be tall.

•The total area under the curve is 1.

•The curve is symmetric about the mean.

•About 68% of the data are within 1 standard deviations from the mean.

•About 95% of the data are within 2 standard deviations from the mean.

•About 99.7% of the data are within 3 standard deviations from the mean.

No matter the shape of the bell curve, the area under it is the same.

ExampleSuppose the mean of a set of data is 60 and the standard deviation is 6.

Boundaries: 1x 1x to

60 1 6 ( ) 60 1 6 ( )to

to54 66

54 6660

68%

68% of the values in this set of data lie within 1 standard deviation of 60, that is, between 54 and 66.

If you randomly select one item from the sample, the probability that the one you pick will be between 54 and 66 is 0.68.

If you repeat this process 1000 times, approximately 68% of those selected will be between 54 and 66.

Ex. 1The lifetimes of 10,000 watch batteries are normally distributed.

Sketch a normal curve that represents the frequency distribution of lifetimes of the batteries.

The standard deviation is 60 days. The mean is 500 days.

Ex. 2Suppose the scores of 500 college freshmen taking Psychology 101 are normally distributed.

Sketch a normal curve that represents the frequency distribution of scores.

The standard deviation is 10. The mean score is 60%

Ex. 3Find the upper and lower limits of the interval about the mean in which 68%, 95%, and 99.7% of the values of a set of normally distributed data can be found if the mean is 124 and σ is 16.

Ex. 4

How many batteries will last between 440-560 days?

How many batteries will last between 380-620 days?

How many batteries will last between 320-680 days?

The lifetimes of 10,000 watch batteries are normally distributed.

The standard deviation is 60 days. The mean is 500 days.

More Examples

• What percent of batteries will last between 500 and 560 days?

• What percent of batteries will last longer than 620 days?

• What percent of batteries last 320 days at the most?