Chapter 7:The Normal Distribution

Important Properties of a Normal Distribution

• The mean, median, and mode have the same value, which is located exactly in the center of the distribution.

• The total area under the Normal curve equals 1. The area to the left of the mean is .5 and the area to the right of the mean is .5.

• Data that lie beyond two standard deviations from the mean are rare, and data that lie beyond three standard deviations from the mean are very rare.

• Recall that a Normal distribution is roughly symmetric, unimodal, and bell-shaped.

• The graphical display of a Normal distribution is a Normal curve.

• Many variables approximate the Normal distribution.• The mean and standard deviation are the two most

important parameters for any normal distribution.

• We know approximately what percent of data fall exactly 1, 2, and 3 standard deviations from the mean. However, how can we find a percent if a value does not fall exactly 1, 2, or 3 standard deviations from the mean.

• The first thing we have to do is standardize our score(s). This is referred to as finding the z-score.

Z-Scores• A z-score is a measure of the relative position of a

data item in terms of the number of standard deviations it is from the mean.

• Z-scores can be negative, positive, or 0.• A positive z-score would indicate the original

value is above the mean. For example, a z-score of 1.24 would mean that the score is 1.24 standard deviations above the mean.

• A negative z-score would indicate the original value is below the mean. For example, a z-score of -0.27 would mean that the score is 0.27 standard deviations below the mean.

• A z-score of 0 would indicate that the original value was the same as the mean.

• Question: Are all negative z-scores bad?

Example: A distribution is approximately Normal with a mean of 26 and a standard deviation of 7. Calculate and interpret the z-score for a value of 21.

The value 21 is 0.71 standard deviations below the mean.

Example: A distribution is approximately Normal with a mean of 100 and a standard deviation of 10. Calculate and interpret the z-score for a value of 117.

The value 117 is 1.7 standard deviations above the mean.

Example: Seth recently took two tests in school. On his history test he scored a 75. The class average on the test was a 63 and the test had a standard deviation of 3. On his biology test, Seth scored a 81. The class average on the test was a 76 and the test had a standard deviation of 7. Relatively speaking, on which test did he perform better?

He performed better on his history test as he was more standard deviations above the class mean.

• In order to find the area under the Normal curve (percentage) for values that do not fall exactly 1, 2, or 3 standard deviations from the mean, we need to utilize the standard Normal table.

• Once scores have been standardized, the mean becomes 0.

Steps to Use the standard Normal table1) Draw the Normal curve and label the mean of 0 along with 3 standard deviations in each direction.2) Standardize the score(s) of interest.3) Plot the z-score(s), draw a vertical line(s), and shade the area of interest.4) Look up the z-score on the standard Normal table.– If the area of interest is shaded to the left, the value in

the table is the desired area.– If the area of interest is shaded to the right, we need to

subtract the area in the table from 1.– If the area of interest is shaded between two z-scores, we

need to look up the area for both z-scores and subtract.

Example A: A data set is Normally distributed with a mean of 259 and a standard deviation of 74. Find the area under the curve less than a score of 180.

• We have to look up the z-score of -1.07 on the table.

• Since the score starts out as “-1.0”, go to the z column and go down until you reach -1.0.

• Since there is a 7 in the hundredths place (.07), go to the right until you reach the .07 column. You should now be located in the spot that has -1.0 to the left, and .07 on the top.

This value should be .1423.

Example B: A data set is Normally distributed with a mean of 259 and a standard deviation of 74. Find the area under the curve less than a score of 350.

Since the z-score is 1.23, look for 1.2 in the z column and .03 on top. This should give us an area of .8907

Example C: A data set is Normally distributed with a mean of 26 and a standard deviation of 2.4. Find the area under the curve more than a score of 29.

A z-score of 1.25 on the table gives an area of .8944. However, this is the area to the LEFT of the score. We want the area to the right of the score. Since the area under the Normal curve totals 1, we need to subtract .8944 from 1. This gives us our desired area, which is .1056.

Example D: A data set is Normally distributed with a mean of 26 and a standard deviation of 2.4. Find the area under the curve above a score of 21.

1-.0188=.9812

Example E: In the 2008 Wimbledon tennis tournament, Rafael Nadal averaged 115 miles per hour on his first-serves. Assuming that the distribution of his first-serve speeds is Normal with a standard deviation of 6 mph, find what proportion of his first-serves you would expect to be between 110 and 125 mph.

Look up both areas and subtract..9525-.2033=.7492

Example F: On the 2009 PGA tour, Tiger Woods had an average driving distance of 298 yards. Assuming that the distribution of his driving distances is Normal with a standard deviation of 12 yards, find what proportion of his drives you would expect to be between 290 and 310 yards.

.8413-.2514=.5899