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5.1 What Is Normal?
LEARNING GOAL
Understand what is meant by a normal distribution and be able to identify situations in which a normal distribution is likely to arise.
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Suppose a friend is pregnant and due to give birth on June 30. Would you advise her to schedule an important business meeting for June 16, two weeks before the due date?
Figure 5.1 is a histogram for a distribution of 300 natural births. The left vertical axis shows the number of births for each 4-day bin. The right vertical axis shows relative
frequencies.
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Figure 5.1
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We can find the proportion of births that occurred more than 14 days before the due date by adding the relative frequencies for the bins to the left of -14.
These bins have a total relative frequency of about 0.21, which says that about 21% of the births in this data set occurred more than 14 days before the due date.
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Figure 5.1
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TIME OUT TO THINK
Suppose the friend plans to take a three-month maternity leave after the birth. Based on the data in Figure 5.1 and assuming a due date of June 30, should she promise to be at work on October 10?
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The Normal Shape
The distribution of the birth data has a fairly distinctive shape, which is easier to see if we overlay the histogram with a smooth curve (Figure 5.2).
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For our present purposes, the shape of this smooth distribution has three very important characteristics:
The distribution is single-peaked. Its mode, or most common birth date, is the due date.
• The distribution is symmetric around its single peak; therefore, its median and mean are the same as its mode. The median is the due date because equal numbers of births occur before and after this date. The mean is also the due date because, for every birth before the due date, there is a birth the same number of days after the due date.
• The distribution is spread out in a way that makes it resemble the shape of a bell, so we call it a “bell-shaped” distribution.
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TIME OUT TO THINK
The histogram in Figure 5.2, which is based on natural births, is fairly symmetric. Today, doctors usually induce birth if a woman goes too far past her due date. How would the shape of the histogram change if it included induced labor births?
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Figure 5.3 Both distributions are normal and have the same mean of 75, but the distribution on the left has a larger standard deviation.
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Definition
The normal distribution is a symmetric, bell-shaped distribution with a single peak. Its peak corresponds to the mean, median, and mode of the distribution. Its variation can be characterized by the standard deviation of the distribution.
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The Normal Distribution and Relative Frequencies
Relative Frequencies and the Normal Distribution
The area that lies under the normal distribution curve corresponding to a range of values on the horizontal axis is the relative frequency of those values.
Because the total relative frequency must be 1, the total area under the normal distribution curve must equal 1, or 100%.
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Figure 5.5 The percentage of the total area in any region under the normal curve tells us the relative frequency of data values in that region.
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TIME OUT TO THINK
According to Figure 5.5 (previous slide), what percentage of births occur between 14 days early and 18 days late? Explain. (Hint: Remember that the total area under the curve is 100%.)
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EXAMPLE 2 Estimating Areas
Look again at the normal distribution in Figure 5.5 (slide ).
Estimate the percentage of births occurring between 0 and 60
days after the due date.
Solution:
About half of the total area under the curve lies in the region
between 0 days and 60 days. This means that about 50% of the births in the sample occur between 0 and 60 days after
the due date.
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EXAMPLE 2 Estimating Areas
Look again at the normal distribution in Figure 5.5 (slide ).
Estimate the percentage of births occurring between 14 days before and 14 days after the due date.
Solution:
Figure 5.5 shows that about 18% of the births occur more than 14 days before the due date. Because the distribution is symmetric, about 18% must also occur more than 14 days after the due date. Therefore, a total of about 18% 18% 36% of births occur either more than 14 days before or more than 14 days after the due date. The question asked about the remaining region, which means between 14 days before and 14 days after the due date, so this region must represent 100% - 36% = 64% of the births.
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When Can We Expect a Normal Distribution?
Conditions for a Normal Distribution
A data set that satisfies the following four criteria is likely
1. Most data values are clustered near the mean, giving the distribution a well-defined single peak.
2. Data values are spread evenly around the mean, making the distribution symmetric.
3. Larger deviations from the mean become increasingly rare, producing the tapering tails of the distribution.
4. Individual data values result from a combination of many different factors, such as genetic and environmental factors.
to have a nearly normal distribution:
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EXAMPLE 3 Is It a Normal Distribution?
Which of the following variables would you expect to have a normal or nearly normal distribution?
a. Scores on a very easy test
Solution:
a. Tests have a maximum possible score (100%) that limits the size of data values. If the test is easy, the mean will be high and many scores will be close to the maximum possible. The few lower scores may be spread out well below the mean. We therefore expect the distribution of scores to be left-skewed and non-normal.
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EXAMPLE 3 Is It a Normal Distribution?
Which of the following variables would you expect to have a normal or nearly normal distribution?
b. Heights of a random sample of adult women
Solution:
b. Height is determined by a combination of many factors (the genetic makeup of both parents and possibly environmental or nutritional factors). We expect the mean height for the sample to be close to the mode (most common height). We also expect there to be roughly equal numbers of women above and below the mean, and extremely large and small heights should be rare. That is why height is nearly normally distributed.
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TIME OUT TO THINK
Would you expect scores on a moderately difficult exam to have a normal distribution? Suggest two more quantities that you would expect to be normally distributed.
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The End
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