1. The Normal Distribution
Unit 6C
The Normal Distribution
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2. The Normal Distribution
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.
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3. Conditions for a Normal Distribution
A data set satisfying the following criteria is likely to have a nearly normal distribution.
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 are increasingly rare, producing the tapering tails of the distribution.
4. Individual data values result from a combination of many different factors.
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4. The 68-95-99.7 Rule for a Normal Distribution
About 68.3% of the data points fall within 1 standard deviation of the mean. About 95.4% of the data points fall within 2 standard deviation points of the mean. About 99.7% of the data points fall within 3 standard deviation points of the mean. This is a wonderful opportunity to talk about what makes a distribution normal. The authors suggest in the reading that when any quantity is a result of many factors, such as genetic or environmental, there is a definite tendency for normal distribution. Apply the rule to male or female heights of your own class or some other quantitative variable that results from many factors to check out that the percentages are reasonable.
Ms. Young

5. Standard Scores
The number of standard deviations that a data value lies above or below the mean is called its standard score (or z-score), defined by
Stress with the students that the formula should not be a mystery to them but rather it simply tells us how many standard deviations the data value is away from the mean.
Data Value
above the mean
below the mean →
Standard Score
positive
negative
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6. Standard Scores
Example: If the mean were 21 with a standard deviation of 4.7 for scores on a nationwide test, find the z-score for a 30. What does this mean?
This means that a test score of 30 would be about 1.91 standard deviations above the mean of 21.
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7. Standard Scores and Percentiles
The nth percentile of a data set is the smallest value in the set with the property that n% of the data are less than or equal to it.
A data value that lies between two percentiles is said to lie in the lower percentile.
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8. Standard Scores and Percentiles
Work through some simple concrete examples to demonstrate how the table works. Try different scenarios that require both adding and subtracting percentages. It may be helpful to emphasize the importance of sketching out a normal curve in the problem-solving process in order to visual what it is we are actually doing.
Ms. Young