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**The Normal Distribution**

Prepared by E.G. Gascon

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**Properties of Normal Distribution**

Peak Image text page 487 It’s peak occurs directly above the mean The curve is symmetric about the vertical line through the mean. The curve never touches the x-axis The area under the curve is always = 1. (This agrees with the fact that the sum of the probabilities in any distribution is 1.)

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**Variations in Normal Curves**

One standard deviation is smaller than normal One standard deviation is equal to the normal One standard deviation is larger than normal

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**The Area Under the Standard Normal Curve**

1 standard deviation A B Image from text p 487 The area of the shaded region under a normal curve form a point A to B is the probability that an observed data value will be between A and B Between -1 and +1 standard deviations there is 68% of the region, therefore the probability of an observed data value being within 1 standard deviation is 68%, etc.

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**Problem solved using the Standard Normal Curve**

The area under a normal curve to the left of x (the data) is the same as the area under the standard normal curve to the left of the z-score for x. What does that mean? The z-score is the formula that converts the raw data (x) from a normal distribution into the lookup values of a STANDARD NORMAL CURVE. [See table in appendix of text or use Excel function =NORMSDIST(Z)] First find the z-score Example: sales force drives an average of 1200 miles, with a standard deviations of 150 miles miles is the mileage in question.

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**What is the probability that a salesperson drives less than 1600 miles?**

Ans: It is the area to the left of the standard normal curve. Look up 2.67 in the Table of Normal Distributions. There is a 99.62% probability that the salesperson drives less than 1600 miles. 2.67

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**Using Table of the Normal Distribution**

Z = 2.67 Table found in text page A-1 back of book Look up 2.6 in the row, and .07 in the column. The intersection is the area to the left, or probability

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Or Use Excel function Enter: Results:

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**What is the probability that a salesperson drives more than 1600 miles?**

2.67 Ans: It is the area to the right of the standard normal curve. Since you know the are to the left of 2.67, the area to the right must be = .0038, or .38% probability that a salesperson drives more than 1600 miles.

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**What is the probability that a salesperson drives between 1200 and 1600 miles?**

It is the difference between driving less than 1600 and less than 1200. 2.67 Ans: The area to the left of 2.67 is already known, it is Find the z value for 1200, , then look it up in the table. Between = = .4962 The probability that a salesperson drives between 1200 and 1600 miles is 49.62%

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**Questions / Comments / Suggestions**

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Normal Probability Distributions Chapter 5. § 5.1 Introduction to Normal Distributions and the Standard Distribution.

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