Normal Distribution

Objectives The student will be able to:

identify properties of normal distribution apply mean, standard deviation, and z-

scores to the normal distribution graph

determine probabilities based on z-scores

Length of Right Foot

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Suppose we measured the right foot length of 30 teachers and graphed the results.

Assume the first person had a 10 inch foot. We could create a bar graph and plot that person on the graph.

If our second subject had a 9 inch foot, we would add her to the graph.

As we continued to plot foot lengths, a pattern would begin to emerge.

Length of Right Foot

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If we were to connect the top of each bar, we would create a frequency polygon.

Notice how there are more people (n=6) with a 10 inch right foot than any other length. Notice also how as the length becomes larger or smaller, there are fewer and fewer people with that measurement. This is a characteristics of many variables that we measure. There is a tendency to have most measurements in the middle, and fewer as we approach the high and low extremes.

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4 5 6 7 8 9 10 11 12 13 14Length of Right Foot

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You will notice that if we smooth the lines, our data almost creates a bell shaped curve.

This bell shaped curve is known as the “Bell Curve” or the “Normal Curve.”

Whenever you see a normal curve, you should imagine the bar graph within it.

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Points on a Quiz

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The mean, mode, and median

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Points on a Quiz

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will all fall on the same value in a normal distribution.

Now lets look at quiz scores for 51 students.

867 / 51 = 17

Normal Distribution Curve

A normal distribution curve is symmetrical, bell-shaped curve defined by the mean and standard deviation of a data set.

The normal curve is a probability distribution with a total area under the curve of 1.

The normal distribution and standard deviations

The total area under the curve is 1.

In a normal distribution:

34% 34%

13.5% 13.5% 2.35%2.35%

Normal distributions (bell shaped) are a family of distributions that have the same general shape. They are symmetric (the left side is an exact mirror of the right side) with scores more concentrated in the middle than in the tails. Examples of normal distributions are shown to the right. Notice that they differ in how spread out they are. The area under each curve is the same.

Standard Normal Distribution

A standard normal distribution is the set of all z-scores.

The mean of the data in a standard normal distribution is 0 and the standard deviation is 1.

z-scores

When a set of data values are normally distributed, we can standardize each score by converting it into a z-score.

z-scores make it easier to compare data values measured on different scales.

z-scores

A z-score reflects how many standard deviations above or below the mean a raw score is.

The z-score is positive if the data value lies above the mean and negative if the data value lies below the mean.

z-score formula

Where x represents an element of the data set, the mean is represented by and standard deviation by

.

Analyzing the data

Suppose SAT scores among college students are normally distributed with a mean of 500 and a standard deviation of 100. If a student scores a 700, what would be her z-score?

Analyzing the dataSuppose SAT scores among college students are normally distributed with a mean of 500 and a standard deviation of 100. If a student scores a 700, what would be her z-score?

700 5002

100z

Her z-score would be 2 which means her score is two standard deviations above the mean.

Analyzing the data

• A set of math test scores has a mean of 70 and a standard deviation of 8.

• A set of English test scores has a mean of 74 and a standard deviation of 16.

For which test would a score of 78 have a higher standing?

Analyzing the data

78-70math -score = 1

8z

To solve: Find the z-score for each test.

A set of math test scores has a mean of 70 and a standard deviation of 8. A set of English test scores has a mean of 74 and a standard deviation of 16. For which test would a score of 78 have a higher standing?

The math score would have the highest standing since it is 1 standard deviation above the mean while the English score is only .25 standard deviation above the mean.

Analyzing the data

What will be the miles per gallon for a Toyota Camry when the average mpg is 23, it has a z value of 1.5 and a standard deviation of 5?

Analyzing the data

What will be the miles per gallon for a Toyota Camry when the average mpg is 23, it has a z value of 1.5 and a standard deviation of 5?

The Toyota Camry would be expected to use 30.5 mpg of gasoline.

xz

Using the formula for z-scores: