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CHAPTER 6

The Normal Distribution

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Objectives

Identify distributions as symmetrical or skewed.

Identify the properties of the normal distribution.

Find the area under the standard normal distribution, given various z values.

Find the probabilities for a normally distributed variable by transforming it into a standard normal distribution.

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Objectives (cont’d.)

Find specific data values for given percentages using the standard normal distribution.

Use the central limit theorem to solve problems involving sample means for large and small samples.

Use the normal approximation to compute probabilities for a binomial variable.

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Introduction

Many continuous variables have distributions that are bell-shaped and are called approximately normally distributed variables.

A normal distribution is also known as the bell curve or the Gaussian distribution.

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Normal and Skewed Distributions

The normal distribution is a continuous, bell-shaped distribution of a variable.

If the data values are evenly distributed about the mean, the distribution is said to be symmetrical.

If the majority of the data values fall to the left or right of the mean, the distribution is said to be skewed.

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Left Skewed Distributions

When the majority of the data values fall to the right of the mean, the distribution is said to be negatively or left skewed. The mean is to the left of the median, and the mean and the median are to the left of the mode.

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Right Skewed Distributions

When the majority of the data values fall to the left of the mean, the distribution is said to be positively or right skewed. The mean falls to the right of the median and both the mean and the median fall to the right of the mode.

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Equation for a Normal Distribution

The mathematical equation for the normal distribution is:

ye

X

( )

2 22

2

where e 2.718 3.14 = population mean = population standard deviation

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Properties of the Normal Distribution

The shape and position of the normal distribution curve depend on two parameters, the mean and the standard deviation.

Each normally distributed variable has its own normal distribution curve, which depends on the values of the variable’s mean and standard deviation.

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Normal Distribution Properties

The normal distribution curve is bell-shaped.

The mean, median, and mode are equal and located at the center of the distribution.

The normal distribution curve is unimodal (i.e., it has only one mode).

The curve is symmetrical about the mean, which is equivalent to saying that its shape is the same on both sides of a vertical line passing through the center.

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Normal Distribution Properties (cont’d.)

The curve is continuous—i.e., there are no gaps or holes. For each value of X, here is a corresponding value of Y.

The curve never touches the x axis. Theoretically, no matter how far in either direction the curve extends, it never meets the x axis—but it gets increasingly closer.

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Normal Distribution Properties (cont’d.)

The total area under the normal distribution curve is equal to 1.00 or 100%.

The area under the normal curve that lies within one standard deviation of the mean is approximately 0.68, or 68%; within two standard deviations, about 0.95, or 95%; and within three standard deviations, about 0.997 or 99.7%.

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Standard Normal Distribution

Since each normally distributed variable has its own mean and standard deviation, the shape and location of these curves will vary. In practical applications, one would have to have a table of areas under the curve for each variable. To simplify this, statisticians use the standard normal distribution.

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

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z Values

The z value is the number of standard deviations that a particular X value is away from the mean. The formula for finding the z value is:

z zX value mean

standard deviation or

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Area Between 0 and z

To find the area between 0 and any z value: Look up the z value in the table.

0 z

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Area in Any Tail

Look up the z value to get the area.

Subtract the area from 0.5000.

0 –z

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Area Between Two z Values

Look up both z values to get the areas.

Subtract the smaller area from the larger area.

0 z2z1

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Area Between z Values—Opposite Sides

Look up both z values to get the areas.

Add the areas.

0 z2 –z

1

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Area To the Left of Any z Value

Look up the z value to get the area.

Add 0.5000 to the area.

0 z

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Area To the Right of Any z Value

Look up the z value in the table to get the area.

Add 0.5000 to the area.

0-z

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Area Under the Curve

The area under the curve is more important than the frequencies because the area corresponds to the probability!

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Calculating the Value of X

When one must find the value of X, the following formula can be used:

X z

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Distribution of Sample Means

A sampling distribution of sample means is a distribution obtained by using the means computed from random samples of a specific size taken from a population.

Sampling error is the difference between the sample measure and the corresponding population measure due to the fact that the sample is not a perfect representation of the population.

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Properties of Distribution of Sample Means

The mean of the sample means will be the same as the population mean.

The standard deviation of the sample means will be smaller than the standard deviation of the population, and will be equal to the population standard deviation divided by the square root of the sample size.

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The Central Limit Theorem

As the sample size n increases, the shape of the distribution of the sample means taken with replacement from a population with mean and standard deviation will approach a normal distribution.

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Central Limit Theorem (cont’d.)

If all possible samples of size n are taken with replacement from the same population, the mean of the sample means equals the population mean or: .

The standard deviation of the sample means

equals: and is called the standard

error of the mean.

X

X n

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Central Limit Theorem (cont’d.)

The central limit theorem can be used to answer questions about sample means in the same manner that the normal distribution can be used to answer questions about individual values.

A new formula must be used for the z values:

zX

n

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Finite Population Correction Factor

The formula for standard error of the mean is accurate when the samples are drawn with replacement or are drawn without replacement from a very large or infinite population.

A correction factor is necessary for computing the standard error of the mean for samples drawn without replacement from a finite population.

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Finite Population Correction Factor

The correction factor is computed using the following formula:

where N is the population size and n is the sample size.

N nN

1

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Correction Factor Applied to Standard Error

The standard error of the mean must be multiplied by the correction factor to adjust it for large samples taken from a small population.

X n

N nN

1

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Correction Factor Applied to z Value

The standard error for the mean must be adjusted when it is included in the formula for calculating the z values.

zX

nN nN

1

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A Correction for Continuity

A correction for continuity is a correction employed when a continuous distribution is used to approximate a discrete distribution.

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Characteristics of a Binomial Distribution

There must be a fixed number of trials.

The outcome of each trial must be independent.

Each experiment can have only two outcomes or be reduced to two outcomes.

The probability of a success must remain the same for each trial.

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Binomial NormalWhen finding Use

P X a( ) P a X a( . . ) 05 05

P X a( ) P X a( . ) 05

P X a( ) P X a( . ) 05

P X a( ) P X a( . ) 05

P X a( ) P X a( . ) 05

Normal Approximation to Binomial Distribution

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Procedure for Normal Approximation

Step 1 Check to see whether the normal approximation can be used.

Step 2 Find the mean and the standard deviation .

Step 3 Write the problem in probability notation, using X.

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Procedure for Normal Approximation (cont’d.)

Step 4 Rewrite the problem using the continuity correction

factor, and show the corresponding area under the normal distribution.

Step 5 Find the corresponding z values.

Step 6 Find the solution.

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Summary

The normal distribution can be used to describe a variety of variables, such as heights, weights, and temperatures.

The normal distribution is bell-shaped, unimodal, symmetric, and continuous; its mean, median, and mode are equal.

Mathematicians use the standard normal distribution which has a mean of 0 and a standard deviation of 1.

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Summary (cont’d.)

The normal distribution can be used to describe a sampling distribution of sample means.

These samples must be of the same size and randomly selected with replacement from the population.

The central limit theorem states that as the size of the samples increases, the distribution of sample means will be approximately normal.

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Summary (cont’d.)

The normal distribution can be used to approximate other distributions, such as the binomial distribution.

For the normal distribution to be used as an approximation to the binomial distribution, the conditions np 5 and nq 5 must be met.

A correction for continuity may be used for more accurate results.

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Conclusions

The normal distribution can be used to approximate other distributions to simplify the data analysis for a variety of applications.