Adapted from the forthcoming CyberStats introductory statistics course.
The Normal Curve
| The essential terms for this Unit: |
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The normal curve is the familiar symmetric, bell shaped curve that's often used to approximate the distribution of measurements in a population.
Example:
The distribution of the number of hours that
college students sleep on a week night is approximated by the normal curve displayed in
the figure below. Characteristics of the model were determined from data collected
in a statistics class at Penn State University.
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Probability Equals Area Under the Curve
The vertical axis in the drawing of the normal curve above is a density scale. When a density scale is used, probability equals area under the curve.
Example:
The proportion of college students who sleep
between 5.5 and 8.5 hours is the area under the normal curve between 5.5 and 8.5 hours, an
area shown in the following figure.
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Characteristics of the Normal Curve
Some important features of the normal curve are:
- The shape is a symmetric bell curve shape.
- Measurements relatively close to the mean are more probable than measurements relatively
far from the mean.
- The center of the distribution is the mean.
- The spread of the bell is determined by the standard deviation.
There actually are several probability distribution models that have these characteristics. The normal curve is by far the most commonly used model with these features.
Interactivity - How the Mean and Standard Deviation Affect the CurveIn the following discovery activity, the two sliders
below the graph can be used to see how the mean and the standard deviation
influence the normal curve.
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Notation for the Population Mean and Standard Deviation
The normal curve is a model for the distribution of measurements in a population.
- The symbol m is used to represent the population mean.
- The symbol s is used to represent the population standard deviation.
Standardized Scores
A standardized score, also called a z-score, measures how many standard deviations a value is from the mean.
A formula for calculating a standardized score is
.
Example:
Suppose that the mean pulse rate in a population is
75 beats per minute and the standard deviation is 10.
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Standardize to Solve Normal Curve Problems
Every normal curve problem can be solved by converting the problem to a question about standardized scores.
Example:
| Suppose that the distribution of pulse rates in a
population is described by a normal curve with a mean of 75 beats per minute and a
standard deviation equal to 10. The standardized score for a pulse rate of 85 is z = 1. The proportion of the population that has a pulse rate less than 85 is also the proportion that has a z-score less than 1. |
The Standard Normal Curve
The standard normal curve is a normal curve model for z-scores. In this model, the mean is 0 and the standard deviation is 1.
The standard normal curve is used to solve every normal curve problem, whether the problem is about heights, IQs, hours of sleep, or any other variable.
A Probability Calculator for the Standard Normal Curve
Below, we introduce a calculator that can be used to determine either of two quantities:
- For a given Z score, the probability less than that Z .
(In other words, the percentile rank of the Z score. ) - For a given percentile rank, the corresponding Z score.
The relationship between Z and Prob<Z is shown in this graph-
The calculator -
Example: For Z = 1, what is the percentile rank?
Example: What Z score is the 25th percentile?
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General Rules for Using the Calculator
To find Prob<Z for a given Z score :
- Enter a value under Z Score.
- Click the Calc P button
- The answer is under Prob<Z.
To find the Z score for a given Prob<Z :
- Enter a value under Prob<Z
- Click the Calc Z button.
- The answer is under Z Score.
Copyright © 1999 CyberGnostics, Inc. All rights reserved.