Download Normal Distribution: Understanding the Meaning of a Normal Curve and Standardized Scores and more Study notes Statistics in PDF only on Docsity!
8.4 The Normal Curve
The essential terms:
density scale normal curve standard normal curve standardized score (z-score)
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.
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.
Characteristics of the Normal Curve
Some important features of the normal curve are:
l The shape is a symmetric bell curve shape. Measurements relatively close to the mean are more probable than measurements relatively far from the mean.
l
l The center of the distribution is the mean. l 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.
Notation for the Population Mean and Standard Deviation
The normal curve is a model for the distribution of measurements in a population.
l The symbol μ is used to represent the population mean. l The symbol σ is used to represent the population standard deviation.
Page 2- The Standard Normal Curve
Standardized Scores
A standardized score , also called a z-score , measures how many standard deviations a value is from the mean.
l Graphing calculators, like the TI-83, also have commands that can be used.
Solving Normal Curve Problems
We now demonstrate how to solve general normal curve problems. As an example, we use the normal curve model for college students' hours of sleep on a week night that we described in 'The Basics' section.
l The mean is μ = 7 hours l The standard deviation is σ = 1.7 hours.
How to Find the Proportion Less Than a Value
To find the proportion of a population with a value less than a specified value:
- Calculate a z-score for the specified value.
- Determine the area to the left of the z-score.
Example
About what proportion of students sleep less than 5 hours on a week night? This will be the area to the left of 5 hours.
The solution:
1. Calculate the standardized score for 5 hours of sleep. The formula is
For 5 hours, z = ( 5 - 7 ) / 1.7 = -2 / 1.7 = -1.
2. Use the table on page 232 to find the area to the left of -1.18.
The answer is about 0.12 (about 12% ). About 12% of college students sleep less than 5 hours on a week night.
Note : If the proportion sleeping less than 5 hours is 0.12 the proportion sleeping more than 5 hours is 1-0.12 = 0..
How to Find the Proportion Greater Than a Value
To find the proportion of a population with a value greater than a specified value:
- Calculate a z-score for the specified value.
- Determine the area to the left of the z-score.
- Area to the right = (1-area to the left).
Example
About what proportion of students sleep more than 10 hours on a week night? This is the area to the right of 10 hours.
The solution:
To the left of z =-1.18, the area is about 0.12. The answer is 0.96 - 0.12 = 0.84. About 84% of college students sleep between 5 and 10 hours on a week night.
How to Find a Percentile
The term "percentile rank" refers to the area (probability) to the left of a value. For instance, we found that the area to the left of 5 hours of sleep is 0.1190. This means that:
l the percentile rank for 5 hours is 0.1190 (about 12%) l the 12th percentile is 5 hours
To find the value corresponding to a specified percentile rank:
- Determine the z-score that has the given percentile rank.
- Figure out what value has that z-score.
Example
What is the 98th percentile of the distribution of hours of sleep? The probability to the left of the answer is 0.98.
The solution:
1. Find the z-score for which the area to the left is 0.98. Look for 0.98 under Prob<=Z in the table on page 232. The value under Z Score is 2.05. The 98th percentile is 2.05 standard deviations above the mean.
l
2. Determine the sleep value that has a z-score of 2.05.
2.05 standard deviations is ( 2.05 )( 1.7 ) = 3.5 hours.
l
3.5 hours above the mean of 7 hours = 10.5 hours.
l
The 98th percentile of the sleep distribution is 10.5 hours.
A formula to calculate the value that has a known z-score is
X = Z σ + μ
For z = 2.05 , x = (2.05) (1.7) + 7 = 3.5 + 7 = 10.5 hours.
The Empirical Rule and Outliers
The following three statements constitute the empirical rule.
About 68% of the values in a population described by a normal curve are within one standard deviation of the mean.
About 95% of the values in a population described by a normal curve are within two standard deviations of the mean.
About 99.7% of the values in a population described by a normal curve are within three standard deviations of the mean.
The empirical rule is also called the 68 - 95 - 99.7% rule.
Example:
The number of hours that college students sleep on a week night is approximated by a normal curve with a mean of 7 hours and a standard deviation of 1.7 hours. About 68% of college students sleep between 5.3 and 8.7 hours on a week night. The calculation is 7 ± 1.
l
About 95% of college students sleep between 3.6 and 10.4 hours on a week night. The calculation is 7 ± (2•1.7)
l
About 99.7% of college students sleep between 1.9 and 12.1 hours in a week night. The calculation is 7 ± (3•1.7)
l
When is a Value an Outlier?
There is no universally accepted criterion for declaring a point to be an outlier, but most data
