Chapter 13
Normal Distributions
Figure 13.3 A perfectly symmetric Normal curve used to describe the
distribution of
sample proportions. (This figure was created using the Stata software
package.)
Density Curve
1. Curves show the proportion of observations in any region by areas under the curve.
2. The scale is chosen so that the area under the curve is 1.
3. It is intended to reflect the idealized shape of the populations distribution.
4. The median is the equal areas point; i.e. half the area under the curve is to the left and the remaining
half is to the right.
5. The mean is the balance point, or center of gravity, at which the curve would balance if made of solid
material.
6. Note: For a symmetric density curve, the mean and median are the same.
Bell-Shaped Curve:
The Normal Distribution of Population Values







Symmetric
Single-peaked
Bell-shaped
Mean = Median = Peak (center of the curve)
The point at which the curvature changes is one standard deviation on either side of the
mean
Mean – fixes the center of the curve
Standard Deviation – determines its shape
Empirical Rule for Any Normal Curve = “68-95-99.7 Rule”



68% of the values fall within one standard deviation of the mean
95% of the values fall within two standard deviations of the mean
99.7% of the values fall within three standard deviations of the mean
Figure 13.8 The 68–95–99.7 rule for Normal distributions.
Example 1:
Scores on a national achievement exam have a mean (µ) = 480 and a standard deviation( σ) = 90.
Scores are normally distributed.
68% of scores will fall between ________________________________________
95% of scores will fall between ________________________________________
99.7% of scores will fall between _______________________________________
Example 2:
The playing life of a radio is normally distributed with a µ = 600 hours and a σ = 100 hours. What is the
probability that a radio selected at random will last from 600 to 700 hours?
Example3:
The yearly yield of wheat per acre on a farm is normally distributed with a µ = 35 bushels, and σ = 8
bushels. Shade the area on a curve that represents the probability that an acre will yield between 19
and 35 bushels. This represents the proportion that will be between 19 and 35 bushels.
With the Mean and Standard Deviation of the Normal Distribution We Can
Determine:



What proportion of individuals fall into any range of values
At what percentile a given individual falls, if you know their value
What value corresponds to a given percentile
Standardized Scores
standardized score = (observed value - mean) / (std dev)




z is the standardized score
x is the observed value
µ is the population mean
σ is the population standard deviation
z
x

Percentiles of Normal Distributions
Standard scores translate directly into percentiles



The cth percentile of a distribution is a value such that c percent of the observations lie
below it and the rest lie above it
Examples: the median(Q2) = the 50th percentile, Q1 = 25th percentile, Q 3 = 75th
percentile
Every standard score for a Normal distribution translates into a specific percentile.
Table B: Percentiles of the Standardized Normal Distribution:
 See Table B (the “Standard Normal Table”) in back of the text (or back of the
supplement). Page 599
 Look up the closest standardized score in the table.
 Find the percentile corresponding to the standardized score (this is the percent of values
below the corresponding standardized score or z-value).
Observed Value for a Standardized Score
observed value = mean + [(standardized score) * (standard deviation)]
x    z
Examples:
1. Tina scored a 74 on the last test. Her class mean was 64, and the standard deviation was 5. Jack
scored an 82 on the test, his class average was 72, and the class standard deviation was7.25. Both
students scored 10 points above their respective class averages, but how well did they do with respect
to the others in their class?
2. Rod’s average commute to college is 17 minutes. This includes driving, parking, and walking to class.
The standard deviation is 3 minutes.
a) One day he clocked it at 21 minutes. Is this possible? Why or why not?
b) Another day he woke up late and had only 9.5 minutes to get to school. Calculate the standardized
score. Is it likely that he will make it on time? Find the probability that corresponds to this standardized
score.
3. Find the corresponding percentiles for the following standardized (z) scores:
a) z = -1.0
b) z = -1.60
c) z = +2.5
4. The time it takes to complete a test is normally distributed with µ = 1.5 hours and a standard
deviation of 15 minutes. After how many minutes can we expect 97% of the tests to be completed?
5. The average height of males ages 18–24 years old was 70.0 inches with a standard deviation of 2.8
inches. It is also known that this distribution of heights follows a normal or bell-shaped curve.
a) What proportion of men are between 68 inches tall and 74 inches tall?
b) What proportion of men are less than 74 inches tall?