Download The Normal Distribution

The Normal Distribution
Before we think about the “normal distribution”, we will discuss the notion of describing a
distribution with a smooth curve. Picture a relative frequency histogram of our class height
distribution. The “bin width” (how wide the interval is that creates the bars) is set to three,
then two, then one.
Imagine if we collected heights in smaller increments than single inches (going out to several
decimal places). We could keep making the bin width smaller and smaller until the histogram
started looking like a smooth curve. We can draw in which curve we think might describe the
class height distribution.
Question: In a relative frequency histogram, if we added
up the heights of all the bars, what would the total be?
The proportion of observations in any given interval of
values is the area under the curve within that interval, so
the total area under the curve should equal________.
(You actually have to use calculus to find area under a
curve, so you’ll always be given a chart or table of areas in
this class…) Once we smooth a histogram with a curve that
has area _____ under it, we call this a density curve.
Describing the center of a distribution when using a density curve
The mean and median have the same interpretations as when we studied them using
histograms.
median 
mean 
on a symmetric curve
 mean = median
 both lie at the center of the curve
on a skewed curve
 mean is pulled away from the median in the direction of the long tail
.
Figure 13.5 The median and mean for two density curves: (a) a symmetric Normal
curve and (b) a curve that is skewed to the right.
Figure 13.6 The mean of a density curve is the point at which it would balance.
Identify the mean and median:
A. Median Mean
B. Mean Median
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Consider the density curve shown at the right.
The total area under the curve is:
A. 0.5
B. 1.0
C. 1.5
D. 2.0
The Normal Distribution
A normal distribution is a unimodal and symmetric (“bell-shaped”) distribution.
The distribution is determined by the mean mu, , and the standard deviation sigma, .
 The mean, , controls the center
 The standard deviation, , controls the spread.
- the standard deviation is the distance from the mean to the inflection point (the
place where the curve changes from concave down to concave up) on the normal
curve
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Pictured at the right are two different
normal distributions. Which is different
between the two distributions?
A. Mean
B. Standard deviation
C. Both
Which is different between the two
normal distributions to the right?
A. Mean
B. Standard deviation
C. Both
Which is different between the two
normal distributions to the right?
A. Mean
B. Standard deviation
C. Both
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68-95-99.7% Rule for the Normal Distribution
We’ll use the 68-95-99.7% Rule (often called the empirical rule) to find probabilities associated
with the normal distribution.
We’ll use the 68-95-99.7% Rule (often called the empirical rule) to find probabilities associated
with the normal distribution. Start each question by labeling your normal curve with the
following percents….
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The Health and Nutrition Examination Study of 1976-1980 (HANES) studied the heights of
adults (aged 18-24) and found that the heights follow a normal distribution with the following:
Women Mean (): 65.0 inches standard deviation (): 2.5 inches
Men
Mean (): 70.0 inches standard deviation (): 2.8 inches
Find the proportion of men with heights less than 72.8 inches.
Standard Scores
Observations expressed in terms of standard deviations above or below the mean are called
Standard Scores.
The standard score is the number of standard deviations above or below the mean at which an
observation is located.
If the observation is below the mean, the standard score will be negative.
If the observation is above the mean, the standard score will be positive.
What is the standard score for a height of 75.6 inches from our example above?
How will we find a standard score if not given a picture? We’ll use a formula to calculate it.
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑠𝑐𝑜𝑟𝑒 =
𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛 − 𝑚𝑒𝑎𝑛
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
Sometimes we call a “standard score” a “Z-score” and write the formula this way
𝑍=
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𝑥−𝜇
𝜎
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When we standardize a normal variable, then Z has a standard normal distribution; that is Z is
a normal variable with a mean of 0 and a standard deviation of 1.
Jennie scored 600 on the verbal part of the SAT. Her friend Gerald took the ACT and scored a
21 on the verbal part. SAT scores are normally distributed with mean 500 and standard
deviation 100. ACT scores are normally distributed with mean 18 and standard deviation 6.
Assuming that both tests measure the same kind of ability, who has the higher score?
A woman is told her weight has a standard score of 1. This means her weight is
A. 1 pound above the mean
B. 1 pound below the mean
C. 1 standard deviation above the mean
D. 1 standard deviation below the mean
Math SAT scores follow a normal distribution with a mean of 500 and standard deviation of
100.
Calculate the standard score for a score of 630.
A. 1.3
B. 1.1
C. -1.3
D. -1.1
Two students get a 65 on two different tests. Student A has a standard score of -1 while
Student B has a standard score of -2. Which student had the better performance on the test?
A. Student A
B. Student B
C. Both students gave equal performances.
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Percentiles
Percentile refers to the proportion of a distribution below a given value.
Ex: The 10th percentile has 10% of the distribution below it and 90% above it.
Ex: The 95th percentile has 95% of the distribution below it and 5% above it.
Formally, the cth percentile of a distribution is a value such that c percent lie below it and the
rest lie above it.
Q1 is the __________ percentile of a distribution
A. 25th
B. 50th
C. 75th
Recall the distribution of SAT math scores follows a normal distribution with a mean of 500
and a standard deviation of 100.
Find the math score that is the (approximate) 98th percentile for this distribution.
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