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MDM 4U1
8.2 Properties of the Normal Distribution
Date:__________
Many physical quantities like height and mass are distributed symmetrically and unimodally
about the mean. Statisticians observe this bell curve so often that its mathematical model is known as
the normal distribution. Other examples are human resources-employee performance is sometimes
considered to be normally distributed and students marks in university.
A population that follows a normal distribution can be completely described by its mean, μ and its
standard deviation, σ. The symmetric unimodal form of a normal distribution makes both the mode
and median equal to the mean. The smaller the value of σ, the more the data cluster about the mean, so
the narrower the bell shape. Larger values of σ correspond to more dispersion and a wider bell shape.
In a normal distribution, it can be said that half the data will lie above the mean and half the data will
lie below the mean. The mean represents the highest peak in this unimodal distribution.
68%,95%,99.7% Rule
For any given normal distribution X, the percentage of data;
 between    and    is approximately 68%. (i.e. within one standard deviation of the mean.)
 between   2 and   2 is approximately 95%
 between   3 and   3 is approximately 99.7%
  3   2   

68%
95%
99.7%
 
  2   3
MDM 4U1
Example 1: Sally is 164 cm tall. In her school, the girls heights are normally distributed with a
mean of 168cm and a standard deviation of 4cm.
a. What is the probability that Sally’s friend Joanne is taller than she is?
b.
What is the probability that Joanne is between 164cm and 172cm tall?
Equations and Probabilities for Normal Distributions
The curve of a normal distribution with mean and standard deviation is given by the equation
f ( x) 
1
 2
e
1  x 
 

2  
2
 Computers and calculators use this equation to calculate
probabilities
Standard Normal Distribution
Probabilities are found between 0
and z. Negative z-scores have the
same probability.
MDM 4U1

An applet can be found online to show the normal distribution
 http://www.oame.on.ca/main/files/Gr12-2007/MDM4U/MDM4U-U5L7Applet/index.html
TRY IT to check your solutions!!
No simple formula exists for the areas under normal distribution curves. Instead these areas have to be
calculated using a more accurate version of “counting grid squares”. However, you can simply such
calculations using z-scores.
Recall from 2.6 that the z-score for a value of the random variable is
z
x

.
The distribution of the z-scores of a normally distributed variable is a normal distribution with
mean 0 and standard deviation 1.
This particular distribution is often called the standard normal distribution
Areas under this normal curve are known to a very high degree of accuracy and have been precalculated and be read from the table on pg.606 and 607.
Example 2:
The mean score on a test (out of 60) was found to be 50.6 and the standard deviation was determined
to be 2.3.
a) Determine the probability that a student’s score chosen at random is less than 55.
b) Determine the probability that a student’s score chosen at random is greater than 55.
c) Determine the probability that a student’s score is between 47 and 53 P(47  x  53)
Solutions:
*Use the Standard Normal Distribution table to solve Normal Distribution problems.
a)
50.6
55
0
b)
55
1.91
MDM 4U1
c)
47
53
Helpful tips:
P(z > a) = 1 - P(z < a)
P(a < z < b) = P(z < b) - P(z < a)
Example 3: Standardized Test Scores
To qualify for a special program at university, Martin has to write a standardized test. The test has a
maximum score of 750, with a mean score of 540 and a standard deviation of 70. Scores on this test
were normally distributed. Only those applicants scoring above the third quartile (the top 25%) are
admitted to the program. Martin scored 655 on this test. Will he be admitted to the program?
Solution:
MDM 4U1
Homework:
1. Handout:
* for #1,2, use the 68%,95%,99.5% rule
1. The daily sales of Gary’s chip truck has a mean of $675.00 and a standard deviation of $35.50.
a. What percent of time will the daily sales be greater than $639.50?
b. What percent of time will the daily sales be less than $746.00?
2. The mean household income in Kingston is $45000 with a standard deviation of $15000.
Household incomes below $30000 will receive a tax credit, household incomes between
$30000 and $75000 will have to pay a 2% tax, and household incomes over $75000 will have
to pay a 5% tax.
a. What percentage of households will have to pay a 2% tax
b. What percentage of households will not have to pay tax
c. What percentage of households will pay tax.
3. Calculate the z-scores for the following data points if the mean for the data set is 6 and the
standard deviation is 2.
a) 4
b) 7
c) 1.5
4. Use the Standard Normal Distribution table to solve Normal Distribution problems. Include a
sketch of the distribution with your solution including each of the standard deviation intervals.
A data set is normally distributed with a mean of 25 and a standard deviation of 5.
Using the z-score tables determine the following probabilities for a data point chosen at random.
a) P ( x  25)
b) P ( x  30)
c) P (20  x  30)
d) P(15  x  35)
e) P(17  x  22)
Solutions:
1. a) 84% b) 97.5% 2. a) 81.5% b) 16% c) 84% 3. a) z = -1 b) z = 0.5 c) z = -2.25
4. a) 0.5 b) 0.1587 c) 0.68 d) 0.95 e) 0.2195
2. Textbook: pg. 430 # 2,3,5,8,9