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MATH 11008:
Standard Normal
Distribution (Ch 16)
• approximately normal distribution: A distribution of data that roughly fits a bellshaped curve is called approximately normal. A distribution of data that has a
perfect bell shape is called a normal distribution.
• z scores: z-scores allow us to transform a random variable X with mean µ and standard
deviation σ into a random variable z with mean 0 and standard deviation 1.
• Standardizing Rule: In a normal distribution with mean µ and standard deviation
σ, the standardized value of the data point x is z, where
z=
x−µ
σ
is normally distributed with mean µ = 0 and standard deviation σ = 1. The random
variable z is said to have the standard normal distribution.
• Properties of the Standard Normal Curve:
◦ It is symmetric about its mean, µ = 0, and has standard deviation σ = 1.
◦ The mean = median = mode = 0. Its highest point occurs at z = 0.
◦ The area under the curve is 1.
• Empirical Rule: In every normal distribution,
◦ approximately 68% of the data lie within 1 standard deviation of the mean. That
is, approximately 68% of the data lie between z = −1 and z = 1.
◦ approximately 95% of the data lie within 2 standard deviations of the mean. That
is, approximately 95% of the data lie between z = −2 and z = 2.
◦ approximately 99.7% of the data lie within 3 standard deviations of the mean.
That is, approximately 99.7% of the data lie between z = −3 and z = 3.
34%
34%
13.5%
13.5%
-3
2.35%
-2
2.35%
-1
0
1
2
3
MATH 11008: STANDARD NORMAL
DISTRIBUTION (CH 16)
2
• Properties of a Normal Distribution:
◦ In a normal distribution, M = µ.
◦ In a normal distribution, the standard deviation σ equals the distance between a
point of inflection and the line of symmetry of the curve.
◦ In a normal distribution,
Q3 ≈ µ + (0.675)σ
and
Q1 ≈ µ − (0.675)σ.
Example 1: Consider the normal distribution shown below and assume that P and P 0 are
two points of inflection of the curve
P
410
pts
P'
650
pts
(a) Find the median M of the distribution.
(b) Find the standard deviation σ of the distribution.
(c) Find the third quartile Q3 of the distribution rounded to the nearest point.
(d) Find the first quartile Q1 of the distribution rounded to the nearest point.
MATH 11008: STANDARD NORMAL
DISTRIBUTION (CH 16)
3
Example 2: Estimate the value of the standard deviation σ (rounded to the nearest tenth
of an inch) of a normal distribution with µ = $18, 565 and Q1 = $15, 514.
Meaning of Normal Curve Areas: In a standard normal curve,
the following three quantities are equivalent:
• Percentage (of total items that lie in an interval.)
• Probability (of a randomly chosen item lying in an interval.)
• Area (under the normal curve along an interval.)
Example 3: Determine the area under the curve that lies to the left of z = 1.38.
Example 4: Determine the percentage of scores that are greater than z = 1.08.
Example 5: Determine the percentage of scores that are between z = −1.34 and z = 0.75.
MATH 11008: STANDARD NORMAL
DISTRIBUTION (CH 16)
4
Example 6: A standardized exam was given to 2500 incoming freshman. The distribution
of scores has the shape of a normal distribution with mean 72 and standard deviation of 12.
(a) What percent of the students scored above 79?
(b) What percent of the students scored below 63?
(c) What percent of the students scored between 65 and 85?
(d) What is the approximate score for a person in the 61st percentile?