Download 68-95-99.7 rule

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Chapter 3: Normal Distribution
Density
curves
Normal distributions
The 68-95-99.7 rule
Finding the normal proportions
Distribution of a Continuous Var.
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It is described by a density curve
The density curve for a continuous var. X is
a curve such that
Proportion of X in between a and b is
the area under it over the interval [a, b]
The properties of a density curve:
–
–
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It is always on or above the horizontal axis
The total area underneath it is one
Normal Distribution
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The “model” distribution of a continuous var.
The normal density curve looks like:
The standard normal density curve centers at
0 with standard deviation 1
Finding Normal Proportions

X: a normal var. with mean m and
standard deviation s
P(X < a)
= P( X  m  a  m )
s
s
z score
= see Table A (p. 690-691)
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Finding Normal Proportions
1.
State the problem and draw a picture
2.
Calculate z scores and mark them on
the picture
3.
Use Table A
Example
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Suppose that the final scores of STAT1000
students follow a normal distribution with m =
70 and s = 10. What is the probability that a
ST1000 student has final score 85 or above
(grade A)?
Between 75 and 85 (grade B)?
Below 50 (F)?
Finding a Value given a Proportion
3 Steps:
1.
2.
3.
State the problem and draw a Z curve picture
Use the table to find the z score
Unstandardize: find the corresponding x score
Example: What is the first quartile of STAT 100
final score?
(for Bell-shaped distributions
only)
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Empirical Rule (68-95-99.7 rule)

If a variable X follows normal distribution, that
is, all X values (the whole population) show
bell-shaped, then:
Mean(X) + 1*SD(X) covers 68% of possible X values
Mean(X) + 2*SD(X) covers 95% of possible X values
Mean(X) + 3*SD(X) covers 99.7% of possible X values
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Empirical Rule (68-95-99.7 rule)

If the data (from a sample) of a variable X
show bell-shaped, then:
X + 1*S covers about 68% of possible X values
X + 2*S covers about 95% of possible X values
X + 3*S covers about 99.7% of possible X values
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How to use Empirical Rule
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
Find the range covering 68%, 95% or 99.7%
of X values

Check if X follows a normal distribution.