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3.5 Chebyshev’s Rule and the
Empirical Rule
Objectives:
By the end of this section, I will be
able to…
1)
Calculate percentages using Chebyshev’s
Rule.
2)
Find percentages and data values using the
Empirical Rule.
CHEBYSHEV’S THEOREM
Chebyshev's Theorem

The Russian mathematician P.L. Chebyshev (18211894) proved a theorem which is valid for any
distribution of data:
𝒙−𝒙
𝒌=
𝒔
Let 𝑘 ≥ 1.
Then the % of distribution that lies within "k"
1
SDs of the 𝑥 is at least 1 − 2 100.
𝑘
NORMAL DISTRIBUTION

IQ scores, heights, weights are all examples of
normal distributions.
Chebyshev’s Theorem
Why is this theorem used?
 It can be used for samples or populations.

-5
75 +5
What is the mean? 75
 What is the standard deviation?

5
Easy Problem
1) Professor Costag is analyzing the grades
from his Statistics course. The average
grade was an 85 with a standard deviation of
3 points. What scores represent the scores
within 2 standard deviations of the mean?
- 2 s.d
79
85
𝒙
+ 2 s.d.
to
91
A little tougher…
k = 3.33
Using the previous example, find the
percentage of students who will score
between 75 and a 95. Remember the
mean is 85 with s = 3.
WE NEED TO FIND k. Clue word is “percentage”
To find k use the following formula:
3.33
3.33
k = 3.33
A little tougher…
Using the previous example, find the
percentage of students who will score
between 75 and a 95. Remember the
mean is 85 with s = 3.
Now use Chebyshev’s Theorem.
k = 3.33
90.999% or 91%
Empirical Rule
Metaphorically, the Empirical Rule is a
Porshe compared to Chebyshev’s Rule,
which is a go-anywhere ATV.
 Basically if you have data that is skewed
or an unknown relationship, use
CHEBYSHEV’S.
 If your distribution is bell shaped (normal)
use EMPIRICAL.

EMPIRICAL RULE
 If
the data distribution is normal (bell
shaped):
 68% of the data values will fall within 1
standard deviation of the x.
 95% will fall within 2 SDs of the x.
 99.7% will fall within 3 SDs of the x.
You still start using Chebyshev’s, it will just
end up being one of these values.
EMPIRICAL RULE
 If
the data distribution is normal (bell
shaped):
 68%
 95%
 99.7%
THESE ARE
YOUR ANSWERS!!!!!
Practice Problem with Empirical Rule

Your GPAs are bell-shaped with a mean of 3.1
and a standard deviation of 0.7. Find the
proportion of GPAs between 2.4 and 3.8.
WE NEED TO FIND k.
To find k use the following formula:
Since k = 1, 68% of
1
the students are within
k = 1 1 standard deviation of
3.1, the mean.
1
Practice

Complete the following two problems.
SHOW ALL YOUR WORK!

ChebyEmpirical