Transcript Measures of Central Tendancy

The Empirical Rule, Chebyshev’s Theorem,
Percentiles and z-scores
Today, we will study The
Empirical Rule and
Chebyshev’s Theorem and
how to apply them.
The Empirical Rule
The Empirical Rule:
•68% of the data under a normal curve lies within one standard deviation of the
mean.
•95% of the data under a normal curve lies within two standard deviations of
the mean.
•99.7% of the data under a normal curve lies within one standard deviation of
the mean.
68%
95%
99.7%
The Empirical Rule
Here is a another way to look at it:
34% 34%
2.35
2.35
13.5%
13.5%
The Empirical Rule
The Empirical Rule:
A normal population has a mean of 50 and a
standard deviation of 5, between what two
numbers does 95% of the data lie.
Answer:
•Between 2 standard deviations from the mean.
50 + 2*5 = 60
50 – 2*5 = 40
The Empirical Rule
The Empirical Rule:
A normal population has a mean of 50 and a
standard deviation of 5, what percent of the data
lies between 45 and 50
Answer:
•68% of the data lies within 1 sd of the mean.
•The percent between 45 and 50 is half of that: 68/2
= 34%
The Empirical Rule
The Empirical Rule:
A normal population has a mean of 50 and a standard
deviation of 5, what percent of the data lies between 40 and
45
Answer:
•½ of 95% = 47.5% of the data lies between 40 and 50.
•½ of 68% = 34% of the data lies between 45 and 50.
•47.5 – 34 = 13.5% of the data between 40 and 45
The Empirical Rule
Remember the Empirical Rule
only applies to the normal
distribution.
68%
95%
99.7%
Chebyshev’s Rule
For K> 1, the portion of any data set
lying within K standard deviations of
the mean is at least:
1
1 2
k
For k = 2:
1 3
1 2 
2
4
For k = 3:
1 8
1 2 
3
9
Chebyshev’s Rule
A non-normal data set has a mean of 50 and
a standard deviation of 5, between what two
numbers does 75% of the
data lie?
We can’t use the Empirical Rule for nonnormal data, so we use Chebyshev’s:
75% of the data is within two standard
deviations of the mean. So, 75% of the data
is between 40 and 60.