Download Empirical Rule

Empirical Rule
If Chebychev Inequality gives the lower bound for probability. For the
present problem the lower bound for the probability is 0.75 and actual
probability is 0.87. There is no contradiction between empirical rule
and Chebychev Inequality
1
of the observations in a set of data lie
k2
within k standard deviations of the mean.” For this problem you are concerned with the
proportion of observations within k  2 standard deviations of the mean. Plugging into
the Chebychev inequality:
The Chebychev Inequality says “At least 1 
p  1
1
1
 1  2  .75
2
k
2
We know the probability must be greater than or equal to 0.75. The actual probability is
0.87, greater than the lower bound defined by the Chebychev inequality.
Given percentage (87%), does the empirical rule apply?
What does the Empirical Rule say for + 2 standard deviations from the
mean?...What percentage of data should be within those two values?
It is important to understand that Chebychev’s Inequality applies to any distribution, of
any shape. The empirical rule applies only to distributions that have an approximate
normal distribution (mound shaped symmetric distributions). The empirical rule states
that approximately 68% percent of the observations in the distribution are within 1
standard deviation of the mean, approximately 95% of the observations are within 2
standard deviations, and approximately 100% of the observations are within 3 standard
deviations of the mean. These percentages can be found by using a table of probabilities
for a normal distribution. In this problem, if you apply the empirical rule, it states 95% of
the observations should be within 2 standard deviations of the mean. The actual percent
in the distribution under consideration is 87%. The actual percentage differs from the
percentage of the empirical rule. We can therefore conclude that the distribution under
consideration does not follow a normal distribution.
In summary, Chebychev’s inequality applies because it will always be true for any
probability distribution. The empirical rule does not apply because it only applies to
approximately normal distributions. Since the empirical rule does not apply, we know
the distribution under consideration is not normally distributed.