The Empirical Rule
If the population distribution is bell-shaped (normal distribution), symmetrical with a
single peak, with mean μ and standard deviation σ, then
1. 68.26% of all population units are within (plus or minus) one standard deviation
of the mean and thus lie in the interval [μ −σ, μ + σ]
2. 95.44% of all population units are within (plus or minus) two standard deviations
of the mean and thus lie in the interval [μ −2σ, μ + 2σ]
3. 99.73% of all population units are within (plus or minus) three standard
deviations of the mean and thus lie in the interval [μ −3σ, μ + 3σ]
1. An interval that contains a specified percentage of the individual measurements in
a population is called a tolerance interval
2. Often we interpret the three-sigma interval [μ ± 3σ] to be a tolerance interval that
contains almost all of the measurements in a normally distributed population
3. Of course, we usually do not know the true values of μ and σ.
• We must estimate the tolerance intervals by replacing μ and σ in these
intervals with the sample mean x and standard deviation s
• We use to estimate μ
Back to Example 2.2
We saw that the distribution of waiting times for Dr. Kim was approximately normal.
We also calculated
x = 14.28 minutes
s = 7.50 minutes
To contain about 66.26% of the possible waiting time;
To contain about 95.44% of the possible waiting times;
14.28+-7.5*3=0~35.78 To contain about 99.73% of the possible waiting times;
What about the sample data for Dr. Jim? What are the tolerance intervals? How well
does the data fit what is expected by the empirical rule?
The distribution of wait time for patients Dr.Jim is not normally distributed.
The data is expected to fit by the empirical rule by 23/24.
Application of Tolerance Intervals
Tolerance intervals are often used to determine whether customer requirements
(specifications) are being met.
• If a process is consistently able to produce output that meets customer requirements,
we say the process is capable
• It is common practice to conclude that a process (that is in statistical control) is
capable if the 3-sigma tolerance interval estimate, [ ± 3s] is within the specification
Factory XYZ has a machine that produces iron bars. A random sample of 35 iron bars gave a
mean of 110.8 cm and a standard deviation of 0.4 cm. Customers who buy iron bars from this
factory require them to not be too long or too short. They are satisfied if the bars have a length
somewhere between 109.5 to 112.5 cms.
Is the factory capable of meeting their customers specifications/requirements? What underlying
assumption did you have to make?
Customers specifications: 109.5 – 112.5 cm
The company’s iron bars produced, 99.72% will have a length between 109.5-112.5cm.
What if a Distribution is NOT Normal?
What if we do not know whether the underlying distribution is bell-shaped and
symmetrical, OR what if we know that it is ske