Empirical Rule:
The total area under any normal curve studied in this course will always be 1. The graph
of the normal distribution is important because the portion of the area under the curve
above a given interval represents the probability that a measurement will lie in that
interval.
Remember when we studied Chebyshev’s Theorem we said that it applies to any and all
distributions. However, we can get a better result when we use the empirical rule for
normal distribution.
Empirical Rule:
For a distribution that is symmetrical and bell-shaped (in particular, for a normal
distribution):
Approximately 68% of the data values will lie within one standard deviation on
each side of the mean.
Approximately 95% of the data values will lie within two standard deviations on
each side of the mean.
Approximately 99.7% of the data values will lie within three standard deviations
on each side of the mean.
The above is called the empirical rule due to the fact that for symmetrical, bell-shaped
distributions, the given percentages are observed in practice. Also, the empirical rule is a
direct consequence of the very nature of the distribution. If you notice the empirical rule
gives a stronger result then Chebyshev because it gives definite percentages, not just
lower limits.
*** View Figure 6-7 (text p. 331).
*** View Example #1 (text p. 332).
*** View Guided Exercise #5 (text p. 332 – 333).
It is extremely important that you memorize the following:
1) About 68.2% of the area under a normal curve falls in the interval between (μ +
σ) and (μ - σ).
2) About 95.4% of the area falls between (μ + 2σ) and (μ - 2σ).
3) About 99.7% of the area falls between (μ + 3σ) and (μ - 3σ).
Also, make sure you understand Figure 6-7 on text p. 331.
Examples:
1) Draw the normal curve when μ = 22 and σ = 4.
a) What is the percentage of area over the interval between 18 & 26?
b) What is the percentage of area over the interval between 14 & 26?
c) What is the percentage of area over the interval between 10 & 22?
d) What is the probability that the data will fall between 14 and 32?
e) What is the probability that the data will fall between 22 and 26?
2) Draw the normal curve when μ = 100 and σ = 16.
a) What is the percentage of area over the interval between 84 & 116?
b) What is the percentage of area over the interval between 68 & 100?
c) What is the percentage of area over the interval between 68 & 84?
d) What is the probability that the data will fall between 84 and 100?
e) What is the probability that the data will fall between 100 and 117?
3) Draw the normal curve when μ = 212 and σ = 12.
a) What is the percentage of area over the interval between 212 & 224?
b) What is the percentage of area over the interval between 212 & 236?
c) What is the percentage of area over the interval between 200 & 224?
d) What is the probability that the data will fall between 200 and 236?
e) What is the probability that the data will fall between 200 and 212?
4) Draw the normal curve when μ = 515 and σ = 13.
a) What is the percentage of area over the interval between 502 & 515?
b) What is the percentage of area over the interval between 502 & 528?
c) What is the percentage of area over the interval between 515 & 528?
d) What is the probability that the data will fall between 528 and 541?
e) What is the probability that the data will fall between 489 and 528?
5) Draw the normal curve when μ = 174 and σ = 17.
a) What is the percentage of area over the interval between 174 & 191?
b) What is the percentage of area over the interval between 191 & 208?
c) What is the percentage of area over the interval between 174 & 208?
d) What is the probability that the data will fall between 157 and 174?
e) What is the probability that the data will fall between 157 and 225?