Download 4.4 – Normal Distributions (Empirical Rule)

4.4 – Normal Distributions (Empirical Rule)
Symmetric probability distributions are often defined as normal.
All normal curves have the following properties, sometimes collectively called the
68 – 95 – 99.7 rule.
•
68% of the data fall within 1 standard deviation of the mean.
•
95% of the data fall within 2 standard deviations of the mean.
•
99.7% of the data fall within 3 standard deviations of the mean.
A normal curve’s symmetry allows you to separate the area under the curve into eight parts and
know the percent of data in each part.
The rule is used for forecasting. It gives an estimate of what data collection might look like if
you surveyed the entire population.
It only works with a normal distribution.
Normal Distributions (Empirical Rule)
Normal Distributions (Empirical Rule)
99.7% between ± 3
95% between ± 2
68% between ± 1
0.15%
2.35%
2.35%
̅ 3
̅ 2
̅
̅
̅
̅ 2
̅ 3
0.15%
Normal Distributions (Empirical Rule)
Steps to Solving Empirical Rule Questions
1.
Draw out a normal curve with a line down the middle and three to either side.
2.
Write the values from your normal distribution at the bottom. Start with the
mean in the middle, then add standard deviations to get the values to the right
and subtract standard deviations to get the values to the left.
3.
Write the percent's for each section (you will need to memorize them!) 34%,
13.5%, 2.35% and 0.15%.
4.
Determine the section of the curve the question is asking for and shade it in.
5.
Add up the percents in the sections that are shaded.
Normal Distributions (Empirical Rule)
Adult women in the US have an average height of 65 inches with a standard deviation of 3.5
inches.
a) 68% of US women are between ________
and ________ inches tall.
b) 16% of US women are shorter than
________ inches tall.
c) 95% of US women are between ________
and ________ inches tall.
d) 2.5% of US women are taller than ________
inches tall.
Normal Distributions (Empirical Rule)
A college entrance exam is designed so that scores are normally distributed with a mean of 500
and a standard deviation of 100.
a) What percentage of the exams is between
400 and 600?
b) What is the probability that a randomly
selected test score is above 600?
c) What is the probability that a randomly
selected test score is less than 300 or greater
than 700?
Normal Distributions (Empirical Rule)
2000 freshmen at State University took a biology test. The scores were distributed normally
with a mean of 70 and a standard deviation of 5. Label the mean and three standard deviations
from the mean. Answer the following questions based on the data:
a) What percentage of scores are between 65
and 75?
b) What is the area under the curve
corresponding to scores between 60 and 70?
c) What is the area under the curve
corresponding to scores between 60 and 85?
d) What percentage of scores are less than a
score of 55?
Normal Distributions (Empirical Rule)
2000 freshmen at State University took a biology test. The scores were distributed normally
with a mean of 70 and a standard deviation of 5. Label the mean and three standard deviations
from the mean. Answer the following questions based on the data:
g) Approximately how many biology students
scored between 55 and 60?
55
60
65
70
75
80
0.15%
2.35%
13.5%
34%
34%
13.5%
2.35%
f) Approximately how many biology students
scored between 60 and 70?
0.15%
e) What percentage of scores are greater than
a score of 80?
85
Normal Distributions (Empirical Rule)
Here are the scores for a recent test in Algebra 2.
90, 90, 95, 100, 80, 80, 75, 80, 70, 60, 95, 100, 100, 100, 75, 80, 90, 90, 90, 70, 70, 80, 85, 90, 90, 85
Find the following using a graphing calculator.
Mean =
Median =
Mode =
Standard Deviation =
Normal Distributions (Empirical Rule)
Mean = 85
Median = 87.5
Mode = 90
Standard Deviation = 10.56
a) How many scores are within 1 standard deviation of the mean?
b) How many scores are within 2 standard deviations of the mean?