2.35% + 13.5% + 34% + 34% + 13.5% + 2.35% =
99.7%

Module 3: Measures of Variability – Ungrouped Data
Solution to Example 4 – Empirical Rule
f. What percent of the test scores will be below 96?
.15% + 2.35% + 13.5% +34% + 34% + 13.5% + 2.35% =
99.85%
Or
100% - .15% =
99.85%

Module 3: Measures of Variability – Ungrouped Data
Example 5 – Empirical Rule
Assume the weights of Roma tomatoes are normally distributed with a mean of 3.6 ounces and a standard
deviation of 0.4 ounces.
Determine the following using the Empirical Rule:
a.
What percent of the tomatoes weigh between 2.8 and 4.8 ounces?
b.
What percent of the tomatoes weigh less than 2.4 ounces?
c.
What value corresponds to the 97.5th percentile of Roma tomato weights?

Module 3: Measures of Variability – Ungrouped Data
Solution to Example 5 – Empirical Rule
The first thing you want to do is draw the bell-curve with line representing the mean and +/- 1, 2, and 3
standard deviations.
Then fill in the numbers which correspond to this particular problem:

Module 3: Measures of Variability – Ungrouped Data
Solution to Example 5 – Empirical Rule
a.
What percent of the tomatoes weigh between 2.8 and 4.8 ounces?
Adding up the shaded area results in : 13.5% + 34% + 34% + 13.5 + + 2.35% = 97.35%

Module 3: Measures of Variability – Ungrouped Data
Solution to Example 5 – Empirical Rule
b.
What percent of the tomatoes weigh less than 2.4 ounces?
The area under the normal curve to the left of 2.4 ounces (3 standard deviations below the mean) is
.15%

Module 3: Measures of Variability – Ungrouped Data
Solution to Example 5 – Empirical Rule
c. What value corresponds to the 97.5th percentile of Roma tomato weights?
Adding .15% + 2.35% + 13.5% + 34% + 34% +13.5% (left to right) = 97.5%. The number associated with that in this problem
is 4.4.
97.5% of the data lie below 4.4, meaning 4.4 is at the 97.5 percentile.
4.4

Module 3: Measures of Variability – Ungrouped Data
Chebyshev’s Theorem (also called
Chebyshev’s Rule)
Chebyshev’s Theorem -- Used if the distribution of the data is unknown. Since we can’t be sure the, that the data is
normally distributed, we can’t assume, for example, that approximately 95% of the data lies within +/- two standard
deviations. We must assume it is some number less than 95%.
In this case, we use Chebyshev’s Theorem:
1-
of the data will fall within +/- k standard deviations of the data (where k > 1)
k = the number of standard deviations.
Suppose we have a data set of unknown distribution with a mean of 40 and a standard deviation of 5. What% of the data will fall between 30 and 50?30 is two standard deviations below the mean and 50 is two standard deviations above the mean. Therefore k = 2
•

Module 3: Measures of Variability – Ungrouped Data
Chebyshev’s Theorem (also called
Chebyshev’s Rule)
Why must k be greater than 1? Using the previous example, suppose we wanted to know what% of the data will fall
between 35 and 45?
In this case, k = 1 since 35 is one standard deviation below the mean and 45 is one above.

#### You've reached the end of your free preview.

Want to read all 48 pages?

- Spring '14
- DebraACasto
- Standard Deviation, Ungrouped Data