WP.2.4: CHEBYSHEV’S THEOREM & THE EMPIRICAL RULE
[WP.2.4]
Application of the standard deviation is a critical question in Statistics. Two ways to preliminarily demonstrate this concept is by examining Chebyshev’s Theorem and The Empirical Rule.
I. CHEBYSHEV’S THEOREM
As evidenced by Russian mathematician Pafnuty Chebyshev (1821-1894), irrespective of shape, the boundaries on the proportion of the data will lie a specified number of standard deviations from the mean. A few examples are as follows:
- At least 75% of the data is within 2 standard deviations of the mean.
- At least 89% of the data is within 3 standard deviations of the mean.
- At least 95% of the data is within 4 1/2 standard deviations of the mean.
Chebyshev’s rule holds for both populations and samples and can be mathematically summarized as follows:
Percentage of Values Surrounding the Mean = 1 - (1/k2)
Note: Chebyshev’s Theorem offers only a rough estimation but serves to establish the relationship that exists between the number of standard deviations from the mean and the percentage/proportion of the data surrounding the mean.
Demonstration 1: On the first test in BA254, the data indicated that the mean score was 125 and the standard deviation was 10. What is the percentage of students that scored between 100 and 150?
Answer: 0.84 or 84%
Problem 1: Chebyshev’s Theorem can be applied to
a. only normal distributions.
b. only symmetrical distributions.
c. any distribution.
d. only negatively skewed distributions.
e. only positively skewed distributions.
Problem 2: According to Chebyshev’s Theorem, at least what proportion of values will lie within two standard deviations of the arithmetic mean?
a. Approximately 75%
b. Approximately 95%
c. Approximately 99.7%
d. At least 75%
Problem 3: The mean weekly income of employees at XYZ Company is $550 and the standard deviation is $75. According to the Chebyshev’s Theorem, at least what percent of the incomes lie within 1.5 standard deviation of the mean?
Problem 4: The mean weigh of a group of male GRCC
students is 180lbs. and the standard deviation is 15 lbs. According to Chebyshev’s Theorem, at least
what percent of the students weigh between 141 lbs and 219 lbs?
Problem 5: The mean IQ scores of typical adults on the Weschler test is 100 with a standard deviation of 15. Using Chebyshev’s Theorem, at least what percentage of adults have a score between 55 and 145?
Problem 6: The mean weight of a package handled by Speedy Delivery Inc. is 18 lbs with a standard deviation of 7 lbs. Using Chebyshev’s Theorem, at least what percentage of packages will lie within 2 standard deviations of the mean?
Problem 7: The mean height of GRCC students is 65 inches with a standard deviation of 3 inches. According to Chebyshev’s theorem, at most, what percentage of students are either greater than 71 inches tall or shorter than 59 inches tall?
II. THE EMPIRICAL RULE
For data that is bell-shaped and symmetrical (Normal):
- Approximately 68% of the data is within +/- 1 standard deviations of the mean.
- Approximately 95% of the data is within +/- 2 standard deviations of the mean.
- More than 99% of the data is within +/- 3 standard deviations of the mean.
Studying the "Normal" or "Gaussian" probability distribution will help to demonstrate this concept in far greater detail and accuracy.
Demonstration 2: A certain Professor was looking to purchase a new pair of skis heading into the winter. His banker spouse was supportive but was interested in how much said Professor intended to spend. The professor said “Eh, should be somewhere around $250.” This did not satisfy the Banker who requested additional information. The Professor went out and did some research and found that the mean cost of the skis he was looking for was $275 and the standard deviation was $50. According to the Empirical rule answer the following:
a. Approximately 68% of all ski prices are between what two values?
b. Approximately 95% of all ski prices are between what two values?
c. Approximately 99%+ of all ski prices are between what two values?
Answer:
a. $225 - 325
b. $175 - $375
c. $125 - $425
Problem 8: The Empirical Rule can be applied to
a. only normal distributions.
b. only asymmetrical distributions.
c. any distribution.
d. only negatively skewed distributions.
e. only positively skewed distributions.
Problem 9: Which of the following is not true about the normal distribution?
a. It is bell shaped.
b. It is symmetrical.
c. The mean, median, and mode are all equal.
d. It is skewed.
Problem 10: The mean age of a manager at a local department store is 35.5 years and the standard deviation is 4.5 years.
a. Assuming that the ages are normally distributed, approximately 99.7% of the managers’ ages will fall between what two values?
b. Approximately 68% of the ages will fall between what two values?
Problem 11: According to the Empirical Rule, what proportion of values will lie within two standard deviations of the arithmetic mean?
a. Approximately 68%
b. Approximately 95%
c. Approximately 99.7%
d. At least 75%
Problem 12: The average number of hours worked by GRCC Business Lab employees is 22 hours with a standard deviation of 7 hours. According to the Empirical Rule, 95% of the employees’ hours worked will fall between what two values?
Problem 13: A survey of 150 adult males revealed that mean number of hours of sleep per night is 7.5 hours with a standard deviation of 0.5 hours. Assuming a normal distribution, approximately what percent of the men sleep between 6 hours and 9 hours per night?
Problem 14: The mean price of a gallon of gasoline is Grand Rapids is currently $1.98 and the standard deviation is $0.02. Assuming that the price of gasoline are normally distributed, approximately what percent of the prices will fall:
a. Between $1.96 and $2.00?
b. Between $1.94 and $2.02?
c. Above $2.00?
d. Between $1.94 and $2.00?
Problem 15: The mean time of a customer service call for an internet service provider is 12 minutes with a standard deviation of 3 minutes. Assuming a normal distribution, what percent of calls are longer than 9 minutes?
Problem 16: The mean number of zucchinis provided from a zucchini plant in a season is normally distributed with a mean of 13 and a standard deviation of 3. If Paul plants 20 zucchini plants, how many plants can he expect to produce between 7 and 19 zucchinis?
Problem 17: The mean number of visits to the local grocery store on a given day is normally distributed with a mean of 250 and a standard deviation of 25. In a given day, approximately what percent of the time will
a. More than 250 people visit the store
b. Between 200 and 250 people visit the store
c. Less than 200 people visit the store
d. Less than 275 people visit the store
Answers:
1. C
2. D
3. 55.6%
4. 85.2%
5. 88.9%
6. 75%
7. 25%
8. A
9. D
10. a. 22 and 49 b. 31 and 40
11. B
12. 8 and 36
13. 99.7%
14. a. 68% b. 95% c. 16% d. 81.5%
15. 84%
16. 19
17. a. 50% b. 47.5% c. 2.5% d. 84%