CHAPTER 4A: NUMERICAL DESCRIPTIVE TECHNIQUES
TRUE/FALSE
1. The mean is affected by extreme values but the median is not.
2. The mean is a measure of variability.
3. In a histogram, the proportion of the total area which must be to the left of the median is more than
0.50 if the distribution is positively skewed.
4. A data sample has a mean of 107, a median of 122, and a mode of 134. The distribution of the data is
positively skewed.
5.
is a population parameter and is a sample statistic.
6. In a bell shaped distribution, there is no difference in the values of the mean, median, and mode.
7. Lily has been keeping track of what she spends to eat out. The last week’s expenditures for meals eaten
out were $5.69, $5.95, $6.19, $10.91, $7.49, $14.53, and $7.66. The mean amount Lily spends on
meals is $8.35.
8. In a negatively skewed distribution, the mean is smaller than the median and the median is smaller
than the mode.
9. The median of a set of data is more representative than the mean when the mean is larger than most of
the observations.
10. The value of the mean times the number of observations equals the sum of the observations.
11. In a histogram, the proportion of the total area which must be to the left of the median is less than 0.50
if the distribution is negatively skewed.
12. In a histogram, the proportion of the total area which must be to the right of the mean is exactly 0.50 if
the distribution is symmetric and unimodal.
13. Suppose a sample of size 50 has a sample mean of 20. In this case, the sum of all observations in the
sample is 1,000.
14. The median of an ordered data set with 30 items would be the average of the 15th and the 16th
observations.
15. If the mean, median, and mode are all equal, the histogram must be symmetric and bell shaped.
MULTIPLE CHOICE
1. Which of the following statistics is a measure of central location?
a.
The mean
c.
The mode
b.
The median
d.
All of these choices are true.
2. Which measure(s) of central location is/are meaningful when the data are ordinal?
a.
The mean and median
b.
The mean and mode
c.
The median and mode
d.
Only mean
3. Which of the following statements about the mean is not always correct?
a.
The sum of the deviations from the mean is zero.
b.
Half of the observations are on either side of the mean.
c.
The mean is a measure of the central location.
d.
The value of the mean times the number of observations equals the sum of all
observations.
4. Which of the following statements is true for the following observations: 9, 8, 7, 9, 6, 11, and 13?
a.
The mean, median, and mode are all equal.
b.
Only the mean and median are equal.
c.
Only the mean and mode are equal
d.
Only the median and mode are equal.
5. In a histogram, the proportion of the total area which must be to the left of the median is:
a.
exactly 0.50.
b.
less than 0.50 if the distribution is negatively skewed.
c.
more than 0.50 if the distribution is positively skewed.
d.
unknown.
6. Which measure of central location can be used for both interval and nominal variables?
a.
The mean
c.
The mode
b.
The median
d.
All of these choices are true.
7. Which of these measures of central location is not sensitive to extreme values?
a.
The mean
c.
The mode
b.
The median
d.
All of these choices are true.
8. In a positively skewed distribution:
a.
the median equals the mean.
b.
the median is less than the mean.
c.
the median is larger than the mean.
d.
the mean can be larger or smaller than the median.
9. Which of the following statements about the median is not true?
a.
It is more affected by extreme values than the mean.
b.
It is a measure of central location.
c.
It is equal to Q2.
d.
It is equal to the mode in a bell shaped distribution.
10. Which of the following summary measures is sensitive to extreme values?
a.
The median
c.
The mean
b.
The interquartile range
d.
The first quartile
11. In a perfectly symmetric bell shaped “normal” distribution:
a.
the mean equals the median.
c.
the mean equals the mode.
b.
the median equals the mode.
d.
All of these choices are true.
12. Which of the following statements is true?
a.
When the distribution is positively skewed, mean < median < mode.
b.
When the distribution is negatively skewed, mean > median > mode.
c.
When the distribution is symmetric and unimodal, mean = median = mode.
d.
When the distribution is symmetric and bimodal, mean = median = mode.
13. In a histogram, the proportion of the total area which must be to the right of the mean is:
a.
less than 0.50 if the distribution is negatively skewed.
b.
exactly 0.50.
c.
more than 0.50 if the distribution is positively skewed.
d.
exactly 0.50 if the distribution is symmetric and unimodal.
14. The average score for a class of 30 students was 75. The 15 male students in the class averaged 70.
The 15 female students in the class averaged:
a.
85.
c.
75
b.
80
d.
70
COMPLETION
1. Another word for the mean of a data set is the ____________________.
2. The size of a sample is denoted by the letter ____________________ and the size of a population is
denoted by the letter ____________________.
3. The sample mean is denoted by ____________________ and the population mean is denoted by
____________________.
4. There are three measures of central location; the mean, the ____________________, and the
____________________.
5. The ____________________ is calculated by finding the middle of the data set, when the data are
ordered from smallest to largest.
6. The ____________________ is the least desirable of all the measures of central location.
7. The ____________________ is not as sensitive to extreme values as the ____________________.
8. The ____________________ mean is used whenever we wish to find the “average” growth rate, or
rate of change, in a variable over time.
9. The ____________________ mean of n returns (or growth rates) is the appropriate mean to calculate
if you wish to estimate the mean rate of return (or growth rate) for any single period in the future.
10. If a data set contains an even number of observations, the median is found by taking the
____________________ of these two numbers.
11. If a data set is composed of 5 different numbers, there are ____________________ modes.
SHORT ANSWER
Strip Mall Rent
Monthly rent data in dollars for a sample of 10 stores in a small town in South Dakota are as follows:
220, 216, 220, 205, 210, 240, 195, 235, 204, and 250.
1. {Strip Mall Rent Narrative} Compute the sample monthly average rent.
2. {Strip Mall Rent Narrative} Compute the sample median.
3. {Strip Mall Rent Narrative} What is the mode?
Pets Survey
A sample of 36 families were asked how many pets they owned. Their responses are summarized in
the following table.
Number of Pets
0
1
2
3
4
5
Number of Families
3
20
5
4
2
2
4. {Pets Survey Narrative} Determine the mean, the median, and the mode of the number of pets owned
per family.
5. {Pets Survey Narrative} Explain what the mean, median, and mode tell you about this particular data
set.
6. How do the mean, median, and mode compare to each other when the distribution is:
a.
symmetric?
b.
negatively skewed?
c.
positively skewed?
a.
mean = median = mode
b.
mean < median < mode
c.
mean > median > mode
7. A basketball player has the following points for seven games: 20, 25, 32, 18, 19, 22, and 30. Compute
the following measures of central location:
a.
mean
b.
median
c.
mode
ANS:
Computers
The following data represent the number of computers owned by a sample of 10 families from
Chicago: 4, 2, 1, 1, 5, 3, 0, 1, 0, and 2.
8. {Computers Narrative} Compute the mean number of computers.
9. {Computers Narrative} Compute the median number of computers.
10. {Computers Narrative} Is the distribution of the number of computers symmetric or skewed? Why?
Weights of Workers
The following data represent the number of employees of a sample of 25 companies: 164, 148, 137,
157, 173, 156, 177, 172, 169, 165, 145, 168, 163, 162, 174, 152, 156, 168, 154, 151, 174, 146, 134,
140, and 171.
11. {Weights of Workers Narrative} Construct a stem and leaf display for the number of workers.
12. {Weights of Workers Narrative} Find the median number of workers.
13. {Weights of Workers Narrative} Find the mean number of workers.
14. {Weights of Workers Narrative} Is the distribution of the number of workers symmetric or skewed?
Why?
15. The number of hours a college student spent studying during the final exam week was recorded as
follows: 7,6, 4, 9, 8, 5, and 10. Compute for the data and the value in an appropriate unit.
Hours Worked per Week
The following data represent the hours worked per week of a sample of 25 employees from a
government department: 31, 43, 56, 23, 49, 42, 33, 61, 44, 28, 48, 38, 44, 35, 40, 64, 52, 42, 47, 39, 53,
27, 36, 35, and 20.
16. {Hours Worked per Week Narrative} Construct a stem and leaf display for the hours.
17. {Hours Worked per Week Narrative} Find the median hours.
18. {Hours Worked per Week Narrative} Compute the sample mean hours.
19. {Hours Worked per Week Narrative} Find the modal hours.
20. {Hours Worked per Week Narrative} Compare the mean and median hours for these employees and
use them to discuss the shape of the distribution.
Salaries of Employees
The following data represent the yearly salaries (in thousands of dollars) of a sample of 13 employees
of a firm: 26.5, 23.5, 29.7, 24.8, 21.1, 24.3, 20.4, 22.7, 27.2, 23.7, 24.1, 24.8, and 28.2.
21. {Salaries of Employees Narrative} Compute the mean salary.
22. {Salaries of Employees Narrative} Compute the median salary.
23. {Salaries of Employees Narrative} Compare the mean salary with the median salary and use them to
describe the shape of the distribution.
24. A sample of 12 construction workers has a mean age of 25 years. Suppose that the sample is enlarged
to 14 construction workers, by including two additional workers having common age of 25 each. Find
the mean of the sample of 14 workers.
25. The mean of a sample of 15 measurements is 35.6 feet. Suppose that the sample is enlarged to 16
measurements, by including one additional measurement having a value of 42 feet. Find the mean of
the sample of the 16 measurements.
26. A mutual fund you purchased in the years 2011-2014 has the following rates of return shown below:
Year
2011
2012
2013
2014
Rate of Return
.50
.30
−.10
−.15
Compute the geometric mean.
2-Year Investment
Suppose you make a 2-year investment of $5,000 and it grows by 100% to $10,000 during the first
year. During the second year, however, the investment suffers a 50% loss, from $10,000 back to
$5,000.
27. {2–Year Investment Narrative} Calculate the arithmetic mean.
28. {2–Year Investment Narrative} Calculate the geometric mean.
29. {2–Year Investment Narrative} Compare the values of the arithmetic and geometric means.
Ages of Senior Citizens
A sociologist recently conducted a survey of citizens over 65 years of age whose net worth is too high
to qualify for Medicaid and who have no private health insurance. The ages of 22 uninsured senior
citizens were as follows: 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 86, 87, 91, 92,
94, and 97.
30. {Ages of Senior Citizens Narrative} Calculate the mean age of the uninsured senior citizens
31. {Ages of Senior Citizens Narrative} Calculate the median age of the uninsured senior citizens.
ANS:
32. {Ages of Senior Citizens Narrative} Explain why there is no mode for this data set.
33. Suppose that a firm’s sales were $2,500,000 four years ago, and sales have grown annually by 25%,
15%, −5%, and 10% since that time. What was the geometric mean growth rate in sales over the past
four years?
34. Suppose that a firm’s sales were $3,750,000 five years ago and are $5,250,000 today. What was the
geometric mean growth rate in sales over the past five years?
35. The value of the standard deviation may be either positive or negative, while the value of the variance
will always be positive.
36. The difference between the largest and smallest observations in an ordered data set is called the range.
37. The standard deviation is expressed in terms of the original units of measurement but the variance is
not.
38.
is a population parameter and s is a sample statistic.
39. While Chebysheff’s Theorem applies to any distribution, regardless of shape, the Empirical Rule
applies only to distributions that are bell shaped.
40. The mean of fifty sales receipts is $65.75 and the standard deviation is $10.55. Using Chebysheff’s
Theorem, 75% of the sales receipts were between $44.65 and $86.85.
41. The data set 10, 20, 30 has the same variance as the data set 100, 200, 300.
42. According to Chebysheff’s Theorem, at least 93.75% of observations should fall within 4 standard
deviations of the mean.
43. Chebysheff’s Theorem states that the percentage of observations in a data set that should fall within
five standard deviations of their mean is at least 96%.
44. The Empirical Rule states that the percentage of observations in a data set (providing that the data set
is bell shaped) that fall within one standard deviation of their mean is approximately 75%.
45. A population with 200 elements has a variance of 20. From this information, it can be shown that the
population standard deviation is 10.
ANS:
46. If two data sets have the same range, the distances from the smallest to largest observations in both
sets will be the same.
47. The coefficient of variation is a measure of variability.
48. The standard deviation is the positive square root of the variance.
49. If two data sets have the same standard deviation, they must have the same coefficient of variation.
50. The units for the variance are the same as the units for the original data (for example, feet, inches,
etc.).
51. The units for the standard deviation are the same as the units for the original data (for example, feet,
inches, etc.).
52. The variance is more meaningful and easier to interpret compared to the standard deviation.
53. The range is considered the weakest measure of variability.
54. Chebysheff’s Theorem applies only to data sets that have a bell shaped distribution.
55. The coefficient of variation allows us to compare two sets of data based on different measurement
units.
56. If the observations are in the millions, a standard deviation of 10 would be considered small. If the
observations are all less than 50, a standard deviation of 10 would be considered large.
57. A sample of 20 observations has a standard deviation of 3. The sum of the squared deviations from the
sample mean is:
a.
20.
b.
23.
c.
29.
d.
171.
58. If two data sets have the same range:
a.
the distances from the smallest to largest observations in both sets will be the same.
b.
the smallest and largest observations are the same in both sets.
c.
both sets will have the same standard deviation.
d.
both sets will have the same interquartile range.
59. The Empirical Rule states that the approximate percentage of measurements in a data set (providing
that the data set has a bell shaped distribution) that fall within two standard deviations of their mean is
approximately:
a.
68%.
b.
75%.
c.
95%.
d.
99%.
60. Which of the following summary measures is affected most by extreme values?
a.
The median.
b.
The mean.
c.
The range.
d.
The interquartile range.
61. Chebysheff’s Theorem states that the percentage of measurements in a data set that fall within three
standard deviations of their mean is:
a.
75%.
b.
at least 75%.
c.
89%.
d.
at least 88.9%.
62. Which of the following is a measure of variability?
a.
The interquartile range.
b.
The variance.
c.
The coefficient of variation.
d.
All of these choices are true.
63. The smaller the spread of scores around the mean:
a.
the smaller the variance of the data set.
b.
the smaller the standard deviation of the data set.
c.
the smaller the coefficient of variation of the data set.
d.
All of these choices are true.
64. Is a standard deviation of 10 a large number indicating great variability, or is it small number
indicating little variability? To answer this question correctly, one should look carefully at the value of
the:
a.
mean.
b.
standard deviation.
c.
coefficient of variation.
d.
mean dividing by the standard deviation.