The mean is the sum divided by the number of values: (4 + 7 + 7 + 9 + 13) / 5 = 40 / 5 = 8. The value 7 is the middle value and the most frequent value, not the mean. The value 9 is the range, 13 - 4. The value 40 is the sum before dividing by 5.
Category: Measures of Center
Question 2
What is the median of the data set 3, 5, 8, 10, 12, 20? Round to two decimal places if needed.
8
9
✓
10
9.67
Correct Answer
9
With six ordered values there is no single middle value, so the median is the average of the third and fourth values: (8 + 10) / 2 = 9. The value 8 is only the third value. The value 10 is only the fourth value. The value 9.67 is the mean, 58 / 6, rounded to two decimal places.
Category: Measures of Center
Question 3
A survey records each student's favorite pizza topping. Which measure of center can be found for these data?
Mean
Median
Mode
✓
Midrange
Correct Answer
Mode
Favorite topping is qualitative data, and the most common category is the only measure of center that makes sense for it, so the mode is the answer. The mean requires numbers that can be added and divided. The median requires values that can be put in a meaningful order. The midrange averages the largest and smallest numbers, which unordered categories do not have.
Category: Measures of Center
Question 4
Ten employees of a small firm earn between 40,000 and 60,000 dollars a year, and the owner earns 2,000,000 dollars. Why does the median describe a typical salary better than the mean here?
The median uses every salary in its calculation
The mean leaves out the largest salary
The median is quicker to compute by hand
The owner's salary pulls the mean far upward
✓
Correct Answer
The owner's salary pulls the mean far upward
The mean adds every salary, so one extreme value drags it far above what almost everyone earns, while the median depends only on the middle of the ordered data. The claim that the median uses every salary in its calculation is wrong, since it is the mean that uses every value's size. The claim that the mean leaves out the largest salary is backward, because the mean includes it, which is the whole problem. Being quicker to compute by hand is not a reason one measure describes the data better.
Category: Measures of Center
Question 5
A data set has a mean of 52 and a median of 45. Which shape is the distribution most likely to have?
Skewed right
✓
Skewed left
Symmetric
Uniform
Correct Answer
Skewed right
A mean well above the median suggests a long tail of high values pulling the mean upward, which is a right (positive) skew. Skewed left would usually put the mean below the median. Symmetric would put the mean and median about equal. Uniform is a symmetric, flat shape, so its mean and median would also be about equal.
Category: Skewness
Question 6
In a distribution that is skewed to the left, how does the mean usually compare with the median?
The mean is greater than the median
The mean is less than the median
✓
The mean equals the median
The mean equals the mode
Correct Answer
The mean is less than the median
A long left tail of low values pulls the mean down, so the mean usually falls below the median. The mean being greater than the median is the typical pattern for a right-skewed distribution. The mean equaling the median is the pattern for a symmetric distribution. The mean equaling the mode is also a symmetric pattern, while in a left-skewed distribution the mode usually sits above the mean.
Category: Skewness
Question 7
Five commute times, in minutes, are 12, 15, 9, 22 and 18. What is their range?
10
9
13
✓
22
Correct Answer
13
The range is the largest value minus the smallest value: 22 - 9 = 13. The value 10 comes from subtracting the first listed value, 22 - 12, instead of the smallest. The value 9 is the minimum. The value 22 is the maximum.
Category: Measures of Spread
Question 8
A data set has Q1 = 20 and Q3 = 34. Using the 1.5 x IQR rule, values above what number are potential outliers?
48
21
-1
55
✓
Correct Answer
55
The IQR is 34 - 20 = 14, and 1.5 x 14 = 21, so the upper fence is 34 + 21 = 55. The value 48 adds only one IQR to Q3 (34 + 14). The value 21 is 1.5 x IQR by itself, before adding it to Q3. The value -1 is the lower fence, 20 - 21, which marks low outliers.
Category: Measures of Location
Question 9
The interquartile range measures the spread of which portion of a data set?
The middle 50% of the data
✓
The lowest 25% of the data
The middle 75% of the data
Every value from minimum to maximum
Correct Answer
The middle 50% of the data
The IQR is Q3 - Q1, and the values between the first and third quartiles make up the middle 50% of the data. The lowest 25% of the data lie below Q1. The middle 75% of the data would need different cutoffs, such as the 12.5th and 87.5th percentiles. Every value from minimum to maximum is described by the range.
Category: Measures of Location
Question 10
A student's exam score is at the 85th percentile. What does this mean?
She answered 85% of the questions correctly
About 85% of scores are at or below hers
✓
About 85% of scores are above hers
Her score was exactly 85 points
Correct Answer
About 85% of scores are at or below hers
A percentile gives the percentage of data values at or below a given value, so about 85% of scores are at or below hers. Answering 85% of the questions correctly describes a percent score, which is a different idea. About 85% of scores being above hers would describe the 15th percentile. A score of exactly 85 points is a raw score, not a position relative to others.
Category: Measures of Location
Question 11
In a box plot, the two ends of the box are drawn at which values?
The minimum and maximum values
The mean and the median
The first and third quartiles
✓
The 10th and 90th percentiles
Correct Answer
The first and third quartiles
The box stretches from Q1 to Q3, so its length shows the interquartile range. The minimum and maximum values are where the whiskers end in a basic box plot. The mean and the median are measures of center, and only the median is marked, by a line inside the box. The 10th and 90th percentiles are not part of a standard box plot.
Category: Measures of Location
Question 12
Which list gives the five-number summary of a data set?
Mean, median, mode, range, IQR
Minimum, Q1, mean, Q3, maximum
Q1, median, Q3, IQR, range
Minimum, Q1, median, Q3, maximum
✓
Correct Answer
Minimum, Q1, median, Q3, maximum
The five-number summary is the minimum, first quartile, median, third quartile and maximum, which are the values a box plot displays. Mean, median, mode, range, IQR mixes measures of center and spread. Minimum, Q1, mean, Q3, maximum wrongly uses the mean in place of the median. Q1, median, Q3, IQR, range includes two differences instead of the minimum and maximum.
Category: Measures of Location
Question 13
A sample has the values 2, 4 and 6. What is the sample standard deviation? Round to two decimal places if needed.
2
✓
1.63
4
8
Correct Answer
2
The mean is 4, the squared deviations are 4, 0 and 4, and their sum is 8. The sample variance divides by n - 1 = 2, giving 4, and the standard deviation is sqrt(4) = 2. The value 1.63 comes from dividing by n = 3 instead of n - 1, giving sqrt(8/3). The value 4 is the variance before taking the square root. The value 8 is the sum of squared deviations.
Category: Measures of Spread
Question 14
A sample of quiz scores has a standard deviation of 5 points. What is the variance? Round to two decimal places if needed.
5
25
✓
2.24
10
Correct Answer
25
The variance is the square of the standard deviation: 5^2 = 25, in squared points. The value 5 is the standard deviation itself. The value 2.24 is sqrt(5), which takes a square root instead of squaring. The value 10 doubles the standard deviation instead of squaring it.
Category: Measures of Spread
Question 15
Test scores have a mean of 70 and a standard deviation of 8. What is the z-score of a score of 82?
12
0.67
1.5
✓
-1.5
Correct Answer
1.5
The z-score is (value - mean) / standard deviation = (82 - 70) / 8 = 12 / 8 = 1.5, so the score is 1.5 standard deviations above the mean. The value 12 is the raw difference before dividing by 8. The value 0.67 comes from dividing 8 by 12, which inverts the fraction. The value -1.5 has the wrong sign, since 82 is above the mean.
Category: Z-Scores and Empirical Rule
Question 16
A data set has a mean of 50 and a standard deviation of 4. Which value lies 2 standard deviations below the mean?
46
48
58
42
✓
Correct Answer
42
A value 2 standard deviations below the mean is 50 - 2(4) = 50 - 8 = 42. The value 46 is only 1 standard deviation below the mean. The value 48 subtracts 2 points instead of 2 standard deviations. The value 58 is 2 standard deviations above the mean.
Category: Z-Scores and Empirical Rule
Question 17
Ana scores 85 on a test with a mean of 75 and a standard deviation of 5. Ben scores 90 on a different test with a mean of 80 and a standard deviation of 10. Who did better relative to the other students taking the same test?
Ana, because she is 2 standard deviations above her mean
✓
Ben, because his raw score is higher than Ana's
Neither, because each is 10 points above the mean
Ben, because his test had more spread in scores
Correct Answer
Ana, because she is 2 standard deviations above her mean
Ana's z-score is (85 - 75) / 5 = 2 and Ben's is (90 - 80) / 10 = 1, so Ana stands farther above her class. Ben's raw score being higher does not matter, since the tests have different means and spreads. Neither doing better is wrong, because 10 points is a bigger jump where the standard deviation is 5. More spread in Ben's scores makes his 10-point lead smaller in relative terms, not larger.
Category: Z-Scores and Empirical Rule
Question 18
Scores on a test have a bell-shaped, symmetric distribution with a mean of 100 and a standard deviation of 15. By the empirical rule, about 95% of scores fall between which two values?
85 and 115
70 and 130
✓
55 and 145
90 and 110
Correct Answer
70 and 130
The empirical rule says about 95% of data in a bell-shaped distribution lie within 2 standard deviations of the mean: 100 - 30 = 70 and 100 + 30 = 130. The interval 85 and 115 is within 1 standard deviation, which holds about 68%. The interval 55 and 145 is within 3 standard deviations, which holds nearly all of the data. The interval 90 and 110 does not match any whole number of standard deviations.
Category: Z-Scores and Empirical Rule
Question 19
Which graph is the best choice for showing how a city's monthly unemployment rate changed over five years?
A pie chart
A box plot
A time series graph
✓
A stem-and-leaf plot
Correct Answer
A time series graph
A time series graph plots values in time order, usually connected by lines, so rises and falls over the five years are easy to see. A pie chart shows parts of a whole at a single moment. A box plot summarizes the distribution of values but loses their time order. A stem-and-leaf plot also shows the distribution of values without showing when each one occurred.
Category: Graphing Data
Question 20
In a stem-and-leaf plot with the key 4 | 2 = 42, one row shows stem 4 with leaves 2, 5 and 8. Which data values does this row represent?
24, 54 and 84
4, 2, 5 and 8
42, 47 and 52
42, 45 and 48
✓
Correct Answer
42, 45 and 48
Each leaf is the final digit placed after the stem, so stem 4 with leaves 2, 5 and 8 gives 42, 45 and 48. The values 24, 54 and 84 put the leaf in front of the stem, which reverses the digits. The values 4, 2, 5 and 8 treat each digit as its own data value. The values 42, 47 and 52 cannot be right because every value in this row shares the stem 4, so all of them must be in the 40s.
Category: Graphing Data