One-variable data distributions and measures of center and spread: worksheet with answers

The number of biography books checked out was 70, and the number of history books checked out was 30. Subtract to find how many more biography books were checked out: \(70-30=40\). Therefore, the correct answer is 40.

The distractors come from common mistakes: 100 is the sum of the two categories, 70 is the biography total only, and 30 is the history total only.

Look at the row labeled Mystery in the table. The number of tickets sold for that movie night is 104.

In a box plot, the median is represented by the center line inside the box. From the five-number summary, the median value is 19.

The Robotics club has 23 students, and the Art club has 15 students. Subtract to find how many more students signed up for Robotics:

\(23-15=8\)

So the correct answer is 8.

The minimum value in a data set is the least value listed. In the table, the scores are 14, 19, 27, 35, and 42. The smallest of these values is 14. The frequencies only show how many times each score appears and do not change the minimum value.

Both distributions are centered around about the same middle interval, 55–59 seconds. However, Team River's times are concentrated near the center intervals, while Team Valley's times are spread more evenly across the outer intervals.

Standard deviation measures how spread out data values are from the mean. Since Team Valley has more data farther from the center, Team Valley has the larger standard deviation.

Adding the same positive number to every value in a data set increases the median by that number because the middle value also increases. However, the range stays the same because both the minimum and maximum values increase equally.

Therefore, the median of Data Set B is greater than the median of Data Set A, and the ranges of the two data sets are equal.

To find the median, first arrange the numbers from least to greatest:

7, 9, 12, 12, 15, 18, 21, 24

There are 8 numbers, so the median is the average of the two middle numbers, which are the 4th and 5th values: 12 and 15.

\(\frac{12+15}{2}=13.5\)

So the median is 13.5. Choice A is the mean of the data set, choice B comes from using a middle value in the unsorted list, and choice D is the range because \(24-7=17\).

Adding the same constant to every value in a data set increases measures of center, such as the median, by that constant. Therefore, the median of the new data set is greater than the median of the original data set.

The range depends on the difference between the greatest and least values. In the original set, the range is \(18-6=12\). After adding 25 to every value, the greatest and least values both increase by 25, so the new range is \((18+25)-(6+25)=43-31=12\). The range stays the same.

Therefore, the correct choice is that the median is greater and the range is equal.

There are 25 food trucks, so the median is the 13th value when the data are ordered from least to greatest.

Use cumulative frequencies to locate the 13th value:

• 20–29: positions 1–4
• 30–39: positions 5–10
• 40–49: positions 11–17

The 13th value falls in the interval 40–49, so the median must be a number in that interval. Of the choices given, only 44 is in that interval.

The original data set has 81 values, so the median is the 41st value when the prices are listed in order. The cumulative frequencies are:

• Up to 8 dollars: 6
• Up to 12 dollars: 15
• Up to 15 dollars: 29
• Up to 18 dollars: 46

Since the 41st value falls in the group priced at 18 dollars, the original median is 18.

After the $60 book is added, the data set has 82 values. The median is now the average of the 41st and 42nd values. Both the 41st and 42nd values are still in the 18-dollar group because the cumulative count through 18 dollars is 46. Therefore, the median remains unchanged.

The added value, 60, is greater than every original value, so it increases the total sum of the data values by a large amount. Since the mean equals the sum divided by the number of values, the mean increases.

Therefore, the mean increases while the median stays the same.

Each data set is symmetric around 44. Because the distributions are symmetric and each set has an odd number of values, the middle value in each set is 44. Therefore, the medians are equal, so statement I is true.

However, the spreads are different. Data set B has more values at the extreme numbers 32 and 56 and fewer values closer to the center compared with data set A. Values farther from the center increase the standard deviation, so the standard deviations are not equal. Therefore, statement II is false.

Thus, only statement I must be true.