Center, Shape, & Spread of Data - ACT Math

1. What is the mode of the data set [7, 9, 7, 5, 9, 9]?
   Answer: 9 The mode is the value that appears most frequently in the dataset. Counting frequencies: 7 appears 2 times, 9 appears 3 times, 5 appears 1 time. Since 9 appears most frequently (3 times), the mode is 9. Choice B gave 7, which appears only twice.
2. A small store tracked the number of customers per hour over 6 hours: 7, 9, 9, 10, 12, 12. What is the mode of the data set $\{7, 9, 9, 10, 12, 12\}$?
   Answer: 9 and 12 The mode is the value or values that appear most frequently in the data set. In {7, 9, 9, 10, 12, 12}, both 9 and 12 appear twice, while others appear once. This data set is bimodal with modes 9 and 12. The modes are 9 and 12. Choice A incorrectly selects only one mode, ignoring the bimodal nature.
3. A data set of 9 values is $\{2, 3, 3, 4, 4, 4, 5, 6, 20\}$. Which best describes the shape of the distribution?​
   Answer: Skewed right The shape of the distribution describes how the data is spread, with skewness indicating a tail on one side. In {2, 3, 3, 4, 4, 4, 5, 6, 20}, most values cluster around 2 to 6, but 20 creates a long tail to the right. This indicates a right-skewed distribution. The distribution is skewed right. Choice A confuses it with left-skewed, where the tail would be to the left.
4. A small club recorded the number of members who attended 8 weekly meetings: 20, 21, 22, 22, 23, 23, 23, 24. If one additional meeting had 32 attendees added to the data set, how would the mean change?
   Answer: The mean would increase. The mean is the average value, and adding a new data point can change it depending on the value added relative to the current mean. The original data [20, 21, 22, 22, 23, 23, 23, 24] sums to 178 with 8 values, so mean = 178 / 8 = 22.25. Adding 32 gives a new sum of 210 and 9 values, new mean = 210 / 9 ≈ 23.33, which is higher than 22.25. The mean would increase. Choice A might assume the added value is below the mean, but 32 is above it.
5. What is the range of the data set [95, 85, 90, 100, 110]?
   Answer: 25 The range is calculated by subtracting the minimum value from the maximum value. In the data set [95, 85, 90, 100, 110], the maximum is 110 and the minimum is 85. Range = 110 - 85 = 25. Choice B gave 20, which would be an incorrect calculation.
6. A runner's mile times (in minutes) over 6 days are: 8, 9, 7, 10, 9, 8. What is the mode of the data set $\{8,9,7,10,9,8\}$?
   Answer: 8 and 9 The mode is the value or values that appear most frequently in a data set. In {8,9,7,10,9,8}, sort or count frequencies: 8 appears twice, 9 appears twice, 7 once, 10 once. Both 8 and 9 appear most often with two occurrences each. The modes are 8 and 9, making it bimodal. Choice D might assume no mode due to ties, but ties mean multiple modes.
7. A data set of 8 values is $\{1, 2, 4, 5, 5, 6, 7, 8\}$. If the value 9 is added to the data set, how does the mean change?​​
   Answer: The mean increases. The mean is the sum divided by the count, and adding a value changes it based on whether the new value is above or below the original mean. Original data {1, 2, 4, 5, 5, 6, 7, 8} sums to 38, mean 38/8 = 4.75. Adding 9 gives sum 47, mean 47/9 ≈ 5.22, which is higher. The mean increases. Choice B might assume the new value decreases it, but 9 is above the original mean.
8. A set of 9 integers is $\{3, 3, 4, 5, 5, 5, 6, 6, 9\}$. What is the median of the data set?
   Answer: 5 The median is the middle value when data is arranged in order. The set {3, 3, 4, 5, 5, 5, 6, 6, 9} has 9 values, so the median is the 5th value when sorted. Counting from the left: 1st=3, 2nd=3, 3rd=4, 4th=5, 5th=5. The median is 5. Choice A incorrectly applies the even-count rule for averaging two middle values.
9. A set of test scores is: 50, 60, 70, 80, 90. If a score of 0 is added to the data set, how does the median change?
   Answer: The median decreases The median is the middle value; adding data may shift it, especially for small sets. Original {50,60,70,80,90} has median 70 (3rd of 5). Adding 0: sorted {0,50,60,70,80,90}, even count, average 3rd (60) and 4th (70) for 65, which is less than 70. The median decreases. Choice C might ignore the shift, but adding a low value pulls it down.
10. What is the mean of the data set [10, 15, 20, 25, 30, 35]?
    Answer: 22.5 The mean is calculated by finding the sum of all values and dividing by the count. Sum = 10 + 15 + 20 + 25 + 30 + 35 = 135. Count = 6 values. Mean = 135 ÷ 6 = 22.5. Choice B gave 25, which is the median of this data set.
11. What is the mean of the data set [2, 4, 6, 8, 10]?
    Answer: 6 The mean is calculated by summing all values and dividing by the count. $Mean = (2 + 4 + 6 + 8 + 10) \div 5 = 30 \div 5 = 6$. This dataset represents an arithmetic sequence where each consecutive value increases by 2.
12. The data set of 7 integers is $\{4, 6, 7, 7, 8, 9, 10\}$. If a value of 13 is added to the data set, how does the mean change?
    Answer: It increases. The original mean is (4 + 6 + 7 + 7 + 8 + 9 + 10) ÷ 7 = 51 ÷ 7 ≈ 7.29. Adding 13 gives a new sum of 51 + 13 = 64, and the new mean is 64 ÷ 8 = 8. Since 8 > 7.29, the mean increases when a value above the original mean is added. Choice A (decreases) would occur if a value below the original mean were added.
13. A set of values is $\{4, 4, 5, 6, 6, 7, 20\}$. Which best describes the shape of the distribution?
    Answer: Skewed right Right-skewed distribution has a tail extending toward higher values, with most data clustered on the left. In {4, 4, 5, 6, 6, 7, 20}, most values (4, 5, 6, 7) are clustered at the lower end, while the value 20 creates a long tail extending to the right. This creates a right-skewed distribution. Choice A (skewed left) would describe the opposite pattern with the tail extending toward lower values.
14. A data set is $\{11, 14, 15, 15, 16, 18\}$. What is the range of the data?
    Answer: 7 The range is calculated as the difference between the maximum and minimum values. From the set {11, 14, 15, 15, 16, 18}, the maximum value is 18 and the minimum value is 11. Range = 18 - 11 = 7. Choice C gave 18, which is the maximum value itself rather than the range calculation.
15. What is the mean of the data set [3, 7, 5, 10, 8]?
    Answer: 6.6 The mean is calculated by finding the sum of all values and dividing by the count. Sum = 3 + 7 + 5 + 10 + 8 = 33. Count = 5 values. Mean = 33 ÷ 5 = 6.6. Choice B gave 7, which would be incorrect arithmetic.
16. The number of texts received each day for a week is: 3, 3, 4, 4, 4, 5, 20. Which best describes the shape of this distribution?
    Answer: Skewed right The shape of a distribution describes how data is spread, such as skewness indicating a tail on one side. For {3,3,4,4,4,5,20}, most values cluster around 3-5 with a tail extending to 20 on the right. This creates a longer tail on the higher side. The distribution is skewed right. Choice A confuses with left skew, where the tail would be on the lower side.
17. The data set $\{2, 3, 3, 4, 4, 4, 5, 20\}$ has one unusually large value. Which best describes the shape of the distribution?
    Answer: Skewed right Right-skewed distribution has a tail extending toward higher values, with most data clustered on the left. In {2, 3, 3, 4, 4, 4, 5, 20}, most values (2, 3, 4, 5) are clustered at the lower end, while the value 20 creates a long tail extending to the right. This creates a right-skewed distribution. Choice A (skewed left) would describe the opposite pattern with the tail extending toward lower values.
18. What is the mode of the data set [3, 6, 9, 6, 15, 6, 10]?
    Answer: 6 The mode is the value that appears most frequently in the data set. Looking at [3, 6, 9, 6, 15, 6, 10], the value 6 appears three times. The values 3, 9, 15, and 10 each appear only once. Therefore, the mode is 6.
19. The numbers of goals scored by a team in 8 games are 2, 0, 3, 1, 2, 4, 1, 2. What is the median of $\{2,0,3,1,2,4,1,2\}$?
    Answer: 2 To find the median, first sort the data: 0, 1, 1, 2, 2, 2, 3, 4. With 8 values (even count), the median is the average of the 4th and 5th values. Median = (2 + 2)/2 = 2. Choice B incorrectly calculated (1 + 2)/2 = 1.5.
20. A cafeteria recorded the number of minutes 8 students waited in line: 2, 3, 3, 4, 4, 5, 6, 15. Which best describes the shape of the distribution of wait times?
    Answer: Skewed right The shape of a distribution describes characteristics like symmetry or skewness based on the data's spread and tails. For the wait times [2, 3, 3, 4, 4, 5, 6, 15], observe that most values cluster between 2 and 6, with a long tail extending to 15 on the higher end. This indicates the tail extends to the right, characteristic of positive skewness. The distribution is best described as skewed right. Choice A confuses this with left-skewed, where the tail would extend to lower values.

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