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ACT Math

ACT Math Help: Center Shape And Spread Of Data

Review real example questions for Center Shape And Spread Of Data in ACT Math.

Question 1

A researcher surveyed 50 families and summarized her findings in the table. Of the families surveyed, what is the mean number of vehicles per family?

  1. 1.9
  2. 2
  3. 2.1
  4. 2.5
Explanation: This is a mean from frequency table question testing the weighted average calculation. Choice A (1.9) is correct — multiply each vehicle count by its frequency, sum the products, then divide by total families. Total vehicles = (1 × 15) + (2 × 25) + (3 × 10) = 15 + 50 + 30 = 95. Mean = 95/50 = 1.9. Choice B (2.0) comes from computing (1 + 2 + 3)/3 = 2 — averaging the vehicle numbers without weighting by frequency. Choice C (2.1) results from an arithmetic error in one product, perhaps computing (2 × 25) = 52 instead of 50: 15 + 52 + 30 = 97, 97/50 = 1.94 ≈ 2.1... or another minor error. Choice D (2.5) averages only the vehicle counts (1 + 2 + 3 + 4)/4 type error, or computes (15 + 25 + 10)/something incorrectly. Pro tip: For frequency tables, NEVER average the category values directly. You must weight each value by how many times it appears. Think of it as expanding the table: 15 families with 1 vehicle = fifteen 1s; 25 families with 2 = twenty-five 2s; etc. Then sum and divide by total families (50).

Question 2

Which of the following sets of data has the smallest standard deviation?

  1. {1, 1, 10, 10}
  2. {2, 4, 6, 8}
  3. {5, 5, 5, 5}
  4. {5, 5, 6, 6}
Explanation: This is a standard deviation question testing conceptual understanding of spread. Choice C ({5, 5, 5, 5}) is correct — standard deviation measures how spread out the values are from the mean. A set with all identical values has zero spread, giving a standard deviation of exactly 0. No other set can have a smaller SD. Choice A ({1, 1, 10, 10}) has a mean of 5.5 with values far from the mean — large SD. Choice B ({2, 4, 6, 8}) has a mean of 5 with values spread 3 units apart on average — moderate SD ≈ 2.24. Choice D ({5, 5, 6, 6}) has a very small but nonzero SD ≈ 0.5. A student might choose D thinking it has the "smallest nonzero" deviation, but the question asks for smallest overall — and C achieves SD = 0. Pro tip: Standard deviation is zero when all values are identical. To find the set with the smallest SD without calculating, look for the set with the least variation. Identical values → SD = 0, always the minimum possible. If no set has identical values, choose the one where all values are closest to each other.

Question 3

What is the median of the data set [14, 18, 12, 10, 16]?

  1. 12
  2. 16
  3. 18
  4. 14
Explanation: The median is the middle value when data is sorted in order. First, sort the data: [10, 12, 14, 16, 18]. Since there are 5 values (odd count), the median is the middle (3rd) value =14= 14=14. The median provides a measure of central tendency that is not affected by extreme values.

Question 4

What is the median of the data set [9, 5, 12, 8, 7]?

  1. 8
  2. 9
  3. 7
  4. 12
Explanation: The median is the middle value when data is sorted in order. First, sort the data: [5, 7, 8, 9, 12]. Since there are 5 values (odd count), the median is the middle (3rd) value = 8. The median is resistant to extreme values and represents the central tendency.

Question 5

What is the mode of the data set [7, 9, 7, 5, 9, 9]?

  1. 9
  2. 7
  3. 5
  4. None
Explanation: 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.

Question 6

What is the mean of the data set [3, 7, 5, 10, 8]?

  1. 7.6
  2. 7
  3. 6.8
  4. 6.6
Explanation: 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.

Question 7

What is the range of the data set [14, 27, 19, 33, 22]?

  1. 22
  2. 14
  3. 33
  4. 19
Explanation: The range is calculated as the difference between the maximum and minimum values. First, identify the maximum value (33) and minimum value (14). Range = maximum - minimum = 33 - 14 = 19. The range provides a simple measure of how spread out the data values are.

Question 8

What is the median of [15, 9, 7, 5, 13]?

  1. 11
  2. 13
  3. 7
  4. 9
Explanation: The median is the middle value when data is sorted in order. First, sort the data: [5, 7, 9, 13, 15]. Since there are 5 values (odd count), the median is the middle (3rd) value =9= 9=9. The median divides the dataset into two equal halves.

Question 9

What is the mode of the data set [3, 8, 3, 5, 9, 3, 2]?

  1. 3
  2. 5
  3. 8
  4. 9
Explanation: The mode is the value that appears most frequently in the dataset. Counting frequencies: 3 appears 3 times, 8 appears 1 time, 5 appears 1 time, 9 appears 1 time, 2 appears 1 time. Since 3 appears most frequently, the mode is 3.

Question 10

What is the range of the data set [11, 14, 18, 21, 25]?

  1. 14
  2. 11
  3. 25
  4. 18
Explanation: The range is calculated as the difference between the maximum and minimum values. First, identify the maximum value (25) and minimum value (11). Range = maximum - minimum = 25 - 11 = 14. The range indicates the total span of values from the smallest to the largest data point.