ACT Statistics Interpretation Practice Questions with Answers

Concept Explanation

ACT Statistics Interpretation is the process of analyzing, synthesizing, and drawing conclusions from data sets, frequency tables, and probability distributions found on the math section of the ACT. This skill requires more than just calculating a mean; it involves understanding how changes in a data set affect measures of center (mean, median) and measures of spread (range, standard deviation). On the exam, you will frequently encounter questions that ask you to identify the most appropriate statistical measure for a specific scenario or to determine how adding or removing a data point shifts the overall distribution. Reliable resources like Khan Academy's Statistics modules provide excellent foundational support for these concepts.

To succeed in ACT Prep, you must be comfortable with the following core concepts:

• Mean: The arithmetic average, calculated by dividing the sum of all values by the number of values.
• Median: The middle value when data is ordered from least to greatest. If there is an even number of values, it is the average of the two middle terms.
• Mode: The value that appears most frequently in a data set.
• Range: The difference between the maximum and minimum values.
• Standard Deviation: A measure of how spread out the numbers are from the mean. A higher standard deviation indicates greater variability.

A common trap on the ACT involves weighted averages. If a class of 10 students averages 80% and a class of 20 students averages 90%, the combined average is not 85%. You must calculate the total points first: $(10 \ \times 80) + (20 \ \times 90) = 2600$, then divide by the total number of students: $\ \frac{2600}{30} \approx 86.7\%$.

Solved Examples

1. Example 1: Impact of Outliers
   A set of five test scores is {82, 85, 88, 90, 92}. If a sixth score of 40 is added to the set, which measure will change the most: the mean or the median?
   1. Calculate the original mean: $\ \frac{82+85+88+90+92}{5} = 87.4$.
   2. Calculate the original median: The middle value of {82, 85, 88, 90, 92} is 88.
   3. Add the new score (40) and recalculate. New set: {40, 82, 85, 88, 90, 92}.
   4. New mean: $\ \frac{40+82+85+88+90+92}{6} = 79.5$. (Change of 7.9)
   5. New median: Average of 85 and 88 is $\ \frac{85+88}{2} = 86.5$. (Change of 1.5)
   6. The mean changed significantly more than the median.
2. Example 2: Frequency Tables
   A survey asked 10 people how many pets they own. 3 people said 0, 4 people said 1, and 3 people said 2. What is the mean number of pets?
   1. Multiply each value by its frequency: $(0 \ \times 3) + (1 \ \times 4) + (2 \ \times 3) = 0 + 4 + 6 = 10$.
   2. Divide the sum by the total number of people: $\ \frac{10}{10} = 1$.
   3. The mean number of pets is 1.
3. Example 3: Standard Deviation Concept
   Set A is {10, 11, 12} and Set B is {5, 11, 17}. Without calculating, which set has a higher standard deviation?
   1. Observe the spread. Both sets have a mean of 11.
   2. In Set A, the values are very close to the mean (distance of 1).
   3. In Set B, the values are further from the mean (distance of 6).
   4. Since Set B has a wider spread, it has a higher standard deviation.

Practice Questions

1. A list of 7 numbers has a mean of 12. If an 8th number, 20, is added to the list, what is the new mean?
2. In a group of 5 integers, the median is 15, the mode is 10, and the range is 10. What is the largest possible value in this set?
3. A data set consists of the values {4, 7, 7, 10, 12, 15}. If each value in the set is increased by 5, how does the standard deviation change?

4. The average of four numbers is 22. If three of the numbers are 15, 25, and 30, what is the fourth number?
5. A set of numbers contains {x, 12, 15, 18, 22}. If the median is 15, what is the maximum possible integer value for x?
6. A basketball player scored 12, 18, 20, and 14 points in four games. How many points must he score in the fifth game to bring his average to 17?
7. Two sets of data are compared. Set X: {2, 4, 6, 8, 10} and Set Y: {102, 104, 106, 108, 110}. Which set has a larger range?
8. If the mean of a set of 10 numbers is 50, and one number is removed, the new mean is 48. What was the value of the number that was removed?
9. A frequency table shows that in a class of 20, 5 students scored 100, 10 students scored 80, and 5 students scored 60. What is the median score?
10. If a data set has a standard deviation of 0, what must be true about the numbers in the set?

Answers & Explanations

1. Answer: 13. If 7 numbers have a mean of 12, their sum is $7 \ \times 12 = 84$. Adding 20 makes the new sum $84 + 20 = 104$. The new mean is $104 / 8 = 13$.
2. Answer: 20. Let the ordered set be $a, b, 15, d, e$. Since the mode is 10, both $a$ and $b$ must be 10. The set is {10, 10, 15, d, e}. The range is 10, so $e - 10 = 10$, which means $e = 20$.
3. Answer: It stays the same. Standard deviation measures the distance between points. Adding a constant to every point shifts the entire distribution but does not change the distance between the points.
4. Answer: 18. The sum of the four numbers is $4 \ \times 22 = 88$. The sum of the known three is $15 + 25 + 30 = 70$. Thus, $88 - 70 = 18$.
5. Answer: 15. For the median of 5 numbers to be 15, two numbers must be less than or equal to 15, and two must be greater than or equal to 15. The current set has 12 (less than 15) and 18, 22 (greater than 15). Therefore, x must be $\leq 15$. The maximum integer is 15.
6. Answer: 21. To average 17 over 5 games, the total points must be $5 \ \times 17 = 85$. The current total is $12 + 18 + 20 + 14 = 64$. The required score is $85 - 64 = 21$.
7. Answer: They are equal. The range for Set X is $10 - 2 = 8$. The range for Set Y is $110 - 102 = 8$.
8. Answer: 68. The original sum was $10 \ \times 50 = 500$. The new sum of 9 numbers is $9 \ \times 48 = 432$. The removed number is $500 - 432 = 68$.
9. Answer: 80. Listing the scores in order: five 60s, then ten 80s, then five 100s. The 10th and 11th values are both 80, so the median is 80.
10. Answer: All values are identical. Standard deviation measures variation; if there is no variation (0), every number must be the same.

1. A set of data has a mean of 15 and a median of 14. If every value in the set is multiplied by 2, what is the new mean?

• A15
• B28
• C30
• D45

Frequently Asked Questions

What is the difference between mean and median on the ACT?

The mean is the calculated average of all numbers, while the median is the middle value of an ordered list. ACT questions often test your ability to determine which measure changes more when an outlier is added to the set.

Does the ACT provide the formula for standard deviation?

No, the ACT rarely requires you to use the complex standard deviation formula. Instead, you need to understand the concept that standard deviation represents the spread of data points around the mean.

How do I handle frequency tables for statistics questions?

To find the mean from a frequency table, multiply each data value by its frequency to find the total sum, then divide by the total number of entries. You can find more practice on this in our ACT Subject Verb Agreement section for general test-taking logic.

What happens to the range if I add a constant to every number?

The range remains unchanged because both the maximum and minimum values increase by the same amount, keeping the difference between them constant. This is a common property tested alongside ACT Punctuation and other math logic.

Can the mean, median, and mode all be the same number?

Yes, in a perfectly symmetrical distribution, such as a normal bell curve, the mean, median, and mode are all located at the center and share the same value. For more complex data analysis, check out the AI Exam Simulator for realistic practice.