Medium ACT Statistics Interpretation Practice Questions

Concept Explanation

Statistics interpretation on the ACT involves analyzing data sets to identify trends, calculate central tendency, and evaluate the impact of outliers on measures like mean and median. While basic questions might simply ask for the average of five numbers, medium-level ACT statistics interpretation practice questions require you to understand how adding or removing data points shifts the balance of a distribution. You will frequently encounter calculations for the mean (arithmetic average), median (middle value), mode (most frequent value), and range (difference between maximum and minimum). A critical skill for this level is recognizing that the median is more resistant to outliers than the mean, a concept often tested in the context of skewed data sets.

To succeed in ACT Prep, you should be comfortable with the following formulas and properties:

• Mean: The sum of all values divided by the number of values $(\frac{\sum x}{n})$.
• 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.
• Weighted Average: Used when different groups have different sizes, calculated as $\frac{(n_1 \times \text{mean}_1) + (n_2 \times \text{mean}_2)}{n_1 + n_2}$.
• Range: A measure of spread calculated as $\text{Max} - \text{Min}$.

Solved Examples

1. A student has test scores of 82, 85, 91, and 88. What score must the student earn on a fifth test to raise the average score to exactly 88?

   Solution:

   1. Determine the total points needed for five tests to average 88: $5 \times 88 = 440$.
   2. Calculate the sum of the current four scores: $82 + 85 + 91 + 88 = 346$.
   3. Subtract the current sum from the required total: $440 - 346 = 94$.
   4. The student must score a 94.

2. A set of 7 integers has a median of 15 and a range of 20. If the smallest number is 8, what is the largest possible value for the mean of the set?

   Solution:

   1. Identify the fixed values: Smallest = 8, Largest = $8 + 20 = 28$, Median (4th term) = 15.
   2. To maximize the mean, we must maximize the unknown values.
   3. The set looks like: $\{8, x_2, x_3, 15, x_5, x_6, 28\}$.
   4. To maximize, set $x_2$ and $x_3$ to 15 (they cannot exceed the median) and set $x_5$ and $x_6$ to 28 (they cannot exceed the maximum).
   5. Sum = $8 + 15 + 15 + 15 + 28 + 28 + 28 = 137$.
   6. Mean = $\frac{137}{7} \approx 19.57$.

3. The average age of a group of 10 employees is 32. If a 54-year-old manager joins the group, what is the new average age?

   Solution:

   1. Find the total age of the original 10 employees: $10 \times 32 = 320$.
   2. Add the manager's age: $320 + 54 = 374$.
   3. Divide by the new total number of people (11): $\frac{374}{11} = 34$.
   4. The new average age is 34.

Practice Questions

1. A data set consists of the following five values: 12, 17, 17, 22, and 30. If the number 30 is replaced by 40, which of the following statistical measures will change? (I. Mean, II. Median, III. Mode)

2. In a class of 20 students, the average score on a quiz was 80%. In another class of 30 students, the average score was 90%. What is the combined average score for all 50 students?

3. A set of five distinct positive integers has a mean of 20 and a median of 18. What is the largest possible value that any one of these integers could be?

4. The range of a list of 6 numbers is 14. If each number in the list is multiplied by 3 and then increased by 5, what is the range of the new list?

5. A basketball player scored an average of 18 points per game over 12 games. How many points must she score in the 13th game to increase her season average to 19 points?

6. For a set of 5 numbers, the mean is 10 and the median is 12. If the smallest number is removed, the mean of the remaining 4 numbers becomes 11. What was the value of the smallest number?

7. A frequency table shows that 5 students scored a 70, 10 students scored an 80, and 5 students scored a 90. What is the standard deviation relative to a set where 20 students all scored an 80?

8. If the arithmetic mean of $x, 2x,$ and $6$ is $10$, what is the value of $x$?

9. A set of numbers has a mean of 50. If every number in the set is decreased by 8, what is the new mean of the set?

10. An outlier is often defined as a value more than 1.5 times the Interquartile Range (IQR) above the third quartile. If $Q1 = 20$ and $Q3 = 30$, what is the minimum value a number must be to be considered an outlier on the upper end?

Answers & Explanations

1. Answer: I only. The mean is calculated by the sum; increasing 30 to 40 increases the sum and thus the mean. The median remains 17 (the middle of 12, 17, 17, 22, 40). The mode remains 17. Refer to ACT Statistics Interpretation Practice Questions for more on measure changes.
2. Answer: 86%. Total points = $(20 \times 80) + (30 \times 90) = 1600 + 2700 = 4300$. Combined average = $4300 / 50 = 86$. This is a weighted average problem.
3. Answer: 43. Sum = $5 \times 20 = 100$. To maximize one integer, minimize the others: $\{1, 2, 18, 19, x\}$. (Since they are distinct, we use 1, 2 and the next integer after the median, 19). $1 + 2 + 18 + 19 + x = 100 \rightarrow 40 + x = 100 \rightarrow x = 60$. Wait, if we use distinct positive integers: $1 + 2 + 18 + 19 + x = 100 \rightarrow x = 60$. Correction: The set is $\{1, 2, 18, 19, 60\}$. If the question implies they must be unique, 60 is the result.
4. Answer: 42. Range is affected by multiplication but not by addition/subtraction. New Range = $14 \times 3 = 42$. For more on data shifts, see ACT Data Analysis Practice Questions.
5. Answer: 31. Total points needed for 13 games: $13 \times 19 = 247$. Points already scored: $12 \times 18 = 216$. Required: $247 - 216 = 31$.
6. Answer: 6. Original sum = $5 \times 10 = 50$. New sum of 4 numbers = $4 \times 11 = 44$. The removed number is $50 - 44 = 6$.
7. Answer: Greater than zero. A set with all identical values has a standard deviation of 0. Since the first set has variation (70s and 90s), its standard deviation must be higher than the set with no variation. Check ACT Data Interpretation Practice Questions for similar conceptual checks.
8. Answer: 8. $\frac{x + 2x + 6}{3} = 10 \rightarrow 3x + 6 = 30 \rightarrow 3x = 24 \rightarrow x = 8$.
9. Answer: 42. If every value in a set is decreased by a constant, the mean is decreased by that same constant: $50 - 8 = 42$.
10. Answer: 45. $\text{IQR} = Q3 - Q1 = 30 - 20 = 10$. $1.5 \times \text{IQR} = 15$. Outlier threshold = $Q3 + 15 = 30 + 15 = 45$.

1. If a set of five numbers has a mean of 20, what is the new mean if each number is doubled?

• A20
• B40
• C10
• D60

Frequently Asked Questions

How does adding a constant to every value affect the standard deviation?

Adding a constant to every value in a data set does not change the standard deviation because the spread or distance between the numbers remains exactly the same. Only multiplying or dividing the values will change the standard deviation.

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 physical middle point of the data set. The ACT often tests your ability to determine which measure is more appropriate for skewed data, where the median is generally more reliable.

How do I calculate a weighted average?

You calculate a weighted average by multiplying each value (or group mean) by its frequency (or group size), summing those products, and then dividing by the total number of items. This is common in problems involving two different classes or groups of different sizes.

What is the interquartile range (IQR)?

The interquartile range is the difference between the third quartile (75th percentile) and the first quartile (25th percentile). It represents the spread of the middle 50% of the data and is used to identify outliers through the 1.5 IQR rule.

Can the mode be a decimal?

The mode is simply the most frequently occurring value in a data set, so it can be a decimal if the data points themselves are decimals. However, if the data set consists only of integers, the mode must be an integer.