Praxis Math Quiz: Interpret Data Variability
20 questions · exam conditions
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Interpret Data VariabilityQuestion 1 of 20

A factory manager is comparing the performance of two machines that fill bags of flour. The target weight is 500 grams. The weights of bags from Machine X have a standard deviation of 2 grams. The weights of bags from Machine Y have a standard deviation of 5 grams. Which statement is best supported by this data?

The weights of bags filled by Machine X are more consistent than those from Machine Y.
The average weight of bags from Machine Y is greater than the average weight from Machine X.
Machine X is faster at filling bags than Machine Y because its output is less variable.
Machine Y is more likely to produce a bag weighing exactly 500 grams than Machine X.
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Praxis Math Quiz

Praxis Math Quiz: Interpret Data Variability

Practice Interpret Data Variability in Praxis Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Interpret Data Variability, giving you a quick way to practice the rules, question types, and explanations that matter most for Praxis Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A factory manager is comparing the performance of two machines that fill bags of flour. The target weight is 500 grams. The weights of bags from Machine X have a standard deviation of 2 grams. The weights of bags from Machine Y have a standard deviation of 5 grams. Which statement is best supported by this data?

  1. The weights of bags filled by Machine X are more consistent than those from Machine Y. (correct answer)
  2. The average weight of bags from Machine Y is greater than the average weight from Machine X.
  3. Machine X is faster at filling bags than Machine Y because its output is less variable.
  4. Machine Y is more likely to produce a bag weighing exactly 500 grams than Machine X.
Explanation: A smaller standard deviation implies less variability and therefore greater consistency. Since Machine X has a standard deviation of 2 grams, which is smaller than Machine Y's 5 grams, the weights of the bags it produces are more consistent and closer to the average weight. Standard deviation does not provide information about the average (mean) weight, the speed of the machines, or the probability of any specific outcome.

Question 2

The monthly rainfall, in inches, for two cities was recorded for a year. For City A, the data has a range of 4 inches. For City B, the data has a range of 9 inches. Based solely on this information, what can be inferred about the standard deviation of the monthly rainfall for the two cities?

  1. The standard deviation for City B is likely greater than for City A. (correct answer)
  2. The standard deviation for City A is equal to the standard deviation for City B.
  3. The mean monthly rainfall for City B is greater than for City A.
  4. The standard deviation for City B is exactly 5 inches greater than for City A.
Explanation: The range is a measure of variability, calculated as the difference between the maximum and minimum values. Standard deviation is another measure of variability that considers the spread of all data points around the mean. While they are different measures, a significantly larger range (9 inches for City B vs. 4 inches for City A) strongly suggests that the data for City B is more spread out, and thus its standard deviation is also likely to be greater. One cannot determine the mean or the exact standard deviation from the range alone.

Question 3

A researcher is studying the weights of apples from two different orchards. The weights from Orchard F are described as being very uniform, with most apples having weights very close to the average. The weights from Orchard G vary significantly, with some apples being quite small and others quite large. Which of the following statements is most likely true?

  1. The standard deviation of weights from Orchard G is greater than that from Orchard F. (correct answer)
  2. The mean weight of apples from Orchard F is greater than that from Orchard G.
  3. The standard deviation of weights from Orchard F is greater than that from Orchard G.
  4. The range of weights is the same for both orchards, but the means are different.
Explanation: The description indicates that the weights from Orchard F have low variability ('very uniform'), while the weights from Orchard G have high variability ('vary significantly'). Standard deviation is a measure of this variability. Therefore, it is most likely that the standard deviation for the more variable set (Orchard G) is greater than the standard deviation for the less variable set (Orchard F). The information given does not allow for any conclusions about the mean weights.

Question 4

The salaries at a small company have a standard deviation of $12,000. If the company owner decides to give every employee a 5% raise, which statement describes the effect on the standard deviation of the salaries?

  1. The new standard deviation will be 5% greater than $12,000. (correct answer)
  2. The new standard deviation will still be $12,000.
  3. The new standard deviation will be $12,000 plus 5% of the mean salary.
  4. The new standard deviation will increase by $600.
Explanation: A 5% raise means every salary is multiplied by 1.05. When every value in a dataset is multiplied by a positive constant, the standard deviation is also multiplied by that same constant. Therefore, the new standard deviation will be 1.05 times the old standard deviation. This is equivalent to saying it will be 5% greater than $12,000. The change is not a fixed amount like $600 (which is 5% of $12,000), but a multiplicative increase. It does not stay the same as it would if a fixed amount were added to each salary.

Question 5

A researcher has a dataset of 100 measurements. The researcher removes the single highest value and the single lowest value from the dataset. Which of the following effects will this action most likely have on the range and standard deviation?

  1. The range will decrease, and the standard deviation will likely decrease. (correct answer)
  2. The range will decrease, but the standard deviation will likely increase.
  3. The range will stay the same, but the standard deviation will likely decrease.
  4. The range and the standard deviation will both likely stay the same.
Explanation: The range is calculated as the difference between the maximum and minimum values. By removing these two specific values, the new range will be calculated from less extreme values, so the range must decrease (unless all values were identical). The standard deviation measures the overall spread from the mean. Removing the two most extreme values (outliers) will almost always reduce the overall variability of the data, causing the standard deviation to decrease as well.

Question 6

In a bowling league, Player A's scores for five games are 150, 155, 160, 165, 170. Player B's scores for five games are 120, 140, 160, 180, 200. Both players have the same mean score of 160. Which statement accurately describes the variability of their scores?

  1. Player B's scores have a larger standard deviation than Player A's scores. (correct answer)
  2. Player A's scores have a larger standard deviation than Player B's scores.
  3. Since the means are the same, the standard deviations of their scores are also the same.
  4. Since the number of games is the same, the variability of their scores is also the same.
Explanation: Although both players have the same mean score (160), the spread of their scores is different. Player A's scores are tightly clustered around the mean (150, 155, 160, 165, 170). Player B's scores are much more spread out (120, 140, 160, 180, 200). A larger spread of data around the mean results in a larger standard deviation. Therefore, Player B's performance is more variable, and their scores have a larger standard deviation.

Question 7

The number of defective parts per 1,000 produced by a factory is recorded each day for ten days. The dataset is {2, 3, 2, 4, 3, 1, 2, 3, 8, 2}. The manager removes the data point '8' because it was caused by a machine malfunction that has since been fixed. How will removing this data point affect the mean and standard deviation of the dataset?

  1. The mean will decrease, and the standard deviation will decrease. (correct answer)
  2. The mean will decrease, but the standard deviation will increase.
  3. The mean will increase, and the standard deviation will decrease.
  4. The mean will increase, and the standard deviation will increase.
Explanation: The data point '8' is an outlier, as it is significantly higher than the other values. Removing a value that is above the original mean will cause the new mean to decrease. Removing an outlier also reduces the overall spread or variability of the data. Therefore, the standard deviation will also decrease.

Question 8

Two groups of people are surveyed on their daily screen time in hours. Group A's times have a mean of 4 hours and a range of 2 hours. Group B's times have a mean of 4 hours and a range of 6 hours. What can be concluded about the standard deviations of the two groups?

  1. The standard deviation of Group B is greater than that of Group A. (correct answer)
  2. The standard deviation of Group A is greater than that of Group B.
  3. The standard deviations of the two groups are equal because their means are equal.
  4. The standard deviations of the two groups are equal because their sample sizes are equal.
Explanation: Both range and standard deviation are measures of data dispersion. A larger range indicates that the data are more spread out. Since Group B has a larger range (6 hours) than Group A (2 hours), it is certain that the data for Group B is more variable. Therefore, the standard deviation for Group B must be greater than the standard deviation for Group A. The means being equal is irrelevant to the comparison of variability, and sample sizes were not given nor are they relevant to the conclusion.

Question 9

The price of a certain stock is tracked for two consecutive weeks. In week 1, the prices were very stable, with little change from day to day. In week 2, the prices were highly volatile, with large fluctuations up and down. If the mean price was the same for both weeks, which week would have a larger standard deviation for the daily stock price?

  1. Week 2, because the prices had larger fluctuations around the mean. (correct answer)
  2. Week 1, because the prices were more predictable and consistent.
  3. Both weeks would have the same standard deviation because the mean was the same.
  4. It is impossible to tell without knowing the range of prices for each week.
Explanation: Standard deviation measures variability or volatility. 'Highly volatile' and 'large fluctuations' directly translate to a large spread of data points around the mean. 'Very stable' and 'little change' translate to a small spread of data points around the mean. Therefore, Week 2, the week with high volatility, would have a larger standard deviation.

Question 10

The standard deviation of a dataset is 2.5. A new dataset is created by multiplying each number in the original dataset by 4 and then adding 7. What is the standard deviation of the new dataset?

  1. 10 (correct answer)
  2. 17
  3. 2.5
  4. 10.5
Explanation: This transformation involves two steps: multiplication and addition. Adding a constant (7) to every data point shifts the dataset but does not change its spread, so it has no effect on the standard deviation. Multiplying every data point by a constant (4) scales the spread of the data. The new standard deviation will be the original standard deviation multiplied by the absolute value of that constant. So, the new standard deviation is 2.5 * 4 = 10.

Question 11

A company wants to determine which of its two delivery drivers is more reliable in terms of delivery time. Driver 1's delivery times have a mean of 40 minutes and a standard deviation of 15 minutes. Driver 2's delivery times have a mean of 45 minutes and a standard deviation of 5 minutes. Which driver is more reliable?

  1. Driver 2, because the smaller standard deviation indicates more consistent delivery times. (correct answer)
  2. Driver 1, because the lower mean delivery time indicates faster service overall.
  3. Both drivers are equally reliable because the difference in their mean times is small.
  4. It cannot be determined without knowing the total number of deliveries each driver made.
Explanation: In this context, 'reliability' refers to consistency and predictability, not speed. Consistency is measured by variability. Driver 2 has a much smaller standard deviation (5 minutes) compared to Driver 1 (15 minutes), meaning their delivery times are more tightly clustered around their average. Even though Driver 2's average time is slightly longer, the performance is far more consistent and therefore more reliable.

Question 12

A set of five test scores is {82, 85, 88, 90, 95}. The mean of this set is 88. A sixth score is added to the dataset, and this new score is also 88. How does the addition of this sixth score affect the variability of the dataset?

  1. The standard deviation will decrease. (correct answer)
  2. The standard deviation will increase.
  3. The standard deviation will remain the same.
  4. The range will increase.
Explanation: The standard deviation measures the average distance of data points from the mean. The original mean is 88. By adding another data point that is exactly equal to the mean, you are adding a value with zero deviation from the mean. This pulls the average deviation down, thus decreasing the standard deviation. The range (95 - 82 = 13) will remain the same because the new point is not a new maximum or minimum.

Question 13

Four archers each shoot six arrows at a target. Archer 1's arrows are all in the bullseye. Archer 2's arrows are spread widely all over the target. Archer 3's arrows are tightly clustered, but in the top-left corner of the target. Archer 4's arrows form a straight vertical line through the center of the target. Which archer's shots have the smallest standard deviation in their distances from the center of the bullseye?

  1. Archer 1 (correct answer)
  2. Archer 2
  3. Archer 3
  4. Archer 4
Explanation: Standard deviation measures the spread of data points from a central value (in this case, the center of the bullseye). Archer 1's arrows are all in the bullseye, meaning their distance from the center is zero. A dataset with all identical values has a standard deviation of 0, which is the smallest possible value. Archer 3's shots have a small standard deviation relative to their own average position, but a large standard deviation relative to the bullseye. Archer 2 and 4 have shots with clear and significant spread from the center.

Question 14

The heights of adult males in a certain population have a mean of 70 inches and a standard deviation of 3 inches. The heights of adult females in the same population have a mean of 65 inches and a standard deviation of 2.5 inches. Which of the following conclusions is best supported by this data?

  1. The heights of the males are more variable than the heights of the females. (correct answer)
  2. The tallest male must be taller than the tallest female.
  3. The heights of the females are more variable than the heights of the males.
  4. A randomly selected male is always taller than a randomly selected female.
Explanation: Variability is measured by the standard deviation. Since the standard deviation for male heights (3 inches) is larger than the standard deviation for female heights (2.5 inches), the heights of males are more spread out and thus more variable. We cannot make definitive conclusions about specific individuals (like the tallest male vs. the tallest female) or compare random individuals without knowing the full distribution.

Question 15

The following dataset represents the number of daily visitors to a small museum over a week: {50, 55, 48, 52, 58, 60, 51}. If on the eighth day, there was a special event and the museum received 150 visitors, how would the inclusion of this eighth day's data affect the variability of the dataset?

  1. Both the range and the standard deviation would increase. (correct answer)
  2. The range would increase, but the standard deviation would decrease.
  3. The standard deviation would increase, but the range would remain the same.
  4. Both the range and the standard deviation would decrease.
Explanation: The original data is tightly clustered. The value of 150 is an outlier, as it is significantly higher than the other data points. Adding an outlier to a dataset increases its overall spread. The range (the difference between the maximum and minimum values) will increase because the maximum value is now much larger. The standard deviation, which measures the average distance of data points from the mean, will also increase because the new point is very far from the original mean.

Question 16

A city's daily high temperatures in degrees Fahrenheit for one week are recorded. The standard deviation of these temperatures is 8 °F. If the temperatures were converted to degrees Celsius using the formula C = (5/9)(F - 32), what would be the effect on the standard deviation?

  1. The new standard deviation would be (5/9) of the original standard deviation. (correct answer)
  2. The new standard deviation would be the original standard deviation minus 32.
  3. The new standard deviation would be the same as the original standard deviation.
  4. The new standard deviation would be (5/9) of the original standard deviation, then subtract 32.
Explanation: The conversion formula involves two steps: subtracting a constant (32) and multiplying by a constant (5/9). Subtracting a constant from every data point shifts the data but does not change its spread, so it does not affect the standard deviation. Multiplying every data point by a constant (5/9) causes the standard deviation to be multiplied by that same constant. Therefore, the new standard deviation in Celsius will be (5/9) times the original standard deviation in Fahrenheit.

Question 17

A teacher grades two different classes on the same 20-point quiz. The scores for Class A have a mean of 15 and a standard deviation of 1.5. The scores for Class B have a mean of 15 and a standard deviation of 3.2. Which of the following is the most accurate conclusion that can be drawn from this information?

  1. The scores in Class A are more clustered around the mean than the scores in Class B. (correct answer)
  2. The highest score in Class B is greater than the highest score in Class A.
  3. The average score is the same for both classes, so the overall performance was identical.
  4. A student chosen at random from Class A is more likely to have a score of 15 than a student from Class B.
Explanation: Standard deviation is a measure of the spread or dispersion of a set of data from its mean. A smaller standard deviation (1.5 for Class A) indicates that the data points tend to be very close to the mean. A larger standard deviation (3.2 for Class B) indicates that the data points are spread out over a wider range of values. Therefore, the scores in Class A are more clustered around the mean. The other choices make assumptions that cannot be supported by the given information; standard deviation does not determine the highest score or the likelihood of any single score.

Question 18

A student's scores on four tests are 70, 80, 90, and 100. The mean score is 85 and the standard deviation is approximately 11.2. The student wants to take a fifth test to make their overall scores as consistent as possible. Which score on the fifth test would best achieve this goal?

  1. 85 (correct answer)
  2. 70
  3. 100
  4. 90
Explanation: To make a set of scores more consistent means to reduce its variability, i.e., to lower its standard deviation. The standard deviation measures the average distance from the mean. Adding a new data point that is equal to the current mean (85) will add a deviation of zero, which will lower the overall average deviation. Adding any other score would introduce a non-zero deviation from the mean, and would not reduce the standard deviation as effectively.

Question 19

The cumulative frequency graph shown represents the exam scores of 200 students. Based on the cumulative frequency graph, what is the interquartile range of the exam scores?

  1. 20
  2. 30 (correct answer)
  3. 50
  4. 70
Explanation: IQR = Q3 − Q1. Q1 corresponds to cumulative frequency 50 (25% of 200), and Q3 corresponds to cumulative frequency 150 (75% of 200). Reading the graph: at cumulative frequency 50, score ≈ 60; at cumulative frequency 150, score ≈ 90. IQR = 90 − 60 = 30. (A) 20 likely comes from misreading quartile positions. (C) 50 is the median value, not IQR. (D) 70 is the full range, not IQR.

Question 20

The box plot shown represents the weekly hours worked by 100 employees at a company. Refer to the box plot. Approximately how many employees worked between 30 and 45 hours per week?

  1. 25
  2. 50 (correct answer)
  3. 60
  4. 75
Explanation: In a box plot, Q1 to Q3 contains 50% of the data (the interquartile range). Here Q1 = 30 and Q3 = 45, so approximately 50% of 100 = 50 employees worked between 30 and 45 hours. (A) 25 confuses IQR with one quartile. (C) 60 and (D) 75 overestimate by misreading the quartile definition. This tests conceptual understanding that box plots partition data into quartiles of equal count, not equal range.