6th Grade Math Quiz: Understand Measures Of Center And Variation

20 questions · exam conditions

0:00

Understand Measures Of Center And VariationQuestion 1 of 20

A librarian tracked the number of books checked out daily for two weeks: 45, 52, 48, 51, 47, 49, 50, 46, 53, 48, 51, 49, 47, 52. She wants to report a single number that best represents typical daily circulation. However, she's concerned that using the wrong measure might mislead the library board about daily operations. Which approach best addresses her concern about accurately representing the data?

Calculate the mean (49.1) since it uses every data point, but also mention the range (8) to show variation is low.

Use the median (49) because it's not affected by outliers, even though this data set contains no extreme values.

Report the mode (several values appear twice) since it shows the most common daily circulation patterns for the library.

Use the range (8) as the primary statistic since measures of variation are more informative than measures of center.

← Back to quizzes

6th Grade Math Quiz

6th Grade Math Quiz: Understand Measures Of Center And Variation

Practice Understand Measures Of Center And Variation in 6th Grade 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 Understand Measures Of Center And Variation, giving you a quick way to practice the rules, question types, and explanations that matter most for 6th Grade 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

Calculate the mean (49.1) since it uses every data point, but also mention the range (8) to show variation is low. (correct answer)
Use the median (49) because it's not affected by outliers, even though this data set contains no extreme values.
Report the mode (several values appear twice) since it shows the most common daily circulation patterns for the library.
Use the range (8) as the primary statistic since measures of variation are more informative than measures of center.

Explanation: The mean (49.1) appropriately summarizes the center using all values, and mentioning the small range (8) shows the data has low variation, making the mean highly representative. This addresses her concern about accuracy. Choice B unnecessarily avoids the mean when there are no outliers. Choice C incorrectly identifies a clear mode when multiple values repeat. Choice D incorrectly prioritizes variation over center when she specifically wants a measure of center.

Question 2

A store manager wants to summarize daily customer counts with a single number that represents a typical day. The data for 7 days is: 120, 125, 118, 340, 122, 119, 124. If the manager wants to minimize the effect of the unusually busy day, which measure of center should be used, and what does this tell us about the data's variation?

Use the mean (152); the high variation shows the store has inconsistent customer traffic patterns throughout the week.
Use the median (122); the high variation indicates one day significantly differs from the store's typical traffic pattern. (correct answer)
Use the mode (there is none); the lack of repeated values indicates low variation in customer traffic.
Use the range (222); this measure of variation also serves as the best measure of center for this data.

Explanation: The median (122) is resistant to the outlier value of 340 and better represents typical daily traffic. The high variation (range = 222) shows that one day was very different from the others. Choice A uses the mean which is affected by the outlier. Choice C incorrectly suggests that no mode means low variation, when actually the outlier creates high variation. Choice D incorrectly treats the range as a measure of center rather than variation.

Question 3

Two classes took the same math test. Class A's scores have a median of 85 and a range of 20. Class B's scores have a median of 85 and a range of 40. A student argues that since both classes have the same median, they performed equally well. What is the best evaluation of this argument using the concept of measures of center and variation?

The argument is correct; median is the most important statistic, and equal medians indicate equivalent class performance overall.
The argument cannot be evaluated; comparing classes requires knowing the mean scores rather than median and range values.
The argument is wrong; Class B performed better because higher variation indicates students achieved more diverse score ranges.
The argument is incomplete; while both classes have the same center, Class A shows more consistent performance due to lower variation. (correct answer)

Explanation: When comparing groups of data, you need to look at both measures of center (like median) and measures of variation (like range) to get the complete picture. The median tells you the middle value, while the range shows how spread out the scores are. Both classes have a median of 85, meaning half the students in each class scored above 85 and half scored below. However, Class A has a range of 20 (meaning scores span 20 points from highest to lowest), while Class B has a range of 40 (scores span 40 points). This tells you that Class A's scores are clustered more tightly around the median, showing consistent performance, while Class B's scores are more spread out, indicating less consistent performance. Answer D correctly identifies that while both classes have the same center (median = 85), Class A performed more consistently due to its lower variation. The argument is incomplete because it ignores this important difference in consistency. Answer A is wrong because median alone doesn't tell the whole story—variation matters too for understanding overall performance. Answer B is incorrect because you can absolutely compare classes using median and range; you don't need the mean. Answer C is flawed because higher variation doesn't indicate better performance—it actually shows less consistency, which is generally less desirable. Remember: When analyzing data sets, always consider both center AND spread. Two groups can have the same average but very different levels of consistency, which affects how you interpret their performance.

Question 4

A teacher says: "The mean of our homework times is 30 minutes, so the times do not vary much." Which statement best explains the mistake?

The teacher should use the median because the median always measures variation.
The teacher should use the range or MAD because the mean is a measure of center (typical value), not a measure of variation (spread). (correct answer)
The teacher should list all homework times instead of using any single-number summary.
The teacher is correct because any measure of center also measures variation.

Question 5

A student wrote: "The range tells the typical value of a data set." Which correction is best?

Correct; range is a measure of center because it uses the largest value.
Correct; range and mean both describe the typical value.
Incorrect; mean and median describe variation, not center.
Incorrect; range is a measure of variation that describes spread, while mean or median are measures of center that describe a typical value. (correct answer)

Explanation: This question tests understanding of measures of center and variation, where a measure of center like mean or median summarizes all values with a single typical number, and a measure of variation like range describes how values vary with a single spread number. Measure of center: single number representing typical value from data (mean averages all to one number, median is middle value). Purpose: summarizes all data values with one number showing 'what is typical?'. Measure of variation: single number describing how spread out values are (range=max-min shows total spread). Purpose: summarizes variability with one number showing 'how much do values differ?'. Different purposes: center tells location (where data centered), variation tells spread (how much varies). The correct answer is choice B, which corrects that range is a measure of variation for spread, not center for typical value, while mean or median describe center.

Question 6

The times (in minutes) it took a group to finish a warm-up were: 8, 9, 10, 12, 16, 17, 18. Find the median (measure of center) and the range (measure of variation).

Median $=12$ and range $=10$ ; 12 is the middle value, and 10 shows the total spread. (correct answer)
Median $=10$ and range $=9$ ; 10 is the middle value, and 9 shows the total spread.
Median $=10$ and range $=10$ ; 10 is the middle value, and 10 shows the total spread.
Median $=12$ and range $=9$ ; 12 is the middle value, and 9 shows the total spread.

Explanation: This question tests understanding of measures of center and variation, where a measure of center like the median summarizes all values with a single typical number, and a measure of variation like the range describes how values vary with a single spread number. Measure of center: single number representing typical value from data (median=middle when ordered: for 8,9,10,12,16,17,18, the 4th value is 12 as center with equal counts above/below; mean could be (8+9+10+12+16+17+18)/7=90/7≈12.86 but median is used here). Purpose: summarizes all data values with one number showing 'what is typical?' (12 represents all seven values as typical). Measure of variation: single number describing how spread out values are (range=max-min: 18-8=10 shows total spread, or MAD=average distance from mean: typical deviation). Purpose: summarizes variability with one number showing 'how much do values differ?' (range 10 means values spread across 10-unit range). Different purposes: center tells location (where data centered), variation tells spread (how much varies). The correct answer is choice A, which states median=12 and range=10 with explanations of middle value and total spread.

Question 7

A student listed daily temperatures (in °F) for 6 days: 62, 65, 67, 70, 70, 73. What is the range (measure of variation), and what does it describe?

Range $=11$ ; it is a single number describing how much the temperatures vary from lowest to highest. (correct answer)
Range $=73$ ; it is a single number describing how much the temperatures vary.
Range $=11$ ; it is a single number describing the typical temperature.
Range $=62$ ; it is a single number describing how much the temperatures vary.

Explanation: This question tests understanding of measures of center and variation, where a measure of center like the mean summarizes all values with a single typical number, and a measure of variation like the range describes how values vary with a single spread number. Measure of center: single number representing typical value from data (mean=sum/count: (62+65+67+70+70+73)/6=407/6≈67.83, or median=average of 67 and 70=68.5). Purpose: summarizes all data values with one number showing 'what is typical?'. Measure of variation: single number describing how spread out values are (range=max-min: 73-62=11 shows total spread, or MAD=average distance from mean: typical deviation). Purpose: summarizes variability with one number showing 'how much do values differ?' (range 11 means values spread across 11-unit range from lowest to highest). Different purposes: center tells location (where data centered), variation tells spread (how much varies). The correct answer is choice A, which gives range=11 and describes it as a single number for how much temperatures vary from lowest to highest.

Question 8

A store recorded the number of customers in 5 hours: 11, 13, 14, 14, 18. Find the mean (measure of center) and explain what it represents using one sentence.

Mean $=11$ ; it is the smallest value so it best represents the typical hour.
Mean $=18$ ; it is the greatest value so it best represents the typical hour.
Mean $=14$ ; it is a single number that represents how spread out the customer counts are.
Mean $=14$ ; it is a single number that represents a typical number of customers per hour for these 5 hours. (correct answer)

Explanation: This question tests understanding of measures of center and variation, where a measure of center like the mean summarizes all values with a single typical number, and a measure of variation like the range describes how values vary with a single spread number. Measure of center: single number representing typical value from data (mean=sum/count: (11+13+14+14+18)/5=70/5=14 averages all values to one number, or median=middle: 14). Purpose: summarizes all data values with one number showing 'what is typical?' (14 represents all five values as typical number of customers). Measure of variation: single number describing how spread out values are (range=max-min: 18-11=7 shows total spread, or MAD=average distance from mean: typical deviation). Purpose: summarizes variability with one number showing 'how much do values differ?'. Different purposes: center tells location (where data centered), variation tells spread (how much varies). The correct answer is choice A, which gives mean=14 and explains it as a single number for typical customers per hour.

Question 9

Two different classes took a quiz.

Class A scores: 50, 50, 50, 50, 50
Class B scores: 40, 45, 50, 55, 60 Which statement best compares the measures of center and variation?

Class A has greater variation because all the scores are the same.
Both classes have the same variation because they both include the score 50.
Class B has a different center because the range is larger.
Both classes have the same center (mean 50), but Class B has greater variation (range 20 vs. 0). (correct answer)

Explanation: This question tests understanding of measures of center and variation, where a measure of center like the mean summarizes all values with a single typical number, and a measure of variation like the range describes how values vary with a single spread number. Measure of center: single number representing typical value from data (for Class A: mean=50, for Class B: (40+45+50+55+60)/5=50). Purpose: summarizes all data values with one number showing 'what is typical?' (50 represents typical score for both). Measure of variation: single number describing how spread out values are (Class A range=50-50=0, Class B range=60-40=20). Purpose: summarizes variability with one number showing 'how much do values differ?' (range 0 means no spread in A, range 20 means greater spread in B). Different purposes: center tells location (same center at 50), variation tells spread (different variations). The correct answer is choice A, which compares same center mean 50 but greater variation in B with range 20 vs. 0.

Question 10

A student measured how many minutes they practiced an instrument each day for a week: 20, 25, 25, 30, 30, 35, 40. What are the median (measure of center) and the range (measure of variation)?

Median $=30$ and range $=20$ . (correct answer)
Median $=25$ and range $=20$ .
Median $=30$ and range $=15$ .
Median $=35$ and range $=20$ .

Explanation: This question tests understanding of measures of center and variation, where a measure of center like the median summarizes all values with a single typical number, and a measure of variation like the range describes how values vary with a single spread number. Measure of center: single number representing typical value from data (median=middle when ordered: for 20,25,25,30,30,35,40, the 4th value is 30 as center). Purpose: summarizes all data values with one number showing 'what is typical?' (30 represents all seven values as typical). Measure of variation: single number describing how spread out values are (range=max-min: 40-20=20 shows total spread, or MAD=average distance from mean: typical deviation). Purpose: summarizes variability with one number showing 'how much do values differ?' (range 20 means values spread across 20-unit range). Different purposes: center tells location (where data centered), variation tells spread (how much varies). The correct answer is choice A, which states median=30 and range=20.

Question 11

A student recorded how many pages they read each day for 5 days: 12, 14, 15, 16, 18 pages. Which choice correctly gives a measure of center (mean) and a measure of variation (range), and correctly describes what each number means?

Mean = 15 pages (a single number that summarizes a typical daily amount); Range = 6 pages (a single number that describes how spread out the daily amounts are). (correct answer)
Mean = 15 pages (a single number that describes how spread out the daily amounts are); Range = 6 pages (a single number that summarizes a typical daily amount).
Mean = 14 pages (a single number that summarizes a typical daily amount); Range = 6 pages (a single number that describes how spread out the daily amounts are).
Mean = 15 pages; Range = 12 to 18 pages (these numbers summarize the data).

Explanation: This question tests understanding that a measure of center, such as the mean or median, summarizes all values with a single typical number, while a measure of variation, such as the range or MAD, describes how values vary with a single spread number. Measure of center: single number representing typical value from data (mean=sum/count: (12+14+15+16+18)/5=75/5=15 averages all values to one number, or median=middle when ordered: 15 is center value with equal counts above/below); purpose: summarizes all data values with one number showing 'what is typical?' (15 represents all five values as typical). Measure of variation: single number describing how spread out values are (range=max-min: 18-12=6 shows total spread, or MAD=average distance from mean: typical deviation); purpose: summarizes variability with one number showing 'how much do values differ?' (range 6 means values spread across 6-unit range); different purposes: center tells location (where data centered), variation tells spread (how much varies). For example, in this data 12,14,15,16,18, center: mean=15 (single number summarizes typical value), variation: range=6 (single number describes spread, values vary 6 units total); or MAD: deviations from 15 are 3,1,0,1,3, average (3+1+0+1+3)/5=1.6 (typical variation 1.6 from center). The correct choice is mean=15 pages (a single number that summarizes a typical daily amount) and range=6 pages (a single number that describes how spread out the daily amounts are), matching their purposes as center for typical value and variation for spread. A common error is swapping the purposes, like using mean for spread or range for typical (as in choice B), or miscalculating mean as 14 (choice C), or not using a single number for range (choice D listing 12 to 18). Center measures include mean (averages all, sensitive to outliers) and median (middle, resistant); variation measures include range (simple max-min) and MAD (average deviation); using them, center answers 'what is typical?' (mean 15), variation 'how much variability?' (range 6), and datasets can have same center but different variation or vice versa; mistakes include confusing purposes or arithmetic errors.

Question 12

Two classes each took a 5-question mini-quiz.

Class A scores: 8, 9, 10, 11, 12
Class B scores: 6, 8, 10, 12, 14

Which statement is true about center and variation for these two data sets?

Both classes have the same measure of center (mean 10), but Class B has greater variation (larger range). (correct answer)
Class A has a larger mean than Class B, and both have the same range.
Both classes have the same range, so they must also have the same mean.
Class B has a smaller range, so it must have a smaller mean.

Question 13

A teacher recorded quiz scores (out of 20) for a small group: 8, 12, 14, 14, 16, 18. Which value is the range (a measure of variation)?

18
10 (correct answer)
14
26

Explanation: This question tests understanding that a measure of variation like the range describes how values vary with a single spread number, specifically max minus min. The range is 18-8=10, a single number showing the spread from least to most scores. Its purpose is to summarize variability with one number answering 'how much do values differ?' (scores vary across 10 points). Measures of center like mean or median summarize typical values, but here we focus on variation. Example: for data 8,12,14,14,16,18, mean=(8+12+14+14+16+18)/6=82/6≈13.7 (center), but range=10 (variation). Different purposes: center tells location (around 13.7), variation tells spread (10 units). Mistakes include calculating mean instead (≈13.7, not an option) or subtracting wrong values; correct is 10, choice A.

Question 14

The heights (in inches) of 5 plants are: 10, 12, 14, 16, 18. What is the mean absolute deviation (MAD) from the mean? (MAD is the average of the distances from the mean.)

2
4
2.4 (correct answer)
8

Explanation: This question tests understanding that a measure of variation like the mean absolute deviation (MAD) describes how values vary with a single number, averaging distances from the mean. First, mean=(10+12+14+16+18)/5=70/5=14; deviations: 4,2,0,2,4; MAD=(4+2+0+2+4)/5=12/5=2.4, a single number showing typical deviation. Its purpose is to summarize variability answering 'how much do values differ from the center?' (typically 2.4 inches). Measures of center like mean summarize typical values (14 inches), but here it's variation. Example: range=18-10=8 (another variation measure, total spread). Different purposes: center tells 'what is typical?' (14), variation tells spread (MAD 2.4). Mistakes: confusing with range (8) or half (4); correct is 2.4, choice B.

Question 15

The heights (in inches) of 5 plants are: 10, 12, 14, 16, 18. What is the mean absolute deviation (MAD) from the mean? (MAD is the average of the distances from the mean.)

4
8
2
2.4 (correct answer)

Question 16

A student recorded the number of pages they read each day for 5 days: 12, 14, 15, 16, 18. Which choice correctly gives a measure of center (mean) and a measure of variation (range), and what each one tells you?

Mean $=15$ pages and Range $=12$ to $18$ pages (variation must be written as two numbers).
Mean $=6$ pages (a single number that shows the spread); Range $=15$ pages (a single number that represents a typical daily amount).
Mean $=15$ pages (a single number that represents a typical daily amount); Range $=6$ pages (a single number that shows the spread from least to most). (correct answer)
Mean $=14$ pages (a single number that represents a typical daily amount); Range $=6$ pages (a single number that shows the spread from least to most).

Explanation: This question tests understanding that a measure of center like the mean summarizes all values with a single typical number, and a measure of variation like the range describes how values vary with a single spread number. The measure of center here is the mean, calculated as (12+14+15+16+18)/5=75/5=15, which is a single number representing the typical daily pages read. Its purpose is to summarize all data values with one number showing 'what is typical?' (15 represents the average for the five days). The measure of variation is the range, calculated as 18-12=6, which is a single number describing how spread out the values are from least to most. Its purpose is to summarize variability with one number showing 'how much do values differ?' (a range of 6 means the pages vary across 6 units). Different purposes: the center tells the location where the data is centered (around 15), while variation tells the spread (total of 6 pages). Choice A correctly identifies mean=15 as center and range=6 as variation with proper interpretations, while others swap them, miscalculate, or use two numbers for range.

Question 17

Refer to the dot plot. A coach recorded the number of goals scored by players in a season. She wants to use one number to represent the typical performance and explain to parents how much individual performances varied. What should she report and why?

Report the mean (4.2 goals) and range (6 goals) to show both the average performance and total spread of scores. (correct answer)
Report the median (4 goals) and mode (3 goals) since using two measures of center provides more complete information.
Report only the median (4 goals) since it summarizes all values and parents don't need to know about variation.
Report the mode (3 goals) and range (6 goals) since the most common score and variation are most important.

Explanation: The mean (4.2) summarizes the center using all values, and the range (6) describes the variation from lowest to highest scorer. This combination answers both parts of the coach's goal. Choice B uses two measures of center instead of including variation. Choice C ignores the coach's stated need to explain variation. Choice D uses mode, which is less informative than mean for this continuous data.

Question 18

Mean $=15$ pages (a single number that represents a typical daily amount); Range $=6$ pages (a single number that shows the spread from least to most). (correct answer)
Mean $=6$ pages (a single number that shows the spread); Range $=15$ pages (a single number that represents a typical daily amount).
Mean $=14$ pages (a single number that represents a typical daily amount); Range $=6$ pages (a single number that shows the spread from least to most).
Mean $=15$ pages and Range $=12$ to $18$ pages (variation must be written as two numbers).

Question 19

A quality control inspector measured the lengths of 8 bolts (in millimeters): 50.1, 49.9, 50.0, 49.8, 50.2, 50.0, 49.9, 50.1. The target length is 50.0 mm. To report on the manufacturing process, the inspector needs one number to summarize the center and one to describe the variation. Which combination provides the most meaningful summary?

Mean = 50.0 mm and range = 0.4 mm; this shows the process is perfectly centered with minimal variation. (correct answer)
Median = 50.0 mm and mode = 50.0 mm; using two measures of center provides better information than including variation.
Mode = 50.0 mm and range = 0.4 mm; this shows the most common result matches the target with controlled variation.
Mean = 50.0 mm and median = 50.0 mm; these identical values prove the data has no variation worth measuring.

Explanation: The mean (50.0) shows the process is centered on target, and the range (0.4) describes how much the values vary from 49.8 to 50.2. Choice B uses two measures of center instead of including variation as requested. Choice C uses mode, but mean is more appropriate for continuous measurements. Choice D incorrectly concludes that equal mean and median indicate no variation, and fails to provide a measure of variation.

Question 20

A student measured the lengths (in cm) of 8 paper strips: 11, 12, 12, 13, 13, 14, 15, 20. Which statement correctly explains the difference between a measure of center and a measure of variation?

A measure of center tells the spread; a measure of variation tells the typical value.
A measure of center and a measure of variation always have to be the same number.
A measure of variation lists all the data values, while a measure of center is a sentence.
A measure of center tells a typical or middle value with one number; a measure of variation tells how spread out the values are with one number. (correct answer)

Explanation: This question tests understanding the difference between measures of center (mean, median) that summarize with a single typical number, and measures of variation (range, MAD) that describe spread with a single number. Measure of center: e.g., mean=(11+12+12+13+13+14+15+20)/8=110/8=13.75, or median=13 (average of 4th and 5th:13+13=26/2=13), single number for typical length. Purpose: summarizes data showing 'what is typical?' (around 13-13.75 cm). Measure of variation: e.g., range=20-11=9, single number for spread. Purpose: shows 'how much values differ?' (across 9 cm). Different purposes: center for location, variation for spread; they don't have to match and variation isn't a list. Choice A correctly explains this distinction.