Exercises: Summarize numerical data sets in relation to their context - Common Core Math - Grade 6

A

Warm-Up: Review What You Know

These problems review skills you need for data summaries.

1.

A measure of center summarizes data with a single representative number. Which of these is a measure of center?

A.

Range

B.

Median

C.

IQR

D.

MAD

2.

A student collects 5 test scores: 12, 7, 15, 4, 9. What is the sum of all five scores?

3.

Five students scored 70, 80, 90, 85, and 75 on a quiz. A classmate says 'the typical score is about 80.' Which operation would let you check this claim by finding the exact average?

A.

Multiply all five scores together, then divide by 5.

B.

Add all five scores together, then divide by 5.

C.

Subtract the smallest score from the largest score.

D.

List the scores in order and pick the middle one.

B

Fluency Practice

Compute each summary statistic. Show your work step by step.

1.

Data set: {8, 10, 12, 14, 16}. Find the mean.

2.

Data set (sorted): {5, 8, 11, 14, 17, 20}. This data set has ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ values (even count). The two middle values are ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ and ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ . The median = average of these two = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .

number of values:

lower middle value:

upper middle value:

median:

$MAD computation table with column 'Value' and column 'Absolute Deviation from Mean (7)'. Values 3, 5, 7, 9, 11 listed. Deviation column has question marks to fill in.$

3.

Data set: {3, 5, 7, 9, 11}. Mean = 7.

Compute MAD step by step. Fill in the absolute deviations from the mean:
|3 − 7| = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ , |5 − 7| = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ , |7 − 7| = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ , |9 − 7| = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ , |11 − 7| = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .
Sum of absolute deviations = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ . MAD = sum ÷ 5 = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .

|3 - 7|:

|5 - 7|:

|7 - 7|:

|9 - 7|:

|11 - 7|:

sum:

MAD:

4.

Data set (sorted): {12, 15, 18, 21, 24, 27, 30, 33, 36}. Find the IQR.

5.

A student computes the 'average deviation' by finding each value's distance from the mean but forgets absolute values. She adds the signed deviations: (3−7) + (5−7) + (7−7) + (9−7) + (11−7) = (−4)+(−2)+0+2+4 = 0. She concludes 'the average deviation is 0, meaning the data has no variation.' What went wrong?

A.

She should have divided by n−1 instead of n.

B.

She forgot to take the absolute value of each deviation. Signed deviations always sum to zero, so she needs |value − mean| before averaging.

C.

She computed the mean incorrectly.

D.

She is correct — the data has no variation.

C

Mixed Practice

Apply summary statistics in varied contexts.

D

Word Problems

Use summary statistics to answer questions about real-world data.

$Dot plot of books checked out. Nine dots cluster between 1 and 8. One outlier dot at 19 is far to the right.$

1.

A librarian recorded how many books 10 students checked out last month: 1, 2, 3, 4, 5, 5, 6, 7, 8, 19.

1.

The librarian wants to write a complete description of the data. Fill in: n = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ observations. Attribute = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ . Unit = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .

n:

attribute:

unit:

2.

The data is: {1, 2, 3, 4, 5, 5, 6, 7, 8, 19}. The mean is 6.0 and the median is (5+5)/2 = 5.0. The outlier is 19. Which statement is correct?

A.

Use the mean (6.0) — it is always the best measure of center.

B.

Use the median (5.0) — the outlier at 19 pulls the mean up, making 5.0 more representative of a typical student.

C.

Use the range — it captures the outlier.

D.

Ignore the outlier since it is just one value.

3.

Describe one 'striking deviation from the pattern' in this data set and explain what it means in context.

2.

Two classes each have 8 students. Their test scores are: Class A: {72, 74, 75, 76, 77, 78, 79, 81}. Class B: {40, 55, 70, 78, 80, 85, 92, 100}.

1.

Find the mean score for Class A.

2.

Class A range = 81−72 = 9. Class B range = 100−40 = 60. Both classes have similar means (~76). What does this tell you about the two classes?

A.

Class A and Class B performed the same — they have the same mean.

B.

Class A is more consistent — students performed similarly to each other. Class B has much more spread — some scored very high, some very low.

C.

Class B performed better because it includes a perfect score of 100.

D.

Class A performed better because its scores are all close to the mean.

E

Find the Mistake

Each problem shows a student's work with an error. Find and explain the mistake.

1.

Sorted data: {10, 14, 18, 22, 26, 30}

Student's work:
"There are 6 values (even). The median is the middle value. The middle is the 3rd value = 18. Median = 18."

What mistake did the student make?

A.

The student is correct — for 6 values, the median is the 3rd value.

B.

For an even number of values, the median is the average of the two middle values (3rd and 4th), not just the 3rd.

C.

The student should have used the mean instead of the median.

D.

The student should have sorted the data first.

2.

A data set of house prices in a neighborhood (in thousands): {120, 135, 140, 145, 150, 155, 160, 850}.

Student writes: "The mean is $231,875. This is the best measure of the typical house price."

What error did the student make in choosing the mean?

A.

The student computed the mean incorrectly.

B.

The student chose the wrong measure. The outlier ($850K mansion) pulls the mean far above most home prices; the median is more representative.

C.

The student should have reported the range instead.

D.

The mean is always the best measure; the student is correct.

F

Challenge Problems

These problems require multi-step reasoning and complete statistical summaries.

1.

A data set of 7 students' heights (cm): {148, 150, 152, 155, 158, 162, 195}. (a) Compute the mean and median. (b) Which is more representative of a typical student's height, and why? (c) What is the striking deviation, and what might explain it?

2.