Exercises: Recognize that a measure of center summarizes all values with a single number - Common Core Math - Grade 6

A

Warm-Up: Review What You Know

These problems review key vocabulary.

1.

A distribution has center, spread, and shape. Which feature describes 'the typical value in the data set'?

A.

Spread

B.

Center

C.

Shape

D.

Range

2.

Which phrase best describes the spread of a distribution?

A.

The most common value.

B.

The middle value when data is sorted.

C.

How far apart or how varied the data values are.

D.

The highest point on a dot plot.

B

Fluency Practice

Use the definitions of center and variation to analyze data sets.

1.

A data set has five values: 3, 5, 7, 9, 11. What is the mean?

2.

A data set has five values: 1, 2, 7, 12, 13. What is the mean?

$Two data sets plotted on number lines. Both have a mean of 7, but Set A clusters tightly around 7 while Set B is widely spread.$

3.

Both data sets {3, 5, 7, 9, 11} and {1, 2, 7, 12, 13} have the same mean (7). What does this show about using mean alone to describe data?

A.

The mean is all we need — both sets are identical.

B.

The mean captures the typical value but hides the variation — we also need a measure of variation.

C.

A higher mean would tell us more about the data.

D.

The mean is never useful for comparing data sets.

4.

Data set A: {5, 5, 5, 5}. Data set B: {2, 4, 6, 8}. Both have a mean of 5. Without computing, which data set has more variation? Explain how you know.

C

Mixed Practice

Apply your understanding of center and variation in different contexts.

D

Word Problems

Use center and variation to interpret real-world situations.

1.

Two basketball players each played 6 games. Player A scored: 18, 20, 19, 21, 18, 20 points. Player B scored: 10, 30, 5, 35, 22, 14 points.

1.

Both players have the same mean score per game. What is that mean?

2.

Which player would a coach choose for the most consistent performance, and why?

A.

Player A — small variation means consistently close to the mean each game.

B.

Player B — higher maximum score (35) makes them the better player.

C.

They are equal — same mean, same ability.

D.

Player B — more variation means more excitement.

2.

A weather reporter says: 'The average high temperature in our city in July is 88°F.' A tourist says: 'Then I know it will be 88°F every day — I'll pack for 88°F.'

What information is the tourist missing, and how could it affect their packing?

A.

The tourist needs to know the total number of days in July.

B.

The tourist needs to know the variation — how much daily temperatures differ from 88°F. Some days could be 75°F or 100°F.

C.

The tourist has all the information needed — the mean is sufficient.

D.

The tourist needs to know the shape of the distribution, not the variation.

E

Find the Mistake

Each problem shows a student's reasoning that contains an error. Find and explain the mistake.

1.

A teacher reports: "The mean test score is 78."

Student Devon says: "So everyone scored 78 on the test."

What is wrong with Devon's reasoning?

A.

Devon is correct — the mean equals every individual's score.

B.

Devon confused the mean with each person's value. The mean is the balance point; individual scores can be very different.

C.

Devon should have used the median, not the mean.

D.

Devon is wrong because the teacher should have reported the maximum.

2.

Data set: {3, 7, 9, 11, 15}. Mean = 9.

Student Lena says: "The mean is 9 and the variation is also 9 — they are the same number, so they mean the same thing."

What is wrong with Lena's reasoning?

A.

Lena is correct — when they are equal numerically, they mean the same thing.

B.

Lena confused the numerical coincidence with meaning. Center and variation measure different things: center = typical value; variation = how spread out values are.

C.

Lena's calculation of the mean is wrong.

D.

Lena should have used the median instead.

F

Challenge Problems

These problems require deeper reasoning about center and variation.

1.

Create a data set of exactly 5 values where the mean is 10 and the range is 0. Then create a different data set of 5 values where the mean is also 10 but the range is 16. Explain what the difference between these two data sets tells you about the relationship between center and variation.

2.