Exercises: Use measures of center and measures of variability to draw informal comparative inferences about two populations - Common Core Math - Grade 7

A

Recall / Warm-Up

1.

A data set is: 4, 5, 6, 6, 7, 7, 8, 9, 42. Which measure of center best represents a typical value in this data set?

A.

Mean, because it uses every value.

B.

Median, because the outlier 42 pulls the mean far above most values.

C.

Mean, because the data set has an even number of values.

D.

Median, because the data set has values in the single digits.

2.

You decide to use the median to describe the center of a data set. Which measure of variability is the best match for your choice?

A.

MAD (Mean Absolute Deviation)

B.

IQR (Interquartile Range)

C.

Range

D.

Mean

3.

Compute the IQR of this data set: 3, 4, 5, 6, 7, 8, 9.

B

Fluency

1.

Compute the mean of the following data set: 3, 4, 4, 5, 5, 5, 6, 6, 7, 8.

2.

Data set: 4, 5, 6, 6, 7, 7, 8, 9, 10, 13. The mean of this data set is 7.5. Compute the MAD.

3.

A data set has one extremely large outlier. Which measure of variability is more resistant to the outlier's effect?

A.

MAD, because it uses every value in the data set.

B.

IQR, because it is computed from the middle 50% of values and the outlier falls in the upper tail.

C.

Both are equally resistant to outliers.

D.

Neither — both are affected the same way by outliers.

4.

Compute the IQR of the following data set: 4, 5, 6, 6, 7, 7, 8, 9, 10, 13.

5.

A random sample of word lengths from two science textbooks produced these results: 4th-grade — median = 5 letters, IQR = 2 letters; 7th-grade — median = 7 letters, IQR = 3 letters. Which statement best compares the two samples?

A.

Words in the 7th-grade text are exactly 2 letters longer than words in the 4th-grade text.

B.

Words in the 7th-grade text tend to be about 2 letters longer than words in the 4th-grade text; the 7th-grade text also has slightly more variability in word length (IQR = 3 vs. 2).

C.

The 7th-grade text has a higher median, so all of its words are longer than 4th-grade words.

D.

The two samples cannot be compared because the IQR values are different.

C

Varied Practice

D

Word Problems

1.

A reading researcher randomly samples 10 words from a science chapter in a 4th-grade textbook and 10 words from a science chapter in a 7th-grade textbook. The word lengths (in letters) are:

4th-grade: 3, 4, 4, 5, 5, 5, 6, 6, 7, 8

7th-grade median = 7.0 letters, IQR = 3 letters.

1.

Compute the median and IQR for the 4th-grade sample. The median is ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ and the IQR is ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .

median:

IQR:

2.

Using the 4th-grade statistics you computed and the 7th-grade statistics given (median = 7 letters, IQR = 3 letters), write a complete informal comparative inference statement. Your statement must include: (1) the names of both populations, (2) the direction of the difference, (3) actual center values, (4) reference to variability, and (5) hedging language.

2.

A student records the daily study hours of 8 classmates: 2, 3, 3, 4, 4, 5, 5, 6. Compute the MAD.

3.

A survey sampled 10 students from each of two after-school programs and asked how many hours per day they spend on screens. Results:

Program A: mean daily screen time = 3 hours, MAD = 0.8 hours
Program B: mean daily screen time = 5 hours, MAD = 0.9 hours

Write a complete informal comparative inference statement. Include hedging language and reference both center and variability.

4.

A student writes: '7th-grade science books ALWAYS use longer words than 4th-grade science books — this is a fact proven by the data.' Which revision makes the statement statistically appropriate?

A.

7th-grade science books DEFINITELY use longer words than 4th-grade science books in all cases.

B.

Based on our samples, words in 7th-grade science books tend to be longer than words in 4th-grade science books, though a larger sample would give us more confidence in this pattern.

C.

The data might suggest that 7th-grade books have longer words, but we really have no idea since we can't measure every word.

D.

7th-grade science books never have shorter words than 4th-grade science books.

E

Error Analysis

1.

A student records quiz scores for Team B: 7, 8, 8, 9, 9, 10, 42.

The student computes:
Sum $= 7+8+8+9+9+10+42 = 93$
Mean $= 93 \div 7 \approx 13.3$

The student writes: "Team B's typical quiz score is about 13.3 points."

What error did the student make? What should the student do instead, and what value would better represent a typical score?

A.

The student computed the sum incorrectly.

B.

The student should have divided by 6, not 7.

C.

The data has an outlier (42) that pulls the mean well above most values; the student should use the median instead, which equals 9.

D.

The student should have removed the score of 42 before computing any statistics.

2.

A student wants to compare word lengths in 7th-grade books vs. 4th-grade books.

The student randomly samples 10 words from a 7th-grade poetry anthology and 10 words from a 4th-grade science textbook. The student computes:

7th-grade poetry sample: median = 4 letters, IQR = 2
4th-grade science sample: median = 6 letters, IQR = 3

The student concludes: "4th-grade books use longer words than 7th-grade books, on average."

Identify the flaw in the student's reasoning. Why might the conclusion be misleading?

A.

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

B.

The student sampled from non-equivalent text types — poetry typically uses short rhythmic words while science text uses long technical terms. The comparison may reflect genre differences, not grade-level differences.

C.

Ten words is too small a sample to compute any statistics.

D.

The student should have compared means rather than medians to make the comparison valid.

F

Challenge

1.

Two groups of students recorded how many hours they slept last night.
Group M: 5, 6, 7, 8, 9, 10
Group N: 9, 10, 11, 12, 13, 14

(a) Compute the mean, median, MAD, and IQR for each group.
(b) Choose the most appropriate measure of center and measure of variability for this comparison, and explain your choice.
(c) Write a complete informal comparative inference statement using those measures.

2.