Question

3
the dot plots below show the number of pages mandee read per day for two books in a series.
book 1 book 2
20 21 22 23 24 25 26 20 21 22 23 24 25 26
number of pages read number of pages read
the mean absolute deviation for each book is 1.25. the difference between the median number of pages read per day for each book
is how many times the mean absolute deviation?
a. 5
b. 4
c. 2
d. 3

Explanation:

Step1: Find median of Book 1

First, we list the number of dots (frequency) for each page count in Book 1:
- 20: 2 dots
- 21: 0 dots
- 22: 0 dots
- 23: 1 dot
- 24: 3 dots
- 25: 5 dots
- 26: 5 dots

Total number of data points: \(2 + 0 + 0 + 1 + 3 + 5 + 5 = 16\). Since there are 16 data points (even number), the median is the average of the 8th and 9th values when ordered. Let's order the data:
- Pages 20: 2 data points (positions 1, 2)
- Page 23: 1 data point (position 3)
- Page 24: 3 data points (positions 4, 5, 6)
- Page 25: 5 data points (positions 7, 8, 9, 10, 11)
- Page 26: 5 data points (positions 12, 13, 14, 15, 16)

The 8th value is 25, the 9th value is 25. So median of Book 1: \(\frac{25 + 25}{2} = 25\).

Step2: Find median of Book 2

Now, list the number of dots (frequency) for each page count in Book 2:
- 20: 2 dots
- 21: 5 dots
- 22: 3 dots
- 23: 6 dots
- 24: 2 dots
- 25: 2 dots
- 26: 0 dots

Total number of data points: \(2 + 5 + 3 + 6 + 2 + 2 + 0 = 20\). Since there are 20 data points (even number), the median is the average of the 10th and 11th values when ordered. Let's order the data:
- Pages 20: 2 data points (positions 1, 2)
- Page 21: 5 data points (positions 3, 4, 5, 6, 7)
- Page 22: 3 data points (positions 8, 9, 10)
- Page 23: 6 data points (positions 11, 12, 13, 14, 15, 16)
- Page 24: 2 data points (positions 17, 18)
- Page 25: 2 data points (positions 19, 20)

The 10th value is 22, the 11th value is 23. So median of Book 2: \(\frac{22 + 23}{2} = 22.5\)? Wait, no, wait. Wait, let's recount the frequencies:

Wait, maybe a better way: for Book 2, the dot plot:

- 20: 2 dots
- 21: 5 dots (so cumulative: 2 + 5 = 7)
- 22: 3 dots (cumulative: 7 + 3 = 10)
- 23: 6 dots (cumulative: 10 + 6 = 16)
- 24: 2 dots (cumulative: 16 + 2 = 18)
- 25: 2 dots (cumulative: 18 + 2 = 20)

So the 10th and 11th values: the 10th value is the last value in the 22 group (since cumulative up to 22 is 10), and the 11th value is the first value in the 23 group. Wait, no, when data is ordered, the values are:

20, 20,

21, 21, 21, 21, 21,

22, 22, 22,

23, 23, 23, 23, 23, 23,

24, 24,

25, 25.

So the 10th term: let's count:

1:20, 2:20,

3:21, 4:21, 5:21, 6:21, 7:21,

8:22, 9:22, 10:22,

11:23, 12:23, 13:23, 14:23, 15:23, 16:23,

17:24, 18:24,

19:25, 20:25.

Ah, I made a mistake earlier. So the 10th term is 22, the 11th term is 23? No, wait, the 10th term is the 10th data point. Let's list all 20 data points:

1. 20

2. 20

3. 21

4. 21

5. 21

6. 21

7. 21

8. 22

9. 22

10. 22

11. 23

12. 23

13. 23

14. 23

15. 23

16. 23

17. 24

18. 24

19. 25

20. 25

So the 10th term is 22, the 11th term is 23. Therefore, median of Book 2 is \(\frac{22 + 23}{2} = 22.5\)? Wait, no, that's not right. Wait, median for even number of data points is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2} + 1)\)th terms. Here, \(n = 20\), so \(\frac{20}{2} = 10\)th term and \(11\)th term. The 10th term is 22, the 11th term is 23. So median is \(\frac{22 + 23}{2} = 22.5\)? Wait, but that seems odd. Wait, maybe I messed up the dot plot counts. Let's re-examine the dot plot for Book 2:

Looking at the dot plot:

- 20: 2 dots (so two 20s)

- 21: 5 dots (five 21s)

- 22: 3 dots (three 22s)

- 23: 6 dots (six 23s)

- 24: 2 dots (two 24s)

- 25: 2 dots (two 25s)

So total: 2 + 5 + 3 + 6 + 2 + 2 = 20, correct.

Now, let's list the data in order:

20, 20,

21, 21, 21, 21, 21,

22, 22, 22,

23, 23, 23, 23, 23, 23,

24, 24,

25, 25.

Now, let's index them from 1 to 20:

1:20

2:20

3:21

4:21

5:21

6:21

7:21

8:22

9:22

10:22

11:23

12:23

13:23

14:23

15:23

16:23

17:24

18:24

19:25

20:25

So the 10th term is 22, the 11th term is 23. So median is (22 + 23)/2 = 22.5. Wait, but that seems incorrect. Wait, maybe the dot plot for Book 1:

Book 1 dot plot:

- 20: 2 dots

- 21: 0

- 22: 0

- 23: 1

- 24: 3

- 25: 5

- 26: 5

So total: 2 + 0 + 0 + 1 + 3 + 5 + 5 = 16. So data points:

1:20

2:20

3:23

4:24

5:24

6:24

7:25

8:25

9:25

10:25

11:25

12:26

13:26

14:26

15:26

16:26

So median is average of 8th and 9th terms. 8th term: 25, 9th term: 25. So median of Book 1 is 25.

Now, median of Book 2: let's check again. Wait, maybe I made a mistake in the dot plot for Book 2. Let's count the dots:

Looking at the Book 2 dot plot:

- 20: 2 dots (so two)

- 21: 5 dots (so five)

- 22: 3 dots (so three)

- 23: 6 dots (so six)

- 24: 2 dots (so two)

- 25: 2 dots (so two)

So the data is:

20, 20,

21, 21, 21, 21, 21,

22, 22, 22,

23, 23, 23, 23, 23, 23,

24, 24,

25, 25.

Now, the 10th term (since n=20, median is (10th + 11th)/2):

1st:20, 2nd:20,

3rd:21, 4th:21, 5th:21, 6th:21, 7th:21,

8th:22, 9th:22, 10th:22,

11th:23, 12th:23, 13th:23, 14th:23, 15th:23, 16th:23,

17th:24, 18th:24,

19th:25, 20th:25.

So 10th term is 22, 11th term is 23. So median is (22 + 23)/2 = 22.5? Wait, but that seems off. Wait, maybe the dot plot for Book 2 is different. Wait, maybe I miscounted the dots. Let's look at the original problem:

Book 2 dot plot:

- 20: 2 dots

- 21: 5 dots (so five dots above 21)

- 22: 3 dots (three dots above 22)

- 23: 6 dots (six dots above 23)

- 24: 2 dots (two dots above 24)

- 25: 2 dots (two dots above 25)

Yes, that's what's given. So the data is as I listed. Then median of Book 2 is 22.5, median of Book 1 is 25. Then the difference is 25 - 22.5 = 2.5? Wait, but that can't be, because the answer choices are 5,4,2,3. So I must have made a mistake.

Wait, maybe I messed up the median calculation for Book 1. Let's recheck Book 1:

Book 1 data points:

- 20: 2

- 21: 0

- 22: 0

- 23: 1

- 24: 3

- 25: 5

- 26: 5

Total: 2 + 0 + 0 + 1 + 3 + 5 + 5 = 16. So the median is the average of the 8th and 9th terms. Let's list the data in order:

1:20

2:20

3:23

4:24

5:24

6:24

7:25

8:25

9:25

10:25

11:25

12:26

13:26

14:26

15:26

16:26

So the 8th term is 25, the 9th term is 25. So median of Book 1 is 25. Correct.

Now, Book 2: let's list the data in order:

1:20

2:20

3:21

4:21

5:21

6:21

7:21

8:22

9:22

10:22

11:23

12:23

13:23

14:23

15:23

16:23

17:24

18:24

19:25

20:25

So the 10th term is 22, the 11th term is 23. So median is (22 + 23)/2 = 22.5. Then difference is 25 - 22.5 = 2.5. But the mean absolute deviation is 1.25. Then 2.5 / 1.25 = 2. Wait, 2.5 divided by 1.25 is 2. So the difference is 2.5, which is 2 times the mean absolute deviation (1.25 * 2 = 2.5). Wait, but 2.5 / 1.25 = 2. So the answer is 2? But let's check again.

Wait, maybe I made a mistake in the median of Book 2. Wait, maybe the dot plot for Book 2 is:

Wait, looking at the dot plot, maybe the number of dots for each is:

- 20: 2

- 21: 5

- 22: 3

- 23: 6

- 24: 2

- 25: 2

Wait, but maybe the median of Book 2 is 23? Wait, no, because with 20 data points, the median is between the 10th and 11th. Let's count the cumulative frequency:

- Up to 20: 2

- Up to 21: 2 + 5 = 7

- Up to 22: 7 + 3 = 10

- Up to 23: 10 + 6 = 16

- Up to 24: 16 + 2 = 18

- Up to 25: 18 + 2 = 20

So the 10th term is in the 22 group (since cumulative up to 22 is 10), and the 11th term is in the 23 group. So the 10th term is the last 22, and the 11th term is the first 23. So the median is (22 + 23)/2 = 22.5. Then the difference between medians is 25 - 22.5 = 2.5. Then 2.5 / 1.25 = 2. So the answer is 2, which is option C.

Wait, but let's confirm with another approach. Maybe the median of Book 2 is 23? Wait, no, because with 20 data points, the median is the average of the 10th and 11th. Let's list the data points:

1:20

2:20

3:21

4:21

5:21

6:21

7:21

8:22

9:22

10:22

11:23

12:23

13:23

14:23

15:23

16:23

17:24

18:24

19:25

20:25

Yes, so 10th is 22, 11th is 23. So median is 22.5. Book 1 median is 25. Difference is 25 - 22.5 = 2.5. Then 2.5 / 1.25 = 2. So the answer is 2, which is option C.

Answer:

C. 2