Question

question
the following frequency table summarizes a set of data. what is the five - number summary?

value	frequency
13	1
14	1
15	4
17	5
18	2
19	3
21	1

select the correct answer below:
○ min:12, q1:14, median:16, q3:18, max:21
○ min:12, q1:15, median:17, q3:18, max:21
○ min:12, q1:15, median:18, q3:20, max:21
○ min:12, q1:13, median:14, q3:20, max:21
○ min:12, q1:15, median:16, q3:18, max:21

Explanation:

Step1: Find total number of data points

First, we calculate the total number of data points by summing the frequencies.
The frequencies are 2, 1, 1, 4, 5, 2, 3, 1.
So, \(2 + 1 + 1 + 4 + 5 + 2 + 3 + 1 = 19\) data points.

Step2: Determine Min and Max

The minimum value (Min) is the smallest value in the data, which is 12.
The maximum value (Max) is the largest value, which is 21.

Step3: Find the Median (middle value)

Since there are 19 data points (odd number), the median is the \(\frac{19 + 1}{2}=10^{th}\) value when the data is ordered.
Now, we build the cumulative frequency:
- Value 12: cumulative frequency \(2\)
- Value 13: cumulative frequency \(2 + 1 = 3\)
- Value 14: cumulative frequency \(3 + 1 = 4\)
- Value 15: cumulative frequency \(4 + 4 = 8\)
- Value 17: cumulative frequency \(8 + 5 = 13\) (since the 10th value falls here, the median is 17? Wait, no, wait. Wait, let's list the positions:
Wait, let's list the data in order with their frequencies:
- 12 (2 times): positions 1, 2
- 13 (1 time): position 3
- 14 (1 time): position 4
- 15 (4 times): positions 5, 6, 7, 8
- 17 (5 times): positions 9, 10, 11, 12, 13
- 18 (2 times): positions 14, 15
- 19 (3 times): positions 16, 17, 18
- 21 (1 time): position 19

Wait, the 10th position is within the 17s (since after 15, we have 8 cumulative, then 17 starts at position 9). So the 10th value is 17. Wait, but let's check again. Wait, cumulative frequency:

After 12: 2

After 13: 3

After 14: 4

After 15: 8

After 17: 8 + 5 = 13. So the 9th, 10th, 11th, 12th, 13th values are 17. So the median (10th value) is 17.

Step4: Find Q1 (25th percentile)

Q1 is the median of the lower half of the data. The lower half is the first \(\frac{19 - 1}{2}=9\) values? Wait, no. For a dataset with n = 19, the lower half is the first 9 values (since the median is the 10th, so the lower half is positions 1 - 9). Wait, no, the formula for Q1 is the value at the \(\frac{n + 1}{4}\)th position when n is odd? Wait, alternatively, for n = 19, the position of Q1 is \(\frac{19 + 1}{4}=5^{th}\) value? Wait, no, let's use the cumulative frequency.

Wait, the lower half: we have 19 data points. The lower half is the first 9 data points? Wait, no, the median is at position 10, so the lower half is positions 1 - 9, and the upper half is positions 11 - 19.

So for Q1, we need the median of the lower half (positions 1 - 9). Let's list the cumulative frequency up to position 9:

- 12: 2 (positions 1 - 2)
- 13: 3 (position 3)
- 14: 4 (position 4)
- 15: 8 (positions 5 - 8)
- 17: 9 (position 9, since 8 + 1 = 9? Wait, no, the frequency of 17 is 5, so after 15 (cumulative 8), the next values (positions 9 - 13) are 17. Wait, position 9 is the first 17. So the lower half is positions 1 - 9. Let's list the values:

Positions 1 - 2: 12

Position 3: 13

Position 4: 14

Positions 5 - 8: 15 (four times)

Position 9: 17

Wait, so the lower half has 9 values. The median of 9 values is the 5th value (since \(\frac{9 + 1}{2}=5\)). So the 5th value:

Positions 1 - 2: 12 (2)

Position 3: 13 (1)

Position 4: 14 (1)

Positions 5 - 8: 15 (4) → so the 5th value is 15. So Q1 is 15.

Step5: Find Q3 (75th percentile)

Q3 is the median of the upper half of the data. The upper half is positions 11 - 19 (since the median is at position 10, so upper half is positions 11 - 19, which is 9 values). Let's list the upper half:

Positions 11 - 13: 17 (three values, since position 9 - 13 are 17)

Position 14 - 15: 18 (two values)

Positions 16 - 18: 19 (three values)

Position 19: 21 (one value)

Wait, the upper half is positions 11 - 19 (9 values). The median of 9 values is the 5th value in the upper half (since \(\frac{9 + 1}{2}=5\)). Let's index the upper half:

Upper half positions (relative to upper half): 1 (position 11), 2 (12), 3 (13), 4 (14), 5 (15), 6 (16), 7 (17), 8 (18), 9 (19)

Wait, no, the actual positions:

Position 11: 17

Position 12: 17

Position 13: 17

Position 14: 18

Position 15: 18

Position 16: 19

Position 17: 19

Position 18: 19

Position 19: 21

So the upper half (positions 11 - 19) has values: 17, 17, 17, 18, 18, 19, 19, 19, 21.

The median of these 9 values is the 5th value (since (9 + 1)/2 = 5). The 5th value is 18. Wait, let's check:

1st: 17 (pos11)

2nd: 17 (pos12)

3rd: 17 (pos13)

4th: 18 (pos14)

5th: 18 (pos15)

Yes, so Q3 is 18.

Wait, but let's verify with cumulative frequency.

For Q1: the 25th percentile of 19 data points. The position is \(\frac{19 + 1}{4}=5^{th}\) value (since (n + 1)p, where p = 0.25). So the 5th value. Let's list the data in order:

1. 12

2. 12

3. 13

4. 14

5. 15

So the 5th value is 15. So Q1 = 15.

For Q3: the 75th percentile, position is \(\frac{3(19 + 1)}{4}=15^{th}\) value. Let's list the 15th value:

Let's count cumulative frequency:

- 12: 2

- 13: 3

- 14: 4

- 15: 8

- 17: 13 (after 17, cumulative is 13)

- 18: 13 + 2 = 15 (so the 15th value is 18). So Q3 = 18.

Median: position \(\frac{19 + 1}{2}=10^{th}\) value. Cumulative frequency:

- 12: 2

- 13: 3

- 14: 4

- 15: 8

- 17: 8 + 5 = 13 (so the 10th value is within 17s, so median is 17? Wait, no, wait:

Wait, cumulative frequency after 15 is 8. Then 17 has frequency 5, so positions 9 - 13 are 17. So the 10th value is 17. Wait, but earlier when we listed the positions, position 9 is 17, position 10 is 17, etc. So median is 17.

Min is 12, Max is 21.

So the five-number summary is Min = 12, Q1 = 15, Median = 17, Q3 = 18, Max = 21.

Looking at the options, the second option is:

Min: 12, Q1: 15, Median: 17, Q3: 18, Max: 21.

Answer:

B. Min: 12, Q1: 15, Median: 17, Q3: 18, Max: 21 (assuming the second option is labeled B as per the options, but in the given options, the second one is:

Min

Q1

Median

Q3

Max

12

15

17

18

21

So the correct answer is this option (the second one in the list).