Question

the following frequency table summarizes a set of data. what is the five - number summary?
value | frequency
9 | 3
10 | 3
11 | 1
13 | 3
14 | 1
15 | 5
16 | 1
17 | 1
18 | 1
select the correct answer below:
min q1 median q3 max
○ 9 11 14 16 18
○ 9 10 11 17 18
○ 9 11 15 16 18
○ 9 10 11 15 18
○ 9 10 13 15 18

Explanation:

Step1: Find total number of data points

First, we calculate the total number of data points by summing the frequencies.
The frequencies are 3, 3, 1, 3, 1, 5, 1, 1, 1.
Summing them: \(3 + 3 + 1 + 3 + 1 + 5 + 1 + 1 + 1 = 19\) (odd number of data points).

Step2: Determine the ordered data set

We list out the data points with their frequencies:
- 9 (3 times): 9, 9, 9
- 10 (3 times): 10, 10, 10
- 11 (1 time): 11
- 13 (3 times): 13, 13, 13
- 14 (1 time): 14
- 15 (5 times): 15, 15, 15, 15, 15
- 16 (1 time): 16
- 17 (1 time): 17
- 18 (1 time): 18

Combining these, the ordered data set is: 9, 9, 9, 10, 10, 10, 11, 13, 13, 13, 14, 15, 15, 15, 15, 15, 16, 17, 18

Step3: Find the minimum (Min)

The smallest value in the data set is 9. So, Min = 9.

Step4: Find the maximum (Max)

The largest value in the data set is 18. So, Max = 18.

Step5: Find the median (middle value)

Since there are 19 data points, the median is the \((\frac{19 + 1}{2}) = 10^{th}\) value.
Counting the ordered data:
1. 9 (1 - 3)
2. 10 (4 - 6)
3. 11 (7)
4. 13 (8 - 10)
So the 10th value is 13? Wait, no, wait. Wait, let's recount:

Wait, the cumulative frequencies:
- After 9 (3): cumulative = 3
- After 10 (3): cumulative = 3 + 3 = 6
- After 11 (1): cumulative = 6 + 1 = 7
- After 13 (3): cumulative = 7 + 3 = 10
Ah, so the 10th value is the last 13 (since cumulative after 13 is 10). Wait, but wait, the ordered data:

Positions 1 - 3: 9s

Positions 4 - 6: 10s

Position 7: 11

Positions 8 - 10: 13s

Position 11: 14

Positions 12 - 16: 15s (since 5 times: 12 - 16)

Position 17: 16

Position 18: 17

Position 19: 18

Wait, so the 10th value is 13? Wait, no, wait, the 10th value: let's list the indices (1 - 19):

1:9, 2:9, 3:9,

4:10, 5:10, 6:10,

7:11,

8:13, 9:13, 10:13,

11:14,

12:15, 13:15, 14:15, 15:15, 16:15,

17:16,

18:17,

19:18

So the median (10th value) is 13? Wait, no, wait, the median for n = 19 is the \((\frac{19 + 1}{2}) = 10^{th}\) term. So the 10th term is 13? Wait, but let's check the options. Wait, maybe I made a mistake. Wait, let's re - check the cumulative frequencies:

Wait, the frequencies:

9: 3 (total so far: 3)

10: 3 (total: 6)

11: 1 (total: 7)

13: 3 (total: 10)

14: 1 (total: 11)

15: 5 (total: 16)

16: 1 (total: 17)

17: 1 (total: 18)

18: 1 (total: 19)

So the data points are in order:

1 - 3: 9

4 - 6: 10

7: 11

8 - 10: 13

11: 14

12 - 16: 15

17: 16

18: 17

19: 18

So the median is the 10th term, which is 13? Wait, no, wait, the 10th term is the third 13 (since 8th, 9th, 10th are 13). Wait, but let's check the quartiles.

Step6: Find Q1 (first quartile)

Q1 is the median of the lower half of the data. The lower half is the first \(\frac{19 - 1}{2}=9\) data points (since n is odd, we exclude the median). So the first 9 data points:

1:9, 2:9, 3:9,

4:10, 5:10, 6:10,

7:11,

8:13, 9:13

The median of these 9 data points is the 5th term (since \(\frac{9 + 1}{2}=5\)). The 5th term in the lower half (first 9 data points) is 10 (since the first 3 are 9, next 3 are 10, 7th is 11, 8th and 9th are 13; the 5th term is 10).

Step7: Find Q3 (third quartile)

Q3 is the median of the upper half of the data. The upper half is the last 9 data points (excluding the median). The last 9 data points are:

11:14,

12:15, 13:15, 14:15, 15:15, 16:15,

17:16,

18:17,

19:18

Wait, no, the upper half is from the 11th data point to the 19th data point. So the data points are:

11:14,

12:15, 13:15, 14:15, 15:15, 16:15,

17:16,

18:17,

19:18

That's 9 data points. The median of these 9 data points is the 5th term (since \(\frac{9+1}{2} = 5\)). The 5th term in this upper half:

11:14 (1st),

12:15 (2nd),

13:15 (3rd),

14:15 (4th),

15:15 (5th)

So Q3 is 15.

Step8: Summarize the five - number summary

- Min: 9 (smallest value)
- Q1: 10 (median of lower half)
- Median: 13 (10th term)
- Q3: 15 (median of upper half)
- Max: 18 (largest value)

Answer:

Min: 9, Q1: 10, Median: 13, Q3: 15, Max: 18 (corresponding to the option "Min Q1 Median Q3 Max 9 10 13 15 18")