Question

distance from school in blocks
wynn needs to find the center of the data set shown on
the dot plot.
the dot plot has
dots.
the dot plot has
dots to the left of the
center and
dots to the right of the center.
the center of the data set is

Explanation:

Step1: Count total dots

Count dots at each position:
- Position 1: 6 dots
- Position 2: 5 dots
- Position 3: 4 dots
- Position 4: 2 dots
- Position 5: 3 dots
- Position 6: 2 dots

Total: \( 6 + 5 + 4 + 2 + 3 + 2 = 22 \) dots.

Step2: Find center position

For 22 dots, the center is between the 11th and 12th dot (since \( \frac{22}{2} = 11 \), so median is average of 11th and 12th).

Count cumulative dots:
- After position 1: 6
- After position 2: \( 6 + 5 = 11 \)
- After position 3: \( 11 + 4 = 15 \)

11th dot is at position 2, 12th at position 3. Wait, no—wait, let's list all dots:

Positions (dots):
1: [1,1,1,1,1,1] (6)
2: [2,2,2,2,2] (5) → cumulative 11
3: [3,3,3,3] (4) → cumulative 15
4: [4,4] (2) → cumulative 17
5: [5,5,5] (3) → cumulative 20
6: [6,6] (2) → cumulative 22

Wait, 11th dot: position 2 (last of position 2), 12th: first of position 3? No, wait, median for even n is \( \frac{n}{2} \)th and \( (\frac{n}{2}+1) \)th. So 11th and 12th.

11th dot: position 2 (6 + 5 = 11, so 11th is last of position 2: value 2? Wait no, position 1: 6 dots (values 1), position 2: 5 dots (values 2) → first 6: 1s, next 5: 2s (total 11), next 4: 3s (12th to 15th: 3s). So 11th dot: 2, 12th: 3? Wait no, no—wait, the dots are ordered: all 1s, then 2s, then 3s, etc. So the data points are:

1,1,1,1,1,1,
2,2,2,2,2,
3,3,3,3,
4,4,
5,5,5,
6,6

Now list them in order:
1. 1
2. 1
3. 1
4. 1
5. 1
6. 1
7. 2
8. 2
9. 2
10. 2
11. 2
12. 3
13. 3
14. 3
15. 3
16. 4
17. 4
18. 5
19. 5
20. 5
21. 6
22. 6

Ah! So 11th is 2, 12th is 3. Wait, but the problem says "center"—maybe median? Wait, but the question about dots left/right of center. Wait, maybe I miscounted. Wait original dot plot:

Wait the user's dot plot:

Position 1: 6 dots (vertical dots: 6)
Position 2: 5 dots
Position 3: 4 dots
Position 4: 2 dots
Position 5: 3 dots
Position 6: 2 dots

Wait maybe my initial count was wrong. Let's re-count:

Position 1: 6 (correct: 6 dots)
Position 2: 5 (correct)
Position 3: 4 (correct)
Position 4: 2 (correct)
Position 5: 3 (correct)
Position 6: 2 (correct)

Total: 6+5=11, +4=15, +2=17, +3=20, +2=22. Correct.

Now, cumulative:

1: 6
2: 11 (6+5)
3: 15 (11+4)
4: 17 (15+2)
5: 20 (17+3)
6: 22 (20+2)

Now, the center for median (since it's a dot plot, median is center). For 22 data points, median is average of 11th and 12th terms.

11th term: in position 2 (since 6 (pos1) +5 (pos2)=11 → 11th term is last of pos2: value 2? Wait no, the data points are ordered as 1s, then 2s, then 3s, etc. So the 1st to 6th are 1s, 7th to 11th are 2s (5 dots: 7-11), 12th to 15th are 3s (4 dots: 12-15), 16-17: 4s, 18-20: 5s, 21-22: 6s.

Ah! So 11th term: 2 (7-11: 5 terms, so 7=2, 8=2, 9=2, 10=2, 11=2), 12th term: 3 (12=3). Wait, but median is \( \frac{2 + 3}{2} = 2.5 \)? But the problem asks for center, maybe median? Wait, but the question about dots left/right of center. Wait, maybe I made a mistake in counting. Wait, let's check the first part: "The dot plot has [ ] dots." Answer: 22.

Then "dots to the left of center and right". If center is median (2.5), then left of 2.5: positions 1 and 2 (values <2.5), right: positions 3,4,5,6 (values >2.5). Wait, no—dots left of center: number of dots with value < center, right: > center.

Wait, maybe the center is the median, which is between 2 and 3? But the problem's last box is "The center of the data set is [ ]". Maybe the median is 2.5? But let's re-examine the dot counts:

Wait, maybe I messed up the position counts. Let's list the number of dots at each position:

- Position 1: 6
- Position 2: 5
- Position 3: 4
- Position 4: 2
- Position 5: 3
- Position 6: 2

Total: 6+5+4+2+3+2=22. Correct.

Now, to find the median (center), we need the middle value(s). For 22 data points, the median is the average of the 11th and 12th data points.

Let's list the data points in order:

1 (6 times), 2 (5 times), 3 (4 times), 4 (2 times), 5 (3 times), 6 (2 times).

So the first 6 are 1s (positions 1–6: value 1), next 5 are 2s (positions 7–11: value 2), next 4 are 3s (positions 12–15: value 3), next 2 are 4s (positions 16–17: value 4), next 3 are 5s (positions 18–20: value 5), last 2 are 6s (positions 21–22: value 6).

So the 11th data point is 2 (position 11: value 2), the 12th data point is 3 (position 12: value 3). Therefore, the median (center) is \( \frac{2 + 3}{2} = 2.5 \). But wait, the problem might be considering the median as the middle position, but maybe the dot plot's center is the median, and the number of dots left and right. Wait, but 22 dots: left of center (median 2.5) would be dots with value <2.5 (i.e., 1 and 2), right: >2.5 (3,4,5,6).

Number of dots left: 6 (1s) +5 (2s)=11.
Number of dots right: 4 (3s)+2 (4s)+3 (5s)+2 (6s)=11.

Ah! So 11 dots left, 11 right. That makes sense (since 22 is even, split into 11 and 11).

So:

- Total dots: 22
- Dots left of center: 11, right: 11
- Center: 2.5 (or median between 2 and 3, but maybe the problem expects the median as the average, or maybe I miscounted the positions. Wait, maybe the positions are 1,2,3,4,5,6, and the center is the median, which is between 2 and 3, but the problem's last box—maybe the median is 2.5, or maybe the center is the median position. Wait, let's check again:

Wait, the 11th dot is at position 2 (value 2), 12th at position 3 (value 3). So the center (median) is \( \frac{2 + 3}{2} = 2.5 \). But maybe the problem considers the center as the median, and the number of dots left and right are 11 each.

Answer:

The dot plot has \(\boldsymbol{22}\) dots.
The dot plot has \(\boldsymbol{11}\) dots to the left of the center and \(\boldsymbol{11}\) dots to the right of the center.
The center of the data set is \(\boldsymbol{2.5}\) (or median between 2 and 3, but with 11 left and 11 right, the median is the average of 11th and 12th, which are 2 and 3, so 2.5).

(Note: If the problem expects integer, maybe a miscalculation, but per count, 22 total, 11 left/right, center 2.5.)