Question

the dot plot below shows the number of goals jessica scored in each lacrosse game last season. goals scored per game which statement about the dot plot is correct? (1) mean > mode (3) mode = median (2) mean = median (4) median > mean

Explanation:

Step1: Count the number of dots for each goal value

- For 0: 3 dots
- For 1: 3 dots
- For 2: 4 dots
- For 3: 5 dots
- For 4: 2 dots
- For 5: 2 dots
- For 6: 1 dot

Total number of games \( n = 3 + 3 + 4 + 5 + 2 + 2 + 1 = 20 \)

Step2: Find the mode

The mode is the value with the highest frequency. Here, 3 has the highest frequency (5 dots), so mode \( = 3 \)

Step3: Find the median

Since \( n = 20 \) (even), the median is the average of the \( \frac{n}{2} = 10^{th} \) and \( \frac{n}{2}+1 = 11^{th} \) values.
Let's list the cumulative frequencies:
- 0: 3 (cumulative: 3)
- 1: 3 (cumulative: 3 + 3 = 6)
- 2: 4 (cumulative: 6 + 4 = 10)
- 3: 5 (cumulative: 10 + 5 = 15)
- 4: 2 (cumulative: 15 + 2 = 17)
- 5: 2 (cumulative: 17 + 2 = 19)
- 6: 1 (cumulative: 19 + 1 = 20)

The \( 10^{th} \) value is 2 (since cumulative frequency at 2 is 10) and the \( 11^{th} \) value is 3 (since cumulative frequency at 3 starts from 11). Wait, no, wait: Wait, cumulative frequency for 0 is 3 (values 1 - 3: 0), cumulative for 1 is 3 + 3 = 6 (values 4 - 6: 1), cumulative for 2 is 6 + 4 = 10 (values 7 - 10: 2), cumulative for 3 is 10 + 5 = 15 (values 11 - 15: 3), cumulative for 4 is 15 + 2 = 17 (values 16 - 17: 4), cumulative for 5 is 17 + 2 = 19 (values 18 - 19: 5), cumulative for 6 is 19 + 1 = 20 (value 20: 6). So the \( 10^{th} \) value is 2 (since the 10th value is the last value in the 2's group) and the \( 11^{th} \) value is 3 (first value in the 3's group). Wait, no, actually, when we list the data points:

0, 0, 0,

1, 1, 1,

2, 2, 2, 2,

3, 3, 3, 3, 3,

4, 4,

5, 5,

6

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

1:0, 2:0, 3:0,

4:1, 5:1, 6:1,

7:2, 8:2, 9:2, 10:2,

11:3, 12:3, 13:3, 14:3, 15:3,

16:4, 17:4,

18:5, 19:5,

20:6

So the 10th value is 2, the 11th value is 3. Wait, no, wait, the 10th value is the 10th element. Let's count:

1:0

2:0

3:0

4:1

5:1

6:1

7:2

8:2

9:2

10:2

11:3

12:3

13:3

14:3

15:3

16:4

17:4

18:5

19:5

20:6

Ah, I made a mistake earlier. The 10th value is 2 (the 10th element is the fourth 2), and the 11th value is 3 (the first 3). Wait, no, the 10th element is the 10th in the list. Let's list all 20 elements:

1. 0

2. 0

3. 0

4. 1

5. 1

6. 1

7. 2

8. 2

9. 2

10. 2

11. 3

12. 3

13. 3

14. 3

15. 3

16. 4

17. 4

18. 5

19. 5

20. 6

So the 10th value is 2, the 11th value is 3. Wait, no, median for even n is the average of the \( \frac{n}{2} \)th and \( (\frac{n}{2}+1) \)th terms. So \( \frac{20}{2} = 10 \)th and \( 11 \)th terms. The 10th term is 2, the 11th term is 3. Wait, no, looking at the list, the 10th term is 2 (the fourth 2), and the 11th term is 3 (the first 3). Wait, no, let's count again:

After 3 zeros (positions 1 - 3), 3 ones (positions 4 - 6), 4 twos (positions 7 - 10), 5 threes (positions 11 - 15), 2 fours (16 - 17), 2 fives (18 - 19), 1 six (20). So position 10 is the last two (position 10: 2), position 11 is the first three (position 11: 3). So median \( = \frac{2 + 3}{2} = 2.5 \)? Wait, no, that can't be. Wait, no, I think I messed up the cumulative frequency. Wait, cumulative frequency for 0: 3 (so values 1 - 3: 0), cumulative for 1: 3 + 3 = 6 (values 4 - 6: 1), cumulative for 2: 6 + 4 = 10 (values 7 - 10: 2), cumulative for 3: 10 + 5 = 15 (values 11 - 15: 3), cumulative for 4: 15 + 2 = 17 (values 16 - 17: 4), cumulative for 5: 17 + 2 = 19 (values 18 - 19: 5), cumulative for 6: 19 + 1 = 20 (value 20: 6). So the 10th value is 2 (since it's the last value in the 2's group, which is position 10), and the 11th value is 3 (first value in the 3's group, position 11). So median is \( \frac{2 + 3}{2} = 2.5 \)? Wait, no, that's incorrect. Wait, no, the median is the middle value when the data is ordered. For 20 data points, the median is the average of the 10th and 11th values. Let's list all data points:

0, 0, 0,

1, 1, 1,

2, 2, 2, 2,

3, 3, 3, 3, 3,

4, 4,

5, 5,

6

Now, let's count the number of elements:

- 0: 3 (total 3)

- 1: 3 (total 6)

- 2: 4 (total 10)

- 3: 5 (total 15)

- 4: 2 (total 17)

- 5: 2 (total 19)

- 6: 1 (total 20)

So the 10th element is the last 2 (since after 9 elements, we have 3 zeros, 3 ones, and 3 twos? Wait no, 3 zeros (3), 3 ones (3, total 6), 4 twos (4, total 10). So the 7th, 8th, 9th, 10th elements are 2. So the 10th element is 2, and the 11th element is 3 (since the 11th element is the first 3). So median is \( \frac{2 + 3}{2} = 2.5 \)? Wait, that's not right. Wait, no, I think I made a mistake in counting the number of twos. Wait, the dot plot for 2: how many dots? The problem says:

For 0: 3 dots, 1: 3 dots, 2: 4 dots, 3: 5 dots, 4: 2 dots, 5: 2 dots, 6: 1 dot. So 2 has 4 dots, so positions 7 - 10: 2 (since 3 + 3 = 6, so next 4 are 2: positions 7,8,9,10). Then 3 has 5 dots: positions 11 - 15: 3. So 10th position: 2, 11th position: 3. So median is \( \frac{2 + 3}{2} = 2.5 \). Wait, but that seems odd. Wait, maybe I miscounted the number of dots. Let's check the dot plot again:

0: 3 dots (three dots above 0)

1: 3 dots (three above 1)

2: 4 dots (four above 2)

3: 5 dots (five above 3)

4: 2 dots (two above 4)

5: 2 dots (two above 5)

6: 1 dot (one above 6)

Yes, that's correct. So total dots: 3 + 3 + 4 + 5 + 2 + 2 + 1 = 20.

Step4: Calculate the mean

Mean \( = \frac{\sum (x \times f)}{\sum f} \), where \( x \) is the goal value, \( f \) is the frequency.

\( \sum (x \times f) = (0 \times 3) + (1 \times 3) + (2 \times 4) + (3 \times 5) + (4 \times 2) + (5 \times 2) + (6 \times 1) \)

\( = 0 + 3 + 8 + 15 + 8 + 10 + 6 = 50 \)

Mean \( = \frac{50}{20} = 2.5 \)

Step5: Compare mean, median, mode

- Mode: 3 (frequency 5)

- Median: \( \frac{2 + 3}{2} = 2.5 \)? Wait, no, wait, earlier calculation of median was wrong. Wait, no, when n is even, median 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 and \( 11 \)th terms.

Looking at the ordered data:

1:0, 2:0, 3:0,

4:1, 5:1, 6:1,

7:2, 8:2, 9:2, 10:2,

11:3, 12:3, 13:3, 14:3, 15:3,

16:4, 17:4,

18:5, 19:5,

20:6

So the 10th term is 2, the 11th term is 3. So median is \( \frac{2 + 3}{2} = 2.5 \). Wait, but mean is also 2.5? Wait, no, mean was calculated as \( \frac{50}{20} = 2.5 \). Oh! I see, I made a mistake earlier in the median calculation. Wait, no, the 10th term is 2, 11th term is 3, so median is \( \frac{2 + 3}{2} = 2.5 \), and mean is also 2.5. Wait, but mode is 3. Wait, no, let's recalculate the mean:

0*3 = 0

1*3 = 3

2*4 = 8

3*5 = 15

4*2 = 8

5*2 = 10

6*1 = 6

Sum: 0 + 3 = 3; 3 + 8 = 11; 11 + 15 = 26; 26 + 8 = 34; 34 + 10 = 44; 44 + 6 = 50. 50 / 20 = 2.5. Correct.

Median: 10th term is 2, 11th term is 3. So (2 + 3)/2 = 2.5. So mean = 2.5, median = 2.5, mode = 3. Wait, but the options are:

(1) mean > mode (2.5 > 3? No)

(2) mean = median (2.5 = 2.5? Yes)

(3) mode = median (3 = 2.5? No)

(4) median > mean (2.5 > 2.5? No)

Wait, so the correct answer is (2) mean = median.

Wait, but let's check again. Maybe I messed up the median. Wait, the number of data points: 20. So the median is the average of the 10th and 11th values. Let's list the data points in order:

0, 0, 0,

1, 1, 1,

2, 2, 2, 2,

3, 3, 3, 3, 3,

4, 4,

5, 5,

6

Now, let's count the positions:

1:0

2:0

3:0

4:1

5:1

6:1

7:2

8:2

9:2

10:2

11:3

12:3

13:3

14:3

15:3

16:4

17:4

18:5

19:5

20:6

Yes, 10th is 2, 11th is 3. So median is (2 + 3)/2 = 2.5. Mean is 50/20 = 2.5. So mean = median. So option (2) is correct.

Answer:

(2) mean = median