Question

the graph below represents the number of goals scored by the boys soccer team in every game so far this season. goals scored what was the median number of goals scored?

Explanation:

Step1: Calculate total number of games

First, we sum up the frequencies (number of games) for each goal value. For 0 goals: 2 games, 1 goal: 1 game, 2 goals: 1 game, 3 goals: 1 game, 4 goals: 3 games, 5 goals: 5 games. Total number of games \( n = 2 + 1 + 1 + 1 + 3 + 5 = 13 \).

Step2: Determine the position of the median

The median is the middle value when the data is ordered. For \( n = 13 \) (odd), the median is the \( \frac{n + 1}{2} = \frac{13 + 1}{2} = 7^{\text{th}} \) value.

Step3: Order the data and find the 7th value

We list the number of goals in order of their frequency: 0 (2 times), 1 (1 time), 2 (1 time), 3 (1 time), 4 (3 times), 5 (5 times). Let's count the cumulative frequency:

• 0 goals: cumulative frequency 2 (values 1 - 2)
• 1 goal: cumulative frequency 3 (value 3)
• 2 goals: cumulative frequency 4 (value 4)
• 3 goals: cumulative frequency 5 (value 5)
• 4 goals: cumulative frequency \( 5 + 3 = 8 \) (values 6 - 8)
• 5 goals: cumulative frequency \( 8 + 5 = 13 \) (values 9 - 13)

The 7th value falls within the 4 - goal group (since cumulative frequency for 4 goals reaches 8, which includes the 7th value). Wait, no, wait: Wait, let's list the data points:

0, 0, 1, 2, 3, 4, 4, 4, 5, 5, 5, 5, 5. Wait, no, let's count the number of each:

• 0 goals: 2 games → data points: 0, 0
• 1 goal: 1 game → data point: 1
• 2 goals: 1 game → data point: 2
• 3 goals: 1 game → data point: 3
• 4 goals: 3 games → data points: 4, 4, 4
• 5 goals: 5 games → data points: 5, 5, 5, 5, 5

Now, let's list all data points in order: 0, 0, 1, 2, 3, 4, 4, 4, 5, 5, 5, 5, 5. Now, the total number of data points is 13. The median is the 7th value (since \( (13 + 1)/2 = 7 \)). Let's count:

1st: 0

2nd: 0

3rd: 1

4th: 2

5th: 3

6th: 4

7th: 4

Wait, so the 7th value is 4? Wait, no, wait: Wait, 0 (2), 1 (1), 2 (1), 3 (1), 4 (3), 5 (5). So cumulative counts:

After 0: 2

After 1: 3

After 2: 4

After 3: 5

After 4: 5 + 3 = 8

After 5: 13

So the 5th value is 3, 6th value is 4, 7th value is 4? Wait, no, the 6th value is the first 4, 7th is the second 4? Wait, no, let's list the indices:

1: 0

2: 0

3: 1

4: 2

5: 3

6: 4

7: 4

8: 4

9: 5

10: 5

11: 5

12: 5

13: 5

Yes, so the 7th value is 4? Wait, but wait, maybe I made a mistake in cumulative frequency. Wait, let's recalculate the total number of games: 2 (0) + 1 (1) + 1 (2) + 1 (3) + 3 (4) + 5 (5) = 2 + 1 + 1 + 1 + 3 + 5 = 13, correct. So the median is the middle value, which is the 7th term. Let's count the terms:

1: 0

2: 0

3: 1

4: 2

5: 3

6: 4

7: 4

So the median is 4? Wait, but wait, maybe I messed up the order. Wait, no, the data is ordered from least to most goals, so 0, 0, 1, 2, 3, 4, 4, 4, 5, 5, 5, 5, 5. So the 7th term is 4. Wait, but let's check again. Wait, the cumulative frequency up to 3 goals is 2 + 1 + 1 + 1 = 5. So the first 5 terms are 0, 0, 1, 2, 3. Then the next 3 terms are 4, 4, 4 (terms 6, 7, 8). Then the next 5 are 5s (terms 9 - 13). So term 6: 4, term 7: 4, term 8: 4. So the 7th term is 4. Therefore, the median number of goals scored is 4.

Answer:

4