Question

question 6 - dot plots
the box plot below shows the number of goals jessica scored in each lacrosse
game last season.
which statement about the dot plot below is correct?
mean > mode
mode = median
median > mean
mean = median

Explanation:

Step1: Count the number of dots for each 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 \( \frac{n}{2} \)-th and \( (\frac{n}{2} + 1) \)-th values.

• \( \frac{n}{2} = 10 \)-th value
• \( \frac{n}{2} + 1 = 11 \)-th value

Let's list the cumulative frequencies:

• Up to 0: 3
• Up to 1: \( 3 + 3 = 6 \)
• Up to 2: \( 6 + 4 = 10 \)
• Up to 3: \( 10 + 5 = 15 \)

So the 10th value is 2 and the 11th value is 3.
Median \( = \frac{2 + 3}{2} = 2.5 \)? Wait, no, wait. Wait, cumulative frequency up to 2 is 10 (so 10th value is 2), cumulative frequency up to 3 is 15 (so 11th value is 3). Wait, no, actually, when we list the data points:

0 (3 times), 1 (3 times), 2 (4 times), 3 (5 times), 4 (2 times), 5 (2 times), 6 (1 time)

So 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, the 10th value is 2 (since first 3 are 0, next 3 are 1, next 4 are 2: 3+3+4=10, so 10th is 2), 11th value is 3. Wait, no, 3+3+4=10, so the 10th term is the last 2, then 11th term is the first 3. So median is \( \frac{2 + 3}{2} = 2.5 \)? Wait, no, that's a mistake. Wait, no, the number of data points: 3 (0s) + 3 (1s) + 4 (2s) + 5 (3s) + 2 (4s) + 2 (5s) + 1 (6s) = 3+3=6, +4=10, +5=15, +2=17, +2=19, +1=20. So the 10th and 11th terms:

1st - 3rd: 0

4th - 6th: 1

7th - 10th: 2 (since 7,8,9,10: four 2s)

11th - 15th: 3 (five 3s: 11,12,13,14,15)

So 10th term is 2, 11th term is 3. So median \( = \frac{2 + 3}{2} = 2.5 \)? Wait, no, that can't be. Wait, no, the median for even number of observations is the average of the \( \frac{n}{2} \) and \( \frac{n}{2} + 1 \) terms. So \( n = 20 \), so \( \frac{20}{2} = 10 \), \( \frac{20}{2} + 1 = 11 \). So 10th term is 2, 11th term is 3. So median is \( \frac{2 + 3}{2} = 2.5 \). Wait, but mode is 3. So mode (3) is greater than median (2.5). Wait, maybe I made a mistake in calculating the median. Wait, let's list all 20 data points:

1. 0

1. 0

1. 0

1. 1

1. 1

1. 1

1. 2

1. 2

1. 2

1. 2

1. 3

1. 3

1. 3

1. 3

1. 3

1. 4

1. 4

1. 5

1. 5

1. 6

Yes, so 10th term is 2, 11th term is 3. So median is \( \frac{2 + 3}{2} = 2.5 \)

Step4: Calculate the mean

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

• For 0: \( 0 \times 3 = 0 \)

• For 1: \( 1 \times 3 = 3 \)

• For 2: \( 2 \times 4 = 8 \)

• For 3: \( 3 \times 5 = 15 \)

• For 4: \( 4 \times 2 = 8 \)

• For 5: \( 5 \times 2 = 10 \)

• For 6: \( 6 \times 1 = 6 \)

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

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

Wait, so mean is 2.5, median is 2.5, mode is 3.

Wait, let's check the options:

• mean > mode: 2.5 > 3? No.

• mode = median: 3 = 2.5? No.

• median > mean: 2.5 > 2.5? No.

• mean = median: 2.5 = 2.5? Yes.

Wait, that's the correct option.

Answer:

mean = median