Averages (Mean, Median, Mode and Range): Question 5

Worked solution

Reading the bar chart

Each bar tells you how many matches (the frequency) ended with a particular number of goals. Writing this as a frequency table makes every calculation easier:

Check the total: $3 + 5 + 8 + 2 + 2 = 20$ matches, as stated. So there are $n = 20$ data values.

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(a) Mode

The mode is the value that occurs most often: the goals score with the tallest bar.

The tallest bar is above $2$ goals (it reaches $8$ matches), so:

$\text{Mode} = \boxed{2}\text{ goals}$

Be careful: the mode is the goals value $2$, not the frequency $8$.

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(b) Range

The range measures the spread of the goals values (not the frequencies):

$\text{Range} = \text{largest} - \text{smallest} = 4 - 0 = \boxed{4}\text{ goals}$

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(c) Median

The median is the middle value once all $20$ results are put in order. Because the data is already grouped by goals scored, the ordered list runs through the goals values, each repeated by its frequency:

$\underbrace{0,0,0}_{3},\ \underbrace{1,1,1,1,1}_{5},\ \underbrace{2,2,2,2,2,2,2,2}_{8},\ \underbrace{3,3}_{2},\ \underbrace{4,4}_{2}$

With $n = 20$ (an even number), the middle is between the

$\frac{n}{2} = 10\text{th and }\frac{n}{2}+1 = 11\text{th values}.$

Counting with a running total of the frequencies:

• Values $1$–$3$ are $0$
• Values $4$–$8$ are $1$
• Values $9$–$16$ are $2$

So the $10$th and $11$th values both fall in the ”$2$ goals” group. The median is their mean:

$\text{Median} = \frac{2 + 2}{2} = \boxed{2}\text{ goals}$

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(d) Mean

The mean is the total number of goals divided by the number of matches. Multiply each goals value by its frequency, then add:

$\text{Total goals} = (0 \times 3) + (1 \times 5) + (2 \times 8) + (3 \times 2) + (4 \times 2)$

$= 0 + 5 + 16 + 6 + 8 = 35$

Now divide by the number of matches, $n = 20$:

$\text{Mean} = \frac{35}{20} = 1.75$

Correct to $2$ decimal places:

$\text{Mean} = \boxed{1.75}\text{ goals}$

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Summary

The mean ($1.75$) is a little lower than the mode and median ($2$) because the three matches with $0$ goals pull the average down, a useful reminder that the mean is affected by every value, while the mode and median are not.