Averages (Mean, Median, Mode and Range): Question 2

Worked solution

Part (a): Mean from a frequency table

For data in a frequency table the mean is

$\bar{x}=\frac{\sum fx}{\sum f}$

where $\sum fx$ is the total of (value $\times$ frequency) and $\sum f$ is the total frequency (here, the total number of matches).

Step 1: Find $\sum fx$ (total number of goals).

$\sum fx = (0\times 6)+(1\times 11)+(2\times 9)+(3\times 8)+(4\times 4)+(5\times 2)$

$\sum fx = 0+11+18+24+16+10 = 79$

Step 2: Find $\sum f$ (total number of matches).

$\sum f = 6+11+9+8+4+2 = 40$

Step 3: Divide.

$\bar{x}=\frac{79}{40}=1.975$

$\boxed{\text{Mean} = 1.975 \text{ goals per match}}$

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Part (b): Find a missing frequency given the mean

Use the same formula, but now $\sum fx$ and $\sum f$ both contain the unknown $p$.

Step 1: Write $\sum fx$ in terms of $p$.

$\sum fx = (0\times 3)+(1\times 6)+(2\times 12)+(3\times p)+(4\times 5)$

$\sum fx = 0+6+24+3p+20 = 50+3p$

Step 2: Write $\sum f$ in terms of $p$.

$\sum f = 3+6+12+p+5 = 26+p$

Step 3: Form an equation using mean $=2.2$.

$\frac{50+3p}{26+p}=2.2$

Step 4: Solve. Multiply both sides by $(26+p)$:

$50+3p = 2.2(26+p)$

$50+3p = 57.2+2.2p$

$3p-2.2p = 57.2-50$

$0.8p = 7.2$

$p = \frac{7.2}{0.8}=9$

$\boxed{p = 9}$

Check: with $p=9$, total goals $=50+3(9)=77$ and total matches $=26+9=35$, giving $\dfrac{77}{35}=2.2$. ✓

Alternative method: The total number of goals must equal $\text{mean}\times\text{matches}=2.2(26+p)$. Setting this equal to the table total $50+3p$ gives the same equation $50+3p=2.2(26+p)$.

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Part (c): Combined mean

When combining two data sets, add the totals; never average the two means:

$\text{combined mean}=\frac{\text{total goals from both teams}}{\text{total matches of both teams}}$

Step 1: Totals for each team.

• Falcons: $\sum fx = 79$ goals in $40$ matches (from part (a)).
• Kestrels: $\sum fx = 50+3(9)=77$ goals in $35$ matches (from part (b)), or equivalently $2.2\times 35 = 77$.

Step 2: Combine.

$\text{combined mean}=\frac{79+77}{40+35}=\frac{156}{75}=2.08$

$\boxed{\text{Combined mean} = 2.08 \text{ goals per match}}$

(Note that the simple average $\tfrac{1.975+2.2}{2}=2.0875$ is incorrect, because the two teams played different numbers of matches.)