Display categorical and numerical data using a range of statistical graphs, including column graphs, sector graphs, line graphs, divided bar graphs and Pareto charts, and interpret the displays

Each angle is frequency over total, times $360^{\circ}$, with total $90$.

Netball: $\tfrac{45}{90} \times 360 = 180^{\circ}$. Soccer: $\tfrac{30}{90} \times 360 = 120^{\circ}$. Cricket: $\tfrac{15}{90} \times 360 = 60^{\circ}$.

Check: $180 + 120 + 60 = 360^{\circ}$.

Markers award one mark for the correct method (fraction times $360$), marks for the three correct angles, and a mark for the closing check that they total a full turn. A common error is dividing by $360$ instead of multiplying, or forgetting to total to $360^{\circ}$.

Sorted descending already. Cumulative percentages of the total $400$:

Password $\tfrac{180}{400} \times 100 = 45\%$; + login $\tfrac{280}{400} \times 100 = 70\%$; + payment $\tfrac{340}{400} \times 100 = 85\%$; + app crash $\tfrac{380}{400} \times 100 = 95\%$; + other $100\%$.

The cumulative line first reaches at least $80\%$ at the third category (payment, $85\%$), so the top $3$ categories account for at least $80\%$ of tickets.

The 80/20 read tells the manager that a small number of categories (here $3$ of $5$) cause most of the tickets, so focusing effort on password, login and payment issues will resolve the great majority of the workload.

Markers award marks for correct cumulative percentages, the correct count of categories reaching $80\%$, and a sensible interpretation naming the "vital few" causes. Watch for using individual rather than cumulative percentages in part (b).

Write the sector angle as a fraction of the whole turn. The soccer sector is the fraction of the $360^{\circ}$ turn equal to the fraction of people who chose soccer:

Evaluate.

so the soccer sector has an angle of $126^{\circ}$. (Sanity check: $14$ out of $40$ is a bit more than a third, and $126^{\circ}$ is a bit more than $120^{\circ}$, which is a third of a turn, so the size is right.)

Part (a) read the value off the line. Find April on the horizontal axis, go up to the line, then across to the vertical axis. The point is $(\text{Apr}, 110)$, so April had $110$ mm of rain.

Part (b) subtract the two readings. The change from February to April is the later value minus the earlier value:

so the rainfall rose by $15$ mm from February to April. (Because the answer is positive, the line went up between those months, which matches reading a rise on the graph.)

Set up the fraction-of-$360^{\circ}$ rule. Each angle is the category's dollars over the total dollars, times $360^{\circ}$. The total is $1440 (and you can check $600 + 300 + 180 + 240 + 120 = 1440$).

Rent.

Food.

so the rent sector is $150^{\circ}$ and the food sector is $75^{\circ}$. (Cross-check: rent is $600 of $1440, which is just over $40\%$, and $150^{\circ}$ is just over $40\%$ of $360^{\circ}$, so the size is sensible.)

Part (a) sort the causes by frequency, largest first. A Pareto chart always orders the bars from the most common cause down: scratches $24$, dents $13$, paint $7$, loose parts $4$, other $2$. They are already in order here.

Part (b) build the cumulative percentage. Keep a running total of the frequencies, then divide each running total by the grand total $50$ and multiply by $100$.

Part (c) read the 80/20 point. The cumulative percentage first reaches at least $80\%$ at the third bar (paint), where it hits $88\%$. So the top $3$ causes (scratches, dents and paint) account for at least $80\%$ of the faults. Fixing those three would remove the great majority of the problems. (Check: $24 + 13 + 7 = 44$ faults out of $50$, and $44/50 = 88\%$, which matches the cumulative figure.)

Part (a) car angle. Use the fraction-of-$360^{\circ}$ rule with total $60$:

so the car sector is $120^{\circ}$.

Part (b) bicycle angle.

so the bicycle sector is $24^{\circ}$.

Part (c) car segment on a divided bar. A divided bar graph splits a single bar in the same proportions, so the car segment is the car fraction of the $300$ mm length:

so the car segment is $100$ mm long. (The method is identical to the sector angle: replace the $360^{\circ}$ of a full circle with the $300$ mm of the full bar.)

Part (a) cumulative percentages. The causes are already sorted largest first. Keep a running total of frequencies and convert each to a percentage of the total $200$.

Part (b) the two biggest causes. The first two bars, late delivery and wrong item, reach a cumulative $72\%$, which is more than $70\%$. So late delivery and wrong item together make up $72\%$ of all complaints.

Part (c) which two to fix. By the 80/20 principle a small number of causes drives most of the problem, so target the two tallest bars: late delivery ($90$) and wrong item ($54$). Together they are

of all complaints, so fixing these two would address $72\%$ of the month's complaints, far more than tackling any other pair. That is the efficient choice.

Part (d) why the bars are sorted. Sorting the bars from largest to smallest is what makes the chart "Pareto": it puts the most important causes on the left so the cumulative line rises steeply at first and then flattens, letting you read at a glance how few causes account for most of the total (the 80/20 read). If the bars were in any other order the cumulative line would not be increasing in a readable way and the "vital few" would be hidden among the "trivial many". (Check: $45 + 27 = 72$, matching the cumulative $72\%$, and the individual percentages $45, 27, 15, 8, 5$ sum to $100\%$.)

Part (a) total increase. Subtract the first value from the last:

so recycling rose by $30$ tonnes over the five years.

Part (b) the year-to-year fall. Read the step between consecutive years: $40 \to 52$ (up $12$), $52 \to 58$ (up $6$), $58 \to 55$ (down $3$), $55 \to 70$ (up $15$). The only fall is from $2022$ to $2023$:

a fall of $3$ tonnes.

Part (c) percentage increase. Percentage increase is the change over the original, times $100$:

so recycling rose by $75\%$ from $2020$ to $2024$.

Part (d) why a line graph suits the data. The data is a single quantity measured over time, and a line graph is built to show change and trend across time, joining the points so the rise and the one dip are easy to see. A sector graph splits one total into parts of a whole, which is not what this data is, so it could not show the year-on-year movement. (Check: the steps $+12, +6, -3, +15$ sum to $+30$, matching the total increase in part (a).)