Quick questions on Data handling and statistical diagrams explained: O-Level E-Maths

A pie chart shows how a whole divides into parts, with each sector's angle proportional to its frequency. Since a full circle is $360^\circ$, the angle for a category is:

A histogram displays grouped continuous data with bars touching, since the data is continuous. For equal class widths the bar height is the frequency; the area of each bar represents the frequency of that class.

A stem-and-leaf diagram keeps the actual data values while showing their shape. The stem is the leading digit (or digits) and each leaf the final digit. A key explains the place value, and the diagram makes the mode, range and median easy to read.

Pie-chart questions run in both directions: as well as drawing a sector from a frequency, you often have to recover a frequency from a sector angle. Reverse the angle formula: the frequency is $\tfrac{\text{sector angle}}{360^\circ} \times \text{total}$. So a sector of $54^\circ$ in a pie chart of $200$ people represents $\tfrac{54}{360} \times 200 = 30$ people. The same proportion can be read as a fraction or percentage of the whole, since the sector angle, the fraction, and the frequency all carry the same proportion.

A back-to-back stem-and-leaf diagram shares one central column of stems, with one data set's leaves growing leftward and the other's rightward, so two groups can be compared side by side. Reading it, the leaves nearer the stem are the smaller digits in each direction, and the overall shape shows which group tends higher or is more spread out. This display keeps every original value while making a direct comparison easy, for instance contrasting two classes' test marks. Recognising when a comparison calls for a back-to-back diagram, rather than two separate ones, is a useful data-handling judgement.

A category represents $\dfrac{1}{4}$ of the data. State its pie-chart sector angle. [1 mark]

State which diagram best shows the change in a city's population over 50 years. [1 mark]

In a pie chart of $200$ people, a sector has angle $54^\circ$. How many people does it represent? [2 marks]