Noesis
1.3.2 · D5Basic Data & Probability

Question bank — Bar charts, pictograms, pie charts — drawing and reading

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Reminder of the words we lean on: a category is one distinct thing being counted (Apples, Monday, Shop A). A frequency is how many times it occurred. A scale tells you how many real units one drawing-unit stands for. A sector is one pizza-slice of a pie chart, and its angle is the wedge opening measured in degrees — see Angles.

True or false — justify

A taller bar always means a larger value.
False — only if the vertical scale is uniform and starts at zero. A broken or non-uniform axis can make a small difference look huge, so height only maps to value when the scale is honest.
In a bar chart, the width of a bar carries meaning.
False — all bars use equal width; only the length/height encodes the value. Width is just a drawing choice, so wide bars are not "more" than thin ones.
Bars in a bar chart should touch each other with no gaps.
False — gaps signal that each category is distinct and separate. Touching bars are for histograms (continuous data), which is a different chart.
Two pie charts with the same slice angles represent the same actual quantities.
False — equal angles mean equal fractions of their own totals, not equal counts. A 50% slice of 20 people is 10; a 50% slice of 200 people is 100.
A pie chart's sector angles must add up to exactly .
True — the whole circle is one whole dataset (), so the parts must fill it completely; a sum of or signals a rounding or arithmetic slip.
In a pictogram, three icons always mean "three".
False — you must read the key first. If , three icons mean ; the icon is a package, not a single unit.
A pie chart is a good choice when you have 20 categories.
False — many tiny slices become an unreadable clutter; a bar chart handles many categories far more clearly, so pie charts suit only a few parts of a whole.
You can read an exact frequency straight off a pie chart without knowing the total.
False — a pie shows proportions, not counts. A slice is one-quarter of something, but you need the total to turn that quarter into an actual number.
Doubling every category's value changes the pie chart's slice angles.
False — each angle is , and doubling every also doubles , so the fraction is unchanged. The pie looks identical; only a bar chart with counts on the axis would grow.

Spot the error

"Monday's bar is 7 cm, so Monday had 7 ice creams."
The error is ignoring the scale. If , then ice creams; the height is in drawing-units, not data-units.
"Apples are of the total, so the apple sector is ."
The circle is , not . The angle is ; percentages and degrees are two different scales for the same fraction (see Fractions and percentages).
"I drew each bar a different width to make the chart look interesting."
Unequal widths make the eye compare areas, distorting the comparison. Only equal-width bars let length alone represent the value fairly.
"My pie angles came to but that's close enough, I'll leave it."
A physical circle is exactly ; a overflow means a slip somewhere. The fix is to recheck the arithmetic and absorb the difference into the largest sector.
"Shop A has 2.5 car-icons, so Shop A has 2.5 cars."
A partial icon is still scaled by the key. With , that is cars; the half-icon means half of ten, not half a car.
"The soccer slice is bigger than the tennis slice, so more people play soccer at every school in the country."
A single pie describes only this dataset's proportions. It says nothing about other schools or absolute national totals.
"I started my bar chart's y-axis at 50 to save space — the bars still compare fine."
A non-zero baseline exaggerates differences: a bar of value 52 next to one of value 55 looks tiny-vs-huge. For honest visual comparison the axis should start at zero.

Why questions

Why do pie charts use rather than, say, marks around the circle?
Because a full turn is defined as (see Angles), and a protractor measures in degrees; is simply the unit our drawing tool speaks in, standing for the whole .
Why must the bar-chart scale be uniform from bottom to top?
So that equal drawing-distances always mean equal value-changes. If the scale stretched partway up, equal-looking bars would represent unequal amounts and the picture would lie.
Why does dividing the value by the scale factor give the bar height?
The scale says "this many real units fit in one cm", so dividing the value by that rate answers "how many cm long is this many units?" — it converts data-units into drawing-units.
Why can a pictogram show a fractional icon while a bar chart just uses a smooth height?
A pictogram counts in whole "packages", so leftovers must be shown as part of a package. A bar has a continuous length, so any in-between value is just a slightly shorter bar — no fraction needed.
Why is a pie chart natural for "parts of a whole" but awkward for "which is biggest across time"?
A circle visually is one complete whole being divided, matching proportions perfectly. But comparing many separate slices by eye is hard, whereas side-by-side bar heights are easy to rank.
Why do we check that pictogram, bar and pie all come from the same underlying frequency table?
All three are just different pictures of the same counts. If a chart disagrees with the table, the chart is wrong — the table is the source of truth.
Why does a sector represent exactly half the data?
Because , so the wedge fills half the circle; a straight-line boundary through the centre splits the whole into two equal halves.

Edge cases

A category has a frequency of . What does its bar look like?
It has zero height — effectively no bar, just a gap and a label. A missing bar is meaningful: it says "this category occurred, but count is zero", not "we forgot it".
A category has frequency in a pie chart. What is its angle?
— the slice has zero width and simply does not appear. Only categories with non-zero counts get a visible wedge.
One category makes up the entire dataset. What does its pie chart look like?
A single sector of — one full circle with no dividing lines. It's a valid but uninformative pie.
The total of all values is (nothing was collected). Can you draw a pie chart?
No — the angle formula divides by , which is undefined. With no data there are no proportions to slice.
A pictogram value is smaller than one icon-unit, e.g. 3 items with .
You draw a fraction of one icon: of an icon. The key still governs, and the reader must estimate the partial picture.
A value gives an ugly angle like . Do you round the drawing?
You may round to draw with a protractor, but keep the exact value for the check; accumulated rounding is what causes the total to miss , so absorb any leftover into the biggest slice.
Two categories have the same frequency. How should their bars and slices look?
Identical — equal-height bars and equal-angle sectors. Any visible difference would falsely imply the counts differ, which uses proportional reasoning correctly.
Recall One-line self-test

Cover every answer above; if you can justify each in your own words (not just "true/false"), you've beaten the traps.

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