Foundations — Bar charts, pictograms, pie charts — drawing and reading

This is the ground floor of the parent topic. Before we draw a single bar, we must be able to read every symbol the parent note quietly assumes you already know. We start from absolutely nothing.

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1. What is a "value" and a "category"?

Picture a set of buckets. Each bucket has a label (that's the category) and some balls inside (that's the value). Everything in this whole topic starts as a table of label → how many.

Why the topic needs this: every chart is just a drawing of these buckets. A bar chart draws each bucket as a tower; a pictogram draws it as a row of icons; a pie chart draws all buckets as slices of one circle. If you can't name the buckets and count the balls, no chart is possible. These buckets come straight from a frequency table.

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2. The symbols $+$ and $=$, and the total $T$

If Apples $=10$, Bananas $=15$, Oranges $=5$, then: $T=10+15+5=30$

The picture: pour all the buckets into one big bucket. $T$ is how many balls that big bucket holds.

Read $\sum$ out loud as "the sum of." That is all it means — it is not a scary new operation, just a lazy way to avoid writing lots of $+$ signs.

Why the topic needs $T$: a pie chart shows parts of a whole. The "whole" is exactly $T$. You cannot ask "what fraction is Apples?" until you know the size of the whole.

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3. The fraction bar $\frac{\cdot }{\cdot }$ — the heart of everything

The picture below shows why $\frac{10}{30}$ means "10 out of 30" and why that equals $\frac{1}{3}$.

Why "value over total"? Every chart formula in the parent note contains $\frac{\text{value}}{T}$. Look at the two places it appears:

$\theta =\frac{v}{T}\times 360^{\circ }\text{and}\text{Value}=\frac{\theta }{360^{\circ }}\times \text{Total}$

In both, the fraction is asking "what slice of the whole is this?" The bucket's balls ($v$) sit on top; the big bucket's balls ($T$) sit on the bottom. That ratio — reviewed fully in Fractions and percentages and Ratio and proportion — is the "fair swap" our core idea promised.

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4. Multiplication $\times$ and the word "of"

The picture: $3\times 20$ is three identical stacks of $20$ balls. That is exactly what a pictogram does: three icons, each worth $20$, gives $3\times 20=60$.

Why the topic needs it: every "read the chart" step is a multiplication.

• Bar chart: $\text{value}=\text{bar length}\times \text{scale}$.
• Pictogram: $\text{value}=\text{icons}\times \text{value per icon}$.
• Pie chart: $\text{value}=\frac{\theta }{360^{\circ }}\times T$.

They are all the same shape: a count of pieces, times the size of one piece.

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5. Scale and "proportional" — why the swap is fair

$\text{length}=\frac{\text{value}}{k}⟺\text{value}=\text{length}\times k$

Why "uniform" matters: if the scale changed halfway up the axis (1 cm = 5 near the bottom, 1 cm = 10 near the top), then a taller bar would lie — its extra length would not mean extra data. A uniform (unchanging) scale is what keeps the swap fair everywhere on the page.

Recall Test the proportional idea

If 1 cm = 5 units, how tall is a bar worth 0 units? ::: $0÷5=0$ cm — a value of zero must draw as nothing. That is proportionality working at the smallest case.

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6. The circle, degrees, and the symbol $360^{\circ }$

Why $360$ and not $100$? Because a circle's whole turn is agreed to be $360^{\circ }$ (a very old choice — $360$ divides neatly by many numbers, so common fractions give whole-number angles). To find a slice we ask: what fraction of the whole circle is this category? and give it that fraction of $360^{\circ }$:

$\theta =\frac{v}{T}\times 360^{\circ }$

This is our "fair swap" again — now trading a fraction for an angle instead of a length. Measuring $\theta$ needs a protractor; see Angles for the tool itself.

Recall Every case of the swap must add up

If Bananas take $180^{\circ }$ and Oranges $60^{\circ }$ and Apples $120^{\circ }$, what must they total, and why? ::: $180^{\circ }+60^{\circ }+120^{\circ }=360^{\circ }$. All the slices together are the whole circle, so their angles must fill exactly one full turn — no gaps, no overlap.

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7. Percent — a fraction wearing a costume

The picture: imagine every whole cut into exactly $100$ equal slivers. "$25\%$" means "$25$ of those $100$ slivers." A pie chart can be labelled in percent instead of degrees because both describe the same fraction of the whole — one uses $100$ as the whole, the other uses $360^{\circ }$.

Why the topic needs it: the parent note says "the full circle $=360^{\circ }=100\%$." Percent is just the fraction language that many exam questions arrive in. Convert with the same fair-swap machinery: $\text{angle}=\frac{\text{percent}}{100}\times 360^{\circ }$.

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How the foundations feed the topic

Read it top-down: everything begins with categories and values. The total and the fraction branch off toward pie charts; the scale multiply branch heads toward bars and pictograms; all roads meet at reading and drawing the charts themselves.

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Equipment checklist

Say the answer out loud before revealing — if you can't, re-read that section above.

• What is a "category" versus a "value"? ::: A category is the labelled group (e.g. Apples); the value is how many are in it (e.g. $10$).
• What does the symbol $\sum$ tell you to do? ::: Add up everything of that kind — it's shorthand for a long chain of $+$ signs.
• In $\frac{v}{T}$, which number is the whole and where does it go? ::: The total $T$ is the whole and goes on the bottom (denominator).
• Convert a value to a bar length with scale $k$: which operation? ::: Divide the value by $k$ ($\text{length}=v/k$).
• Convert a bar length back to a value: which operation? ::: Multiply the length by $k$ ($v=\text{length}\times k$).
• What English word does $\times$ often stand for in these formulas? ::: "of" — as in "$\frac{1}{4}$ of $120$" $=\frac{1}{4}\times 120$.
• How many degrees is a full circle, and what symbol marks degrees? ::: $360^{\circ }$; the raised little circle ${}^{\circ }$ means "degrees."
• What is $\theta$ in a pie chart? ::: The angle at the centre that measures how wide a sector (slice) opens.
• Why must a uniform scale be used on a bar axis? ::: So length stays proportional to value everywhere — a changing scale would make bars lie.
• Write $25\%$ as a fraction. ::: $\frac{25}{100}=\frac{1}{4}$, because percent means "out of $100$."
• What must every pie chart's sector angles add up to, and why? ::: Exactly $360^{\circ }$, because the slices together are the whole circle.