Pie charts and shares of a whole

A pie shows parts of one whole

A pie chart splits one total into parts. Each sector's share is a fraction of that total, and the shares add up to 100% (printed labels may add to 99% or 101% after rounding). What a pie shows well is the relative size of the parts at one moment; what it never shows on its own is the size of the whole. For that you need a total from the title, a note, a table or a passage.

Three conversions

Share to amount. $\text{amount} = \text{share} \times \text{total}$. If a budget of €2,400,000 has 25% for materials, materials cost $0.25 \times 2\,400\,000 = 600\,000$ euros.

Share to angle. A full circle is $360°$, so a share $s$ is a sector of $s \times 360°$. Learn the friendly ones: 10% is $36°$, 25% is $90°$, 50% is $180°$, and 5% is $18°$. Then 15% is $36° + 18° = 54°$.

Angle to share. Divide by 360: a sector of $72°$ is $\frac{72}{360} = \frac{1}{5} = 20\%$. Some pies are labelled with angles rather than percentages, so be ready to go both ways.

Comparing two pies

Questions love two pies side by side: this year and ten years ago, or two cities. A sector that is smaller in the second pie can stand for a larger amount, if the second whole is bigger. So:

• To compare shares, compare the percentages directly.
• To compare amounts, multiply each share by its own total first.

Example: food was 20% of a household's €30,000 spending in one year (€6,000) and 18% of €40,000 ten years later (€7,200). Food's share fell by 2 percentage points, yet spending on food rose by $\frac{1\,200}{6\,000} = 20\%$. Both statements are true; they answer different questions.

Change in share versus change in amount

A share moving from 45% to 36% has fallen by 9 percentage points, and by $\frac{9}{45} = 20\%$ of its old value. Neither number says what happened to the amount; that also depends on the change in the total. When a question asks about "the number of" or "spending on", compute amounts. When it asks about "the share" or "the proportion", compare percentages, and read whether it wants points or percent.

A share of a share

Pies are sometimes nested: a sector is split again in a second chart or in the text. "60% of the students are in economics, and 25% of the economics students are international" makes international economics students $0.60 \times 0.25 = 0.15$, that is 15% of all students. Multiply the shares; never add them, and be clear which whole the final answer is a share of.

The "other" sector and rounded labels

Many pies end with a sector labelled "other" that lumps small categories together. It is one sector, not a category you can split: a question asking whether some unnamed group exceeds 3% cannot be answered from a 7% "other" sector. Labels are also often rounded, so the printed shares may add to 99% or 101%. If a question's options are close together, the rounding can matter; work from exact figures in a table or passage whenever one is given, and treat a rounded share as an approximation.

Pies without a calculator

Shares are usually round, so the arithmetic is mostly multiplying a round percent by a round total. Write $36\%$ of $2\,500$ as $36 \times 25 = 900$: one percent of 2,500 is 25. For angles, build from $10\% = 36°$ and $1\% = 3.6°$. For comparisons between sectors in the same pie, the totals cancel: the ratio of two amounts is just the ratio of their shares.