Percentages and growth in tables and charts

Growth is a fraction of the starting value

Most data insight questions on time series come down to one formula:

$$\text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%$$

The base is the older value, the one the change starts from. A rise from 40 to 50 is $\frac{10}{40} = 25\%$; a fall from 50 to 40 is $\frac{10}{50} = 20\%$. In a table with many years, "from 2023 to 2024" fixes the base as the 2023 cell, whichever is larger.

Multipliers chain and reverse

A change of $+p\%$ multiplies by $1 + \frac{p}{100}$, and a change of $-p\%$ by $1 - \frac{p}{100}$. This gives two rules you will use constantly:

• Successive changes multiply. Up 20% then down 10% is $1.2 \times 0.9 = 1.08$, a net rise of 8%, not 10%.
• To go back, divide. If a 2025 figure of 1,350 is 8% above 2024, then 2024 was $\frac{1\,350}{1.08} = 1\,250$, not $1\,350 - 8\%$.

When a table gives yearly growth rates instead of levels, you need the multipliers to say anything about the change over several years. Adding the rates is only an approximation, and options are built to catch it.

Index numbers are levels, not changes

An index sets a base period to 100 and expresses other periods relative to it: a price index of 117 in 2022 means prices 17% above the base year. But the change between two later years is not the difference of the indices read as percent. From 117 to 130 is a rise of 13 index points, and $\frac{13}{117} = \frac{1}{9} \approx 11.1\%$. Only changes measured from the base year can be read off directly.

Percentage points or percent

When the figure in the display is itself a rate (an unemployment rate, a market share, a pass rate), a change can be described two ways. A rate going from 35% to 25% has fallen by 10 percentage points and by $\frac{10}{35} \approx 28.6\%$ of itself. And the count behind the rate is a third thing: if the group also grew, the count may have fallen less, or even risen. Ask which of the three the question wants, then compute only that.

Growth of a total is not an average of growths

If three gyms grew by 20%, 5% and $-10\%$, the combined membership did not grow by $\frac{20 + 5 - 10}{3} = 5\%$ unless the gyms were the same size. Bigger units weigh more. Add the levels first, then compute one percent change from the totals.

Comparing growth rates by hand

No calculator means you compare rates the efficient way:

• Find a friendly form. $\frac{54}{360}$: divide by 18 to get $\frac{3}{20} = 15\%$.
• Use 1% and 10%. For 95 out of 1,420, 1% of 1,420 is 14.2, so 6.5% is about 92 and 7% is about 99: the answer is between them, nearer 6.7%.
• Cross-multiply to compare two fractions without dividing.
• Estimate first. Is the answer above or below 10%? Many options die on that question alone.

Tables of growth rates

Some displays give yearly growth rates instead of levels: a column of "change on previous year (%)". Read them with care. A positive rate that shrinks (from 8% to 3%) still means the quantity rose, only more slowly; the level fell only in a year with a negative rate. And the highest level is not necessarily in the year with the largest rate: if every rate is positive the level keeps rising, so the peak is at the end of the last year; once some rates are negative, an earlier peak can stay the highest. When a question asks about levels, sketch them from an assumed start of 100 and apply the multipliers.

A routine for growth questions

Identify the two figures and their order in time. Decide whether the question asks for a change in level, in percent, or in percentage points. For several periods, use multipliers; for a group, add levels first. Then estimate before you choose: a final answer that disagrees with the rough direction of the data has a sign or base error in it.