Percent vs. Percentage Points: What's the Difference?

Start with the base

A percentage is a part divided by a base, multiplied by 100. The arithmetic is simple. Most mistakes come from using the wrong base, so the first question to ask about any percentage is: percent of what?

Everyday percentage problems come in three shapes. The first asks what percent one number is of another: 45 is 75% of 60. The second asks for a percentage of a number: 15% of 80 is 12. The third works backwards from a part to the whole: if 30 is 40% of something, the whole is 75.

The backwards case comes up more often than people expect. Suppose a price of 120 includes 20% VAT and you want the pre-tax price. Taking 20% off 120 gives 96, which is wrong. The tax was charged on the pre-tax price, so 120 is 120% of that price, and the correct answer is 120 / 1.2 = 100. Once you see that the base is the pre-tax price, the mistake goes away.

• What percent is A of B: A / B x 100. Example: 45 / 60 x 100 = 75%.
• What is X% of B: B x X / 100. Example: 80 x 15 / 100 = 12.
• A is X% of what: A / X x 100. Example: 30 / 40 x 100 = 75.

Percent change vs. percentage-point change

When the quantity you are tracking is itself a percentage, such as an approval rating, an interest rate or a market share, there are two ways to describe a change. Suppose approval rises from 40% to 44%. The difference between the two figures is 4 percentage points. The relative change is 4 / 40 = 10%. Both statements are true, but they describe different things.

Writing "approval rose 10%" invites readers to imagine a jump from 40% to 50%. To avoid that, use percentage points (often shortened to points) for the arithmetic difference between two percentages, and percent for the relative change.

Interest rates show how large the gap can be. A rate that moves from 1.0% to 1.5% has risen by 0.5 percentage points, but by 50% in relative terms. Financial news usually reports such moves in points, or in basis points, where one basis point is 0.01 percentage point, so 0.5 points is 50 basis points. When a headline just says a rate went up by some percent, check the before and after numbers to see which meaning is intended.

Calculating percent change

Percent change is (new - old) / old x 100. If revenue grows from 800,000 to 1,000,000, the increase is 200,000, and 200,000 / 800,000 = 25%. Some reports instead give the new value as a percentage of the old: 1,000,000 / 800,000 = 125%, sometimes labelled year-over-year ratio. Subtract 100 from that ratio and you get the percent change, so a ratio of 112% means +12% and a ratio of 95% means -5%.

Notice that increases and decreases use different bases. If revenue falls from 1,000,000 back to 800,000, the decline is 200,000 / 1,000,000 = 20%. The same 200,000 gap is a 25% rise in one direction and a 20% fall in the other.

Percent change is undefined when the starting value is zero, since you cannot divide by zero. Going from 0 to 50 is best reported as an increase of 50, not as a percentage. Very small starting values cause a related problem: growth from 2 to 20 is a 900% increase, which sounds dramatic but involves tiny numbers. Showing the actual values next to the percentage keeps the picture honest.

Why stacked changes don't simply add

A 20% discount followed by an extra 10% off is not 30% off. On a 10,000 item, the first discount brings the price to 8,000, and the second takes 10% of 8,000, leaving 7,200. Relative to the original price you pay 0.8 x 0.9 = 0.72, so the combined discount is 28%. The second discount applies to an already reduced price.

The same logic explains why a rise followed by an equal fall leaves you behind. Increase 100 by 20% and you have 120; decrease 120 by 20% and you have 96, not 100. It is also why an investment that loses 50% needs a 100% gain to get back to where it started.

Growth rates over several periods multiply too. Three consecutive years of 10% growth give 1.1 x 1.1 x 1.1 = 1.331, a total increase of 33.1%, not 30%. Whenever percentages are applied one after another, convert each to a multiplier, multiply the multipliers, and convert back.

• 20% off then 10% off: 0.8 x 0.9 = 0.72, so 28% off in total.
• Up 20% then down 20%: 100 to 120 to 96.
• 10% growth for 3 years: 1.1 cubed = 1.331, so 33.1% in total.

A quick checklist for reading percentages

When you meet a percentage in a report or an article, run through a few questions. What is the base: the whole population, last year's figure, or something else? Is the underlying quantity already a percentage? If so, is the change described in points or in percent? And what are the actual numbers behind it?

Raw numbers keep percentages in proportion. "Users up 200%" may mean 10 users became 30. Conversely, a one-point change in a rate can involve a very large number of people when the base population is large. Neither the percentage nor the raw count tells the full story alone.

Be careful when averaging percentages, too. If 30 of 40 students pass in one class (75%) and 2 of 10 pass in another (20%), the simple average of the two rates is 47.5%, but the combined pass rate is 32 of 50, or 64%. To combine groups of different sizes, add up the underlying counts and divide again.

Checking your numbers with the calculators

The percentage calculator covers the three basic problem types: what percent A is of B, what X% of a number is, and what the whole is when you know a part and its percentage. You can also choose how the result is rounded: to the nearest value, always down, or always up.

The rate of change calculator takes a before value and an after value and returns the difference, the percent change and the after value as a percentage of the before value. Entering two rates, such as 40 and 44, is a quick way to see the 4-point difference and the 10% relative change side by side. It does not return a rate when the before value is zero, since percent change is undefined in that case.