Percent Change and Percentage Points Done Right

Why this matters

"Pressure dropped 20 percent." From what, to what, and 20 percent of which number? A service report that leaves the base out invites the reader to supply one, and the one they pick is often wrong. The same pressure falling from 50 to 40 psi is down 20 percent, and coming back from 40 to 50 psi is up 25 percent, with the same 10 psi in both directions. An efficiency rising from 80 to 95 percent is up 15 percentage points and up 18.75 percent, and the fuel it saves is neither of those. None of this is advanced. It is one formula and one distinction, applied the same way every time, and the payoff is a report whose numbers mean what the writer meant.

The one formula

Percent change compares a new value with an old one, measured against the old:

percent change = (new - old) / old x 100

The denominator is the whole idea. Change is always measured against where you started, so the same difference produces a different percentage depending on which end you start from.

THE BASE DECIDES THE PERCENT

pressure falls    50 -> 40 psi
  (40 - 50) / 50   = -10 / 50   = -20 percent

pressure returns  40 -> 50 psi
  (50 - 40) / 40   = +10 / 40   = +25 percent

down 20 then up 20    50 -> 40 -> 48 psi
  40 x 1.20        = 48, not 50
  (48 - 50) / 50   = -4 percent net

EFFICIENCY        80 -> 95 percent
  in points        95 - 80      = 15 points
  as a change      15 / 80      = +18.75 percent
  fuel for same heat
    at 80: 1 / 0.80  = 1.250
    at 95: 1 / 0.95  = 1.053
    (1.053 - 1.250) / 1.250 = -15.8 percent

Each block has the same shape: name the old value, name the new value, divide the difference by the old. The rest of this card is about what goes wrong when one of those three is left unstated.

Why down 20 and up 20 is not a round trip

The first two blocks show it with pressure. A drop of 20 percent from 50 psi lands at 40 psi. A rise of 20 percent from 40 psi lands at 48 psi, not 50, because the second 20 percent is taken of a smaller number. To get back to 50 from 40, the rise has to be 25 percent.

That asymmetry is behind a common misreading of before-and-after numbers. A pump that lost 20 percent of its head and then had its impeller cleaned "gaining back 20 percent" has not recovered. A static pressure that rose 30 percent when a filter loaded and then fell 30 percent after it was changed is now below where it started: 1.30 x 0.70 = 0.91, so 9 percent lower than the original. Whenever two percent changes are chained, multiply the factors (1.20, 0.80, 1.30, 0.70) rather than adding the percentages, and the round trip either closes or it visibly does not.

Worked example: a trend across three visits

A maintenance log records total external static pressure on an air handler at three visits: 0.50 in WC at the first, 0.65 in WC at the second, 0.80 in WC at the third, with the same filter type each time and nothing else changed that the log records.

• Visit 1 to 2: (0.65 - 0.50) / 0.50 = 0.15 / 0.50 = +30 percent.
• Visit 2 to 3: (0.80 - 0.65) / 0.65 = 0.15 / 0.65 = +23.1 percent.
• Visit 1 to 3: (0.80 - 0.50) / 0.50 = 0.30 / 0.50 = +60 percent.

The rise per visit was the same 0.15 in WC both times, yet the percentages fell from 30 to 23.1, because the second one is measured against a larger base. A reader seeing "+30 percent, then +23 percent" might conclude the climb is slowing. In inches of water it is not slowing at all; it is rising by the same step each visit. And the overall change is neither the sum (30 + 23.1 = 53.1) nor anything else you can get by adding: it is the product of the factors, 1.30 x 1.231 = 1.60, which is the 60 percent computed directly.

So for a trend, report the readings themselves and the change from the first one. "0.50, 0.65, 0.80 in WC, up 60 percent from the first visit" tells the truth. A column of visit-to-visit percentages hides a steady rise behind falling numbers.

Percentage points versus percent

When the thing being measured is itself a percentage (an efficiency, a callback rate, a relative humidity, a duty cycle), there are two ways to describe its change, and they are different numbers.

• Percentage points are the plain difference between two percentages. An efficiency going from 80 percent to 95 percent rises 15 points.
• Percent change applies the formula to the percentages as values. 15 points on a base of 80 is 15 / 80 = 18.75 percent.

Both are correct. Writing "efficiency improved 15 percent" is wrong both ways: it is 15 points, or 18.75 percent. A reader who sees "15 percent" will assume one of them, and on a figure like a callback rate going from 4 percent to 5 percent the two readings are far apart: 1 point, or 25 percent more callbacks. A shop owner reading "callbacks up 1 percent" and one reading "callbacks up 25 percent" are looking at the same jobs and drawing very different conclusions. Say "points" for the difference and keep "percent" for the change.

Worked example: what an efficiency gain does to fuel

A customer replacing a furnace with a nominal 80 percent efficiency by one at 95 percent asks how much less gas they will burn for the same heat.

The tempting answers are 15 percent (the points) and 18.75 percent (the change in efficiency). Neither is the fuel. Fuel needed is heat delivered divided by efficiency, so for the same heat it goes as 1 / efficiency:

• Fuel at 80 percent: 1 / 0.80 = 1.250 units of fuel per unit of heat.
• Fuel at 95 percent: 1 / 0.95 = 1.053 units.
• Change in fuel: (1.053 - 1.250) / 1.250 = -0.197 / 1.250 = -15.8 percent.

Check it another way: the fraction of fuel saved is 1 - (old efficiency / new efficiency) = 1 - 0.80 / 0.95 = 1 - 0.842 = 0.158. Same 15.8 percent.

So the honest sentence is: for the same heat, the new furnace burns about 16 percent less gas than the old one would at its rated efficiency. That carries its own conditions in the same breath: rated efficiencies, not measured ones; the same heat demanded, which a different house, a different winter or a leakier duct system will not supply; and nothing about cost, which depends on the rate on the bill. The arithmetic is exact. What it claims is only as good as those three conditions, and the sentence should name them.

Worked example: amps over nameplate

A condenser fan motor has a nameplate full-load current of 6.2 A. A tech reads 7.1 A.

• Percent over: (7.1 - 6.2) / 6.2 = 0.9 / 6.2 = 14.5 percent over nameplate.
• Write it with its base: "7.1 A against a 6.2 A nameplate, about 15 percent over."

Stated that way, the reader can check it and nobody can misread the denominator. Stated as "running 15 percent high," a reader might take the base to be last year's reading, or the compressor's current, or the breaker rating. Whether 15 percent over nameplate matters depends on the voltage at the motor and its service factor, which is a judgment for a different card; the arithmetic just has to be unambiguous.

Percent error against a reference

Checking an instrument against a reference uses the same formula, with the reference as the base: percent error = (measured - reference) / reference. A meter reading 118.0 V on a 120.0 V calibrated source is (118.0 - 120.0) / 120.0 = -1.67 percent. The base is the reference because the reference is the value you trust; dividing by the instrument's own reading gives a slightly different and less meaningful number (-2.0 / 118.0 = -1.69 percent here, close only because the error is small).

"Thirty percent more" is not "thirty percent less" the other way

If an old motor draws 30 percent more power than its replacement, the replacement does not draw 30 percent less than the old one. Call the new motor's draw 1.00. The old draws 1.30. The new one's saving, measured against the old: (1.00 - 1.30) / 1.30 = -0.30 / 1.30 = -23.1 percent. "Thirty percent more" and "23 percent less" describe the same pair; only the base changed.

Sales copy and spec sheets often choose the base that makes the bigger number, and that is not a reason for a service report to do the same. Pick the base that answers the customer's question (usually the equipment they have now) and name it.

Percentages that cannot be taken

Some quantities have no meaningful percentage change, and a calculator will produce one anyway.

• Temperatures on ordinary scales. A room going from 40 F to 80 F has not doubled in any physical sense; the Fahrenheit zero is arbitrary. Talk about temperature in degrees of difference. Temperature Conversion and the Delta-T Rule covers why.
• A change from zero. A count going from 0 to 3 has no percentage change; dividing by zero is undefined. Give the counts.
• Small bases. Two callbacks last month and three this month is a 50 percent rise built on one job. With a small base, report the counts beside the percentage, or leave the percentage out.

Averaging several percentages raises its own problems, because a plain mean of rates taken over different-sized groups is not the overall rate; that is covered in Averaging Readings: Mean, Median and Weighted.

The reporting habit

Every percentage in a report carries three things in the same sentence: the old value, the new value, and which one is the base. "Supply pressure fell from 50 psi to 40 psi, down 20 percent." "Efficiency rose from 80 to 95 percent, up 15 points." With both values written out, any reader can redo the arithmetic, which is the whole point of writing it down; with the percentage alone, they are trusting a base they cannot see.

References

• Equipment nameplates and manufacturer data for rated efficiency and full-load current
• See related: Averaging Readings: Mean, Median and Weighted; Temperature Conversion and the Delta-T Rule; Write the Calculation Down: The Field Math Habit; Common Unit Conversions + Math Formulas