Percentage Change Calculator

Background.

Sales rise from 100 to 150: that is a 50% increase. Next month they fall back from 150 to 100, and the fall is 33.33%, not 50%. Nothing is wrong with the arithmetic. A percentage change is always measured against the starting value, and the two months start from different places. The same 50 units are half of 100 but only a third of 150.

That dependence on the starting point is the whole definition: subtract the old value from the new, divide by the size of the old value, multiply by 100. A positive answer is an increase and a negative one a decrease. It is the figure behind year-on-year growth, price inflation, a share price's move and a before-and-after comparison, and it is what most people mean when they ask how much something changed.

Alongside it you get two other numbers. The absolute change is the plain difference in the original units, which keeps a large percentage on a tiny base in proportion: going from 2 to 5 is a 150% rise, but only 3 units. The percent difference compares the two values against their average instead of against either one, so it gives the same answer whichever value you enter first; it suits two measurements of equal standing, such as readings from two instruments, where calling one of them the starting point would be arbitrary. For 100 and 150 it is 40%.

Two inputs need care. A starting value of zero has no percentage change at all, because every increase from nothing is infinitely large relative to it, so the calculation is refused rather than shown as infinity. A negative starting value is measured by its size, without its sign, so the sign of the answer still shows which way the value moved: a loss narrowing from −10 to −5 is a rise of 5 on a base of 10, +50%. The FAQs below cover what that convention means for losses, debts and values that cross zero.

What is percentage change calculator?

Percentage change, also called relative change, expresses how far a quantity moved as a share of where it started: (new − old) ÷ |old| × 100, where |old| is the old value without its sign. It is a ratio of two values in the same units, so the units cancel and the result is a pure number of percent. Because the old value is the yardstick, the measure is deliberately one-directional: swapping the two values changes both the size and the sign of the answer. That is what makes percentage changes chain correctly over time, since each period's change is measured against the value at the start of that period. It is not the same as a change in percentage points, which is a plain subtraction between two percentages, and it is not the same as percent difference, a symmetric comparison that divides by the average of the two values and has no direction. Statistical agencies use the same calculation for inflation: the 12-month change in a price index is this month’s index minus the index a year earlier, divided by the earlier index, so an index that moves from 300 to 312 shows 4% inflation.

How to use this calculator.

1. 01Enter the earlier or reference value as the initial value: last year's revenue, the price before the change, the reading before treatment. It cannot be zero.
2. 02Enter the later or comparison value as the final value. Either value may be negative and the range runs to a billion either way.
3. 03For year-on-year comparisons, pair the same month or quarter in both years. Comparing December with the following March mixes the change you care about with ordinary seasonal swings.
4. 04Read the percent change, shown to two decimal places. A plus sign means the value went up and a minus sign that it went down, even when the starting value is negative, because the change is measured against the starting value's size.
5. 05Check the absolute change alongside it, in the same units as your inputs. When the starting value is small, the percentage can look dramatic while the actual movement is modest.
6. 06Use the percent difference instead when neither value is the natural starting point, such as two labs measuring the same sample. It does not change when you swap the values.

The formula.

Percent change = (F − I) ÷ |I| × 100, where I is the initial value, F the final value and |I| the initial value without its sign. For the usual case of a positive starting value this is (F ÷ I − 1) × 100, which shows what it really measures: the multiplier that turns the old value into the new one, minus one. A change of +50% means the value was multiplied by 1.5; −20% means it was multiplied by 0.8. That is why successive changes combine by multiplying their multipliers, not by adding percentages.

The absolute change is simply F − I, in the inputs' own units.

Analysts who want a signed measure in which rises and falls cancel use the log change, ln(F ÷ I), instead. From 100 to 150 it is +0.4055 and from 150 back to 100 it is −0.4055, so the round trip sums to zero, which plain percentages never do. It is not shown here, but it explains why finance and economics papers often quote log returns.

Percent difference = |F − I| ÷ ((|I| + |F|) ÷ 2) × 100. Its denominator is the average size of the two values, so swapping them changes nothing and the result is never negative. For two positive values it relates to the percent change r (as a fraction) by D = |r| ÷ (1 + r/2): a 10% change is a 9.52% difference, a 100% change a 66.67% difference and a 200% change a 100% difference. The two measures agree closely for small changes and drift apart as changes grow. Percent difference also has a ceiling: it reaches 200% whenever one value is zero or the two have opposite signs.

The initial value divides everything, so zero is refused, and with it the whole result, percent difference included. Dividing by its size rather than its signed value keeps the sign of the answer tied to direction when the starting point is negative: −200 to −100 is a rise of 100 on a base of 200, +50%, and −10 to −20 is a fall of 10 on a base of 10, −100%. The multiplier reading does not carry over to a negative start, though: −10 to −5 halves the number's size and shows +50%. Results are shown to two decimal places for the percentages and up to four for the absolute change.

A worked example.

Frequently asked questions.

References& sources.

1. [1]NIST/SEMATECH e-Handbook of Statistical Methods — section on relative-difference measures including percent change and the symmetric percent-difference formula used in inter-method comparisons.
2. [2]U.S. Bureau of Labor Statistics — Handbook of Methods, chapter on percent-change methodology used in CPI, PPI, and employment reporting.
3. [3]Khan Academy — Percent word problems, percent change, and reverse percentages: pedagogical reference for the canonical relative-change formula.
4. [4]Stewart, J. (2015). Calculus: Early Transcendentals, 8th edition. Cengage Learning. Section 2.7 on rates of change and the definition of relative change as the limiting form of the percent-change ratio.
5. [5]ISO 80000-2:2019 — Quantities and units — Part 2: Mathematics. Specifies notation for dimensionless ratios including percent and the conventions for expressing relative differences.
6. [6]NIST Guide to the SI, Special Publication 811, §7.10.2 — guidance on expressing relative differences as dimensionless percentages.