Percentage Difference Calculator

The calculator finds the percentage difference between two numbers. The percentage difference is used to compare two numbers when both describe the same thing – for example, the number of employees in a company.

It is important not to confuse percentage difference with percentage change! Percentage change is used when there is an old value and a new value; there is always a clear reference point in the percentage change calculations. On the other hand, the percentage difference is used when the two numbers are of “equal worth,” and it’s impossible to choose a reference number. Instead, the average of the two numbers is used as a reference point for percentage difference calculations.

Directions for use

To calculate the percentage difference, enter the known values into V₁ (value one) and V₂ (value two) fields, and press “Calculate.” The difference calculator only accepts positive integers or decimal numbers.

Definition

As mentioned above, the percentage difference is used to calculate the difference between two numbers when these two numbers are of equal worth. It is often confused with percentage change, and we will now explain the difference between these two operations.

Percentage change describes the change from the old value to the new value relative to the old value. It is calculated as the absolute difference between the two values divided by the old value. In percentage difference calculations, the values have equal worth. There is no old value or new value. Therefore, the reference point for percentage difference calculations is the average of the two values.

Formula

$$Percentage\ difference=\frac{|V_1-V_2 |}{\frac{(V_1+V_2)}{2}}×100$$

Or,

Percentage difference = 100 × |V₁ – V₂| / {(V₁ + V₂)/2}

Here, V₁ and V₂ are the two compared values, |V₁ – V₂| - is their absolute difference, and (V₁ + V₂)/2 – is the average of the two values. Basically, the percentage difference represents the sum of two percentage change values – percentage change from V₁ to the average of the two values, and percentage change from V₂ to the average of the two values.

Notice how the outcome of the calculation does not depend on which value you choose to be V₁ and which value you choose to be V₂.

Example

Let’s find the percentage difference between two numbers: 6 and 9. Using the percentage difference formula, we get the following:

Percentage difference = 100 × |V₁ – V₂| / {(V₁ + V₂)/2} = 100 × |6 - 9| / {(6 + 9)/2} = 100 × |-3| / {15/2} = 100 × 3 / 7.5 = 300 / 7.5 = 40%

The percentage difference between 6 and 9 is 40%. These 40% result from a 20% percentage change from 6 to 7.5 and a 20% percentage change from 7.5 to 9.

How the Percentage Difference Can Be Confusing

The percentage difference is a powerful tool for comparing two values in situations when it’s not clear which value can be taken as a reference point. But, sometimes, the percentage difference can be confusing. This happens when you use percentage difference to compare two values of very different orders of magnitude. In the example above, we established that the percentage difference between 6 and 9 is 40%. Let’s now calculate the percentage difference between 6 and 90:

Percentage difference = 100 × |V₁ – V₂| / {(V₁ + V₂)/2} = 100 × |6 - 90| / {(6 + 90)/2} = 100 × |-84| / {96/2} = 100 × 84 / 48 = 8400 / 48 = 175%

So far, everything seems to make sense – the absolute difference in numbers increased, and so did the percentage difference.

Now let’s look at the percentage difference between 6 and 900:

Percentage difference = 100 × |V₁ – V₂| / {(V₁ + V₂)/2} = 100 × |6 - 900| / {(6 + 900)/2} = 100 × |-894| / {906/2} = 100 × 894 / 453 = 89400 / 453 = 197.351%

Notice how even though the absolute difference in numbers increased by a whole order of magnitude, the percentage difference increased much less than the previous time. Now let’s look at 6 and 9000:

Percentage difference = 100 × |V₁ – V₂| / {(V₁ + V₂)/2} = 100 × |6 - 9000| / {(6 + 9000)/2} = 100 × |-8994| / {9006/2} = 100 × 8994 / 4503 = 899400 / 4503 = 199.734%

We see that the increase in percentage difference is even smaller, even though the absolute difference between the two numbers increased by another order of magnitude. This happens because V₁ and V₂ are now very far apart from each other, so far that adding or subtracting V₁ to/from V₂ doesn’t change much in the final ratio. Imagine you add 5 to 10 – that’s a significant relative increase. However, adding 5 to 1000000 wouldn’t really change much. Since both values find themselves in the numerator and the denominator of the percentage difference formula, the final result doesn’t convey the idea of how different the numbers are in reality.

Therefore, percentage difference should only be used when comparing values of the same magnitude or differing by one order of magnitude! Otherwise, the final result can be misleading.

Calculation example

You want to buy sneakers and compare the price of a pair of sneakers in two different shops. If a pair of sneakers costs $110 in the first shop and $120 in the second shop, what is the percentage difference in price?

Solution

First, let’s establish the given values.

V₁ = 110

V₂ = 120

Then, let’s calculate the percentage difference using the percentage difference formula:

Percentage difference = 100 × |V₁ – V₂| / {(V₁ + V₂)/2} = 100 × |110 - 120| / {(110 + 120)/2} = 100 × |-10| / {230/2} = 100 × 10 / 115 = 1000 / 115 = 8.69565% ≈ 8.7%

The percentage difference between the price of a pair of sneakers in the two shops is 8.7%.

Note that the percentage difference would be the same if you had visited the shops in a different order, i.e., if you would choose 120 as V₁ and 110 as V₂:

Percentage difference = 100 × |V₁ – V₂| / {(V₁ + V₂)/2} = 100 × |120 - 110| / {(120 + 110)/2} = 100 × |10| / {230/2} = 100 × 10 / 115 = 1000 / 115 = 8.69565% ≈ 8.7%