Percent Change

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Percent Change Calculation:

  •  i =

    Starting value that will be changing

  •  f =

    After change value for comparison

  •  c =

    What is the change between the two values as a percentage of the initial value

  •  d =

    Increase or Decrease?

  •

    Choose the number of decimals to show in your answer. This is also known as significant figures. Select an appropriate amount of significant figures based on the precision of the input numbers.

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Percent Change Formula

Percent change, denoted here as PΔ, is the change between a known initial value and a known final value as a percent of the initial value.

PΔ = ((F - I) / | I |) * 100

The bars "|" in the formula above mean to use the absolute value of that variable. If the variable is negative remove the negative sign to get the absolute value. This is necessary to get the correct change direction, or sign on the final percent, when calculating the percent change of negative values.

Percent Change Variables

Calculating percent change requires and initial and final value, these can also be thought of as the starting and ending values of any measurement that is changing. Values in just about any units can be compared as a percent but both the initial and final values must be measured in the same units.

Percent Change (PΔ)
Change as a percentage of the initial value. Symbol Delta (Δ) is shorthand for change. A possitive result will indicat and increase and a negative result will indicate a decrease.
Initial Value (I)
Original starting value before changing.
Final Value (F)
Value after change. The value being compared to the original value.
Absolute Initial Value (| I |)
Original starting value with any negative signs removed, if I is not negative no change is required.

Percent Change Solution

What is the percent change (PΔ) from an initial value of 50 to a final value of 60?

  • PΔ = ((F - I) / | I |) * 100
  • substitute in the known values
  • PΔ = ((60 - 50) / | 50 |) * 100
  • Initial value of 50 and final value of 60 are entered into the formula
  • PΔ = (10 / | 50 |) * 100
  • Find the difference between the final value and initial value, 10 in this case
  • PΔ = (10 / 50 ) * 100
  • Take the absolute value of the initial value in the denominator. Since it is already unsigned, or not negative, nothing needs to be done
  • PΔ = 0.2 * 100
  • Divided the difference between the final and initial value by the absolute initial value to get the change as a ratio of the initial value
  • PΔ = 20 %
  • Multiplied by 100 to express the change in percent
  • 20%

SOLUTION CONCLUSION

Percent Change Example 5 to 7

Another example is below for when the initial value is 5 and the final value is 7.

  • PΔ = ((F - I) / | I |) * 100
  • PΔ = ((7 - 5) / | 5 |) * 100
  • PΔ = (2 / 5) * 100
  • PΔ = 0.4 * 100
  • 40 %

Changing from a value of 5 to a value of 7 units is a 40% increase. This can be confirmed because 40% of 5 is 2 and 5 + 2 = 7. The answer is output unsigned, not negative, so the change is an increase.

Percent Change Example 80 to 50

Another example is below for when the initial value is 80 and the final value is 50. In the example the final value is decreasing by some percent of the initial value.

  • PΔ = ((F - I) / | I |) * 100
  • PΔ = ((50 - 80) / | 80 |) * 100
  • PΔ = (-30 / 80) * 100
  • PΔ = -0.375 * 100
  • -37.5 %

Changing from a value of 80 to a value of 50 units is a 37.5% decrease. When the final value is less than the initial value the numerator is negative resulting in a negative percent change, which is a descrease.

Percent Change Example -13 to -26

Another example is below for when the initial value is -13 and the final value is -26. This is an example of what happens when both values are negative and also on taking the absolute value of the initial value.

  • PΔ = ((F - I) / | I |) * 100
  • PΔ = (((-26) - (-13)) / | (-13) |) * 100
  • PΔ = (-13 / 13) * 100
  • PΔ = -1 * 100
  • -100 %

Changing from a value of -13 to a value of -26 units is a 100% descrease. The final value is double to initial value in the negative direction. This demonstrates the subtraction of a negative from a negative, recall that two negatives make a possitive, and demonstrates taking the absolute value of a negative.

Percent Change U.S. Monthly Oil Production Example

According to data from the U.S. Energy Information Administration1 oil production in January of 2020 averaged 12,785 thousand barrels per day and one year late, in January of 2021, production averaged 11,056 thousand barrels per day. What is the year-over-year percent change in U.S. monthly oil production from January 2020 to 2021?

  • PΔ = ((F - I) / | I |) * 100
  • Let initial production of 12,785 be I and final production of 11,056 be F
  • PΔ = ((11056 - 12785) / | 12785 |) * 100
  • The values have been substituted into the formula
  • PΔ = (-1729 / | 12785 |) * 100
  • Calculate the difference between the final value (F) and initial value (I)
  • PΔ = (-1729 / 12785) * 100
  • Take the absolute value of the final value, since it is possitive number there is no change
  • PΔ = (-0.13523660539695) * 100
  • Multiply the decimal by 100 to express the change as a ratio of the original value out of one hundred
  • -13.524 %
  • The answer has been rounded

The year-over-year change in oil production was -13.524 %. Fewer barrels of oil were produced in January of 2021 compared to the same month in 2020 resulting in a negetive percent change, indicating the decrease. The absolute change in production was -1,729 thousand barrels per day, which is useful information but does not indicate if this is a large change relative to overall producrtion. Percent change can be a useful tool for quick comparisons because it indicates if the change is large, or small, relative to the initial value.

1"U.S. Field Production of Crude Oil.", U.S. Energy Information Administration, 30 Dec. 2021, https://www.eia.gov/dnav/pet/hist/LeafHandler.ashx?n=PET&s=MCRFPUS2&f=M

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