Pitfalls
Percentage change and percentage-point change differ.
When a variable is in percent terms, we should be careful which change is used:
For example, suppose the unemployment rate increased from 2.0% to 2.2%. The increase is 0.2 in absolute terms, and denominated in
percentage points. To compute the relative change, we divide the absolute change of 0.2 by the initial value of the unemployment rate which is 2.0. This gives us a relative change of 10%. Refer to
Absolute vs Relative Change for more information.
A smaller percentage increase does not mean that the variable has decreased.
Example 4: Working for free?
Car racers in Raceland earned $100 every month in 2010. They earned more in 2011, at $300, and this grew to $600 in 2012. An analyst said that in Raceland, car racers’ percentage wage rise dropped ‘by 100 per cent’ from the previous year. Does this mean that car racers earn nothing in Raceland in 2012?
No. Upon closer inspection, what the analyst meant was that the wage increase dropped from 200% to 100%, which is a difference of 100 (in percentage points, rather than per cent). Although the percentage wage change fell from 200% to 100%, the latter wage change is still positive, i.e. the monthly wage of car racers still increased from 2011 to 2012.
Monthly Wage of Car Racers
Bear in mind what the 100 percent is
Example 4 also shows us the need to keep in mind what the base or the 100 per cent refers to. For the percentage wage increase from 2010 to 2011, the base used is $100, the monthly wage received in 2010. As for the percentage wage increase from 2011 to 2012, the base used is the monthly wage of $300 in 2011 instead.
Percentages may not be additive
Not all percentages are additive. Only those with the same corresponding base value can be summed (like in a
distribution). Otherwise, it would not be meaningful to add up percentages of different bases. A specific example where bases differ is in calculating percentage change. A decrease of 50% and a subsequent 50% increase does not restore the variable to its original value. This is because a different base is used in deriving the percentage decrease and percentage increase. Refer to Examples 5 and 6.
Example 5: Percentages may not be additive
Suppose now that the wages for car racers in Raceland decreased by 50% from 2010 to 2011, and then increased by 50% in 2012. Recalling that the wage in 2010 is $100, does this imply that wages in 2012 are fully restored to its original value in 2010?
If your answer was yes, you probably thought that the total change in wages from 2010 to 2012 was
which implies no change in wages from 2010 to 2012.
However, this is incorrect.
A 50% decrease of $100 implies that wages dropped by $50. (The base here is $100.) Therefore, the wages of car racers in 2011 was $50. An increase of 50% in 2012 relative to the lower base of $50 corresponds to a rise in wages of only $25. As a result, the wage level in 2012 was $75, which is still less than the original wage of $100.
Monthly Wage of Car Racers Over Three Years
Example 6: Percentages may not be additive
Due to business restructuring, Company Megamind has decided to reduce the number of employees by 20% each year from 2011 to 2015. Assuming there are 2,000 workers initially, how many workers will there be at the end of 2015?
Similarly, the reduction in the number of employees for subsequent years can be calculated, with a different base each year.
(Reduction in 2011: base = 2,000.
Reduction in 2012: base = 2,000 – 400 = 1,600.
Reduction in 2013: base = 1,600 – 320 = 1,280 etc.)
Number of Megamind Employees, 2010 to 2015 (December)
Note that this 20% decrease is a year-on-year change, which is computed based on the number of workers in the preceding year, rather than at the start of the entire period (i.e. Dec 2010). If we kept the base at Dec 2010 headcount, each year’s decrease will be fixed at 20% of 2,000, or 400 workers.
