Work with Percentages and Percentage Change
You can now pick an average that tells the truth about your data. This chapter takes on the quantity people use most and misread most: the percentage. Percentages feel familiar, we see them on receipts, weather forecasts, and news headlines, yet the same familiarity hides a trap. A percentage is only meaningful once you know what it is a percentage of, and a change measured from the wrong starting point can turn real growth into a shrinking figure. Your job here is to compute percentages and changes correctly, and to never lose track of what your number is measured against.
Why a Percentage Needs a Second Look
Imagine a friend tells you their savings “went up 50%” last month. It sounds impressive, until you learn they started with 10 euros and now have 15. The percentage was real, but without knowing the starting amount it told you almost nothing useful. Percentages compress two numbers into one, and in doing so they quietly drop the context that gives them meaning.
That is the whole challenge of this chapter. The arithmetic of percentages is simple; the discipline is remembering what sits underneath. Get into the habit of asking “percentage of what?” every single time, and you will avoid the most common numeric mistake there is.
What a Percentage and Its Base Really Are
Start with the two words that anchor everything else.
Suppose you walked 12,000 steps today and 3,000 of them were before breakfast. The part is 3,000, the base is 12,000, so the morning share is 3,000 ÷ 12,000 × 100 = 25%. Change the base and the same part tells a different story: 3,000 steps out of a 6,000-step day would be 50%. The part did not move; only the base did. That is why naming the base is never optional.
Measure Change Over Time
Most interesting questions are not about a single moment but about movement: did it grow, did it shrink, by how much? That calls for a percentage change.
The starting value is the base here, and choosing it correctly is the whole game. If you ran 40 km one month and 50 km the next, the change is (50 - 40) ÷ 40 × 100 = 25%. The base is 40, the distance you started from, because you are measuring growth from that point. Read the formula slowly and you will see the base is always the earlier, reference value, never the value you ended on.
Always Ask: Percentage of What?
Here is the mistake that catches almost everyone at least once: dividing by the wrong base. It is easy to reach for whichever number is closest to hand, but the base has to be the reference value, not the new one.
Both numbers are right there in front of me, so how do I know which one is the base?
Ask what the change is measured from. Growth is measured from where you began, so the base is the starting value. In the running example, dividing by the new distance instead gives (50 - 40) ÷ 50 × 100 = 20%, which quietly understates real growth of 25%. The arithmetic is flawless; the base is simply wrong.
Tell Percentage Points from Percentage Change
One more distinction saves you from a trap that hides inside percentages themselves. When the thing you are tracking is already a percentage, there are two different ways it can move, and mixing them up produces confident nonsense.
So the same movement, 20% up to 24%, is honestly described two ways: “up 4 percentage points” or “up 20% relative to before.” Both are true, but they are not interchangeable, and saying “up 4%” when you mean 4 points is simply wrong.
See It in Action: A Month of Training
You keep a simple running log. In April you ran 40 km; in May you ran 50 km. Let’s work through the numbers such a month raises, naming the base at every step.
First, a plain percentage. Of your 50 km in May, 12 km were long runs. What share is that? The part is 12 and the base is May’s total, 50, so 12 ÷ 50 × 100 = 24%. Long runs made up 24% of May’s distance.
Next, the percentage change in total distance from April to May. The new value is 50, the old value is 40, and the base is April, the month you started from: (50 - 40) ÷ 40 × 100 = 25%. Your training volume grew by 25%.
Finally, watch a percentage move. In April, long runs were 20% of your distance (8 km of 40); in May, they were 24% (12 km of 50). That share rose by 4 percentage points. Measured as a percentage change relative to April’s 20%, it rose (24 - 20) ÷ 20 × 100 = 20%. Notice you now have three honest numbers from one month, 25% more distance, a 4 percentage-point rise in the long-run share, and a 20% relative rise in that share, and each answers a different question. Keeping them straight is exactly the skill this chapter builds.
Your Turn!
Your phone logs how many steps you walk each month. Here is what it recorded.
January: 180,000 steps.
February: 216,000 steps.
It also tells you what share of your steps fell on weekends: 25% in January and 31% in February.
Do the following, writing your answers down before you check:
In February, 31% of your steps were on weekends. How many steps is that?
Compute the percentage change in total steps from January to February. Say clearly which value is the base before you divide.
A friend sees the weekend share rise from 25% to 31% and says “your weekend walking went up 6%.” Is that the right way to describe it? Explain using percentage points and percentage change.
Then check your answer against the model solution found at the end of this chapter.
Let’s Recap!
A percentage expresses a part out of 100, and it is meaningless until you know its base, the whole it is measured against.
To compute a percentage, divide the part by the base and multiply by 100.
A percentage change measures growth or shrinkage from a starting value, which is always the base: new minus old, divided by old, times 100.
Dividing by the wrong base is the most common percentage error, so name the base out loud before you divide.
A percentage point is the gap between two percentages, which is not the same as the percentage change relative to the starting one.
With totals, shares, honest averages, and percentages all in hand, you have met every core single-dataset quantity Part 1 set out to teach. A short Part 1 quiz comes next, your chance to confirm the whole set before Part 2 turns to comparing groups fairly, reading what your results mean, and building the correctness-check habit.
Model Solution
The part is the weekend steps and the base is February’s total, 216,000: 31% of 216,000 is 0.31 × 216,000 = 66,960 steps.
The new value is 216,000 and the old value is 180,000, so January is the base: (216,000 - 180,000) ÷ 180,000 × 100 = 36,000 ÷ 180,000 × 100 = 20% increase. Dividing by February’s 216,000 instead would give about 16.7%, the wrong-base error, because you grew from January, not from February.
“Up 6%” is not the right way to say it. The weekend share rose from 25% to 31%, a rise of 6 percentage points. As a percentage change relative to the January share, it rose (31 - 25) ÷ 25 × 100 = 24%. So the honest descriptions are “up 6 percentage points” or “up 24% relative to January,” and “up 6%” confuses the point gap with a percentage change.
