The median, read at half the total
200 people take a quiz marked out of 60. The cumulative frequencies say how many scored less than each multiple of 10: 12 people scored less than 10, 48 less than 20, 104 less than 30, 156 less than 40, 184 less than 50, and all 200 less than 60.
The median is the score with half of the people below it. Half of 200 is 100, so go up the cumulative frequency axis to 100, across to the curve and down to the score axis. The reading is about 29. The arithmetic behind it: 100 lies between 48 and 104, which are 56 apart, and it is 52 of those 56 above 48, so the curve reaches it of the way from 20 to 30, at about 29.3.
Across at 100, half of 200, then down to a score of about 29: the median.
Percentiles cut the count into hundredths
The median splits the people into two halves and the quartiles split them into quarters. Percentiles split them into hundredths. The kth percentile is the value with k% of the data below it, so it is read across at k% of the total.
For the 90th percentile, 90% of 200 is 0.9 × 200 = 180 people. Read across at 180 and down: the 90th percentile is a score of about 49. The height 180 lies between 156 and 184, and it is 24 of the 28 steps between them, so the reading is of the way from 40 to 50, at about 48.6. So 90% of the people scored less than about 49.
Across at 180, which is 90% of 200, then down to a score of about 49: the 90th percentile.
What “the top 10%” means as a count
Suppose the people in the top 10% of the quiz win a prize. 10% of 200 is 20 people, so the prize goes to the 20 highest scores. The other 200 − 20 = 180 people are below them, so the boundary is where the curve reaches 180: the 90th percentile, a score of about 49. A person is in the top 10% when they score more than about 49.
The curve always counts from the bottom, so a question about the top must be turned into a count from the bottom first. Going in at 20 instead of 180 gives a score of about 12, the 10th percentile: the score that the lowest 20 people are below.
Quartiles are percentiles
A quarter of 200 is 50, and 25% of 200 is also 50, so the lower quartile and the 25th percentile are read at the same height and are the same score, about 20. In the same way, the median is the 50th percentile, read at 100, and the upper quartile is the 75th percentile, read at 75% of 200 = 150, a score of about 39.
So the lower quartile is the 25th percentile, the median is the 50th percentile and the upper quartile is the 75th percentile. The middle half of the people, between the 25th and the 75th percentiles, scored from about 20 to about 39.
Across at 50, a quarter of 200 and 25% of 200 alike, then down to about 20: the lower quartile, which is the 25th percentile.
Across at 150, three quarters of 200, then down to about 39: the upper quartile, which is the 75th percentile.
From a score to its percentile
The curve can also be read the other way, from a score to a percentile. One person scored 35. Go up from 35 on the score axis to the curve, then across to the cumulative frequency axis. The reading is 130: 35 is halfway through the class from 30 to 40, so the curve is halfway from 104 to 156 there, at 130.
So 130 of the 200 people scored less than 35. As a percentage, %, so a score of 35 is at the 65th percentile. This is its percentile rank: 65% of the people who took the quiz scored less.
A percentile is not a percentage of the marks. A score of 35 out of 60 is about 58% of the marks, but it is at the 65th percentile, because a percentile compares one person with everyone else who took the quiz, not with the highest possible score. A baby at the 75th percentile for weight on a growth chart is heavier than 75% of babies of the same age; it says nothing about the percentage of any target weight.
Up from a score of 35 to the curve, then across to 130. That is 130 of 200 people, or 65%, so 35 is at the 65th percentile.