# Percentiles

Percentile: the value below which a percentage of data falls.

### Example: You are the fourth tallest person in a group of 20

80% of people are shorter than you: That means you are at the 80th percentile.

If your height is 1.85m then "1.85m" is the 80th percentile height in that group.

## In Order

Have the data in order, so you know which values are above and below.

• To calculate percentiles of height: have the data in height order (sorted by height).
• To calculate percentiles of age: have the data in age order.
• And so on.

## Grouped Data

When the data is grouped:

Add up all percentages below the score,
plus half the percentage at the score.

### Example: You Score a B!

In the test 12% got D, 50% got C, 30% got B and 8% got A You got a B, so add up

• all the 12% that got D,
• all the 50% that got C,
• half of the 30% that got B,

for a total percentile of 12% + 50% + 15% = 77%

In other words you did "as well or better than 77% of the class"

(Why take half of B? Because you shouldn't imagine you got the "Best B", or the "Worst B", just an average B.)

## Deciles

Deciles are similar to Percentiles (sounds like decimal and percentile together), as they split the data into 10% groups:

• The 1st decile is the 10th percentile (the value that divides the data so 10% is below it)
• The 2nd decile is the 20th percentile (the value that divides the data so 20% is below it)
• etc!

### Example: (continued) You are at the 8th decile (the 80th percentile).

## Quartiles

Another related idea is Quartiles, which splits the data into quarters:

### Example: 1, 3, 3, 4, 5, 6, 6, 7, 8, 8

The numbers are in order. Cut the list into quarters: In this case Quartile 2 is half way between 5 and 6:

Q2 = (5+6)/2 = 5.5

And the result is:

• Quartile 1 (Q1) = 3
• Quartile 2 (Q2) = 5.5
• Quartile 3 (Q3) = 7

The Quartiles also divide the data into divisions of 25%, so:

• Quartile 1 (Q1) can be called the 25th percentile
• Quartile 2 (Q2) can be called the 50th percentile
• Quartile 3 (Q3) can be called the 75th percentile

### Example: (continued)

For 1, 3, 3, 4, 5, 6, 6, 7, 8, 8:

• The 25th percentile = 3
• The 50th percentile = 5.5
• The 75th percentile = 7

## Estimating Percentiles

We can estimate percentiles from a line graph. ### Example: Shopping

A total of 10,000 people visited the shopping mall over 12 hours:

Time (hours) People
0 0
2 350
4 1100
6 2400
8 6500
10 8850
12 10,000

### b) Estimate what percentile of visitors had arrived after 11 hours.

First draw a line graph of the data: plot the points and join them with a smooth curve: a) The 30th percentile occurs when the visits reach 3,000.

Draw a line horizontally across from 3,000 until you hit the curve, then draw a line vertically downwards to read off the time on the horizontal axis: So the 30th percentile occurs after about 6.5 hours.

b) To estimate the percentile of visits after 11 hours: draw a line vertically up from 11 until you hit the curve, then draw a line horizontally across to read off the population on the vertical axis: So the visits at 11 hours were about 9,500, which is the 95th percentile.