# 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 |

### a) Estimate the 30th percentile (when 30% of the visitors had arrived).

### 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, so 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**.