# Percentile Calculator

Welcome to **the percentile calculator**, where we'll go through the topic of **finding percentiles** together. Even if we don't know much about the data set, the percentile in statistics is a useful tool to get a general idea of what we're dealing with. It tells us roughly **how many entries are below and how many are above a certain point**. Moreover, the percentile formula is very simple, which makes the task of how to find a percentile extremely easy. Later on, we can even store that information in a neat percentile chart that proves quite useful during data analysis.

So pour yourself a nice cup of tea, and **let's see how to calculate percentiles**.

## Percentiles in statistics

Let's say that you're trying to **start a small business** that deals with small indoor renovations like room painting and wallpapering. Obviously, the idea is one thing, but you also have to decide on **the price of your services**. On the one hand, you need to make enough so that you make some money after taxes. On the other hand, **you can't be too expensive** as you wont get any customers. So, how do you set the right number?

It's a good strategy to decide **your price based on how it compares to similar businesses**. If you'd like only thirty percent of your competition to do it cheaper than you, you'll there's plenty of people that are renovating their houses so **it should attract quite a few offers**, and would still provide some profit.

Once you do some research, look around on social media, and make a table of what prices your competitors offer, **you want exactly the 30th percentile of that data set**. The *30*, as you might have guessed, corresponds to **the percent of values that are below that number**.

In general, when we have a data set of numbers, **the n-th percentile** in statistics is the value which is larger than at least `n%`

of all the entries. Or, equivalently, smaller than `(100 - n)%`

of them at most. As such, the number `n`

should be an integer between `1`

and `100`

.

Other than economic uses, percentiles often appear in medicine and health-based statistics. **If you have a newborn**, you will measure their height, weight, and BMI percentiles to see if they are developing normally. Feel free to check out some of our calculators that deal with such topics:

- Height percentile calculator;
- Weight percentile calculator;
- BMI percentile calculator; and
- Head circumference percentile calculator.

All of these calculators are based on **long, meticulous studies** that connect each percentile to statistics gathered for many years. These were later mashed together into **handy percentile charts** for each parameter. And how do we read those *percentile charts*? Well, let's waste not a second longer and find out!

## Percentile chart

It's easiest if we look at **an example of such a percentile chart** and explain how things work using it.

Here's a **BMI percentile chart for boys from two to twenty years old** (source: Centers for Disease Control and Prevention):

On the sides, we have **the BMI scale**, and on the bottom there's **age**. The curves, from the bottom up, are the 3rd, 10th, 25th, 50th, 75th, 85th, 90th, 95th, and 97th percentiles (the number is written on the right side along the curve).

Now, let's take a closer look at some specific values to see how the percentile chart works. First, let's say that **our kid is** `12`

**years old and his BMI is** `21`

. We find the `12`

on the bottom axis, `21`

on the side axis, and check where they cross. We can see that, in fact, **it's precisely on one of the curves** - the one with number `85`

. This means that our son's BMI corresponds to the 85th percentile. This, in turn, suggests that **85% of all 12-year-old boys have a lower BMI than him**. Well, maybe we should reconsider those after school swimming classes?

Next, let's try to use the percentile chart **the other way round**. Now that we know our son's current state, let's say that you'd like him to **get down to the 75th percentile** by the age of fourteen. In other words, we'd like to see what his BMI should be by then.

Firstly, **we find the 75th percentile curve**. Then, we find the age (in our case, it's `14`

) on the bottom scale and see where the two cross. Lastly, we read out the BMI number on the horizontal line that goes through the crossing point. This tells us that **we'd like the kid to be at** a BMI of `21.2`

. Arguably, it's very close to the `21`

from before, but we have to keep in mind that most probably he'll grow a few inches in the next two years.

We hope that this example convinced you that learning how to find percentiles is worth your time. Still, knowing what they are is one thing, but seeing **how to calculate the percentiles** is also a handy skill.

## Percentile formula

Suppose that **we want to calculate the** `k`

**th percentile** of a sequence of numbers `a₁`

, `a₂`

, `a₃`

,..., `aₙ`

, and, for simplicity, **we assume that they are ordered** from the smallest to the largest (otherwise, we'd have to order them before moving to the next step).

Finding percentiles will require **a few intermediate calculations** before we see the final percentile formula. They are:

**Percentile rank**- Describes where (in terms of the entries of the data set) we are with the percentile.

`rank = (k / 100) * (n + 1)`

;

- The
**integer part**of the rank - The largest integer not larger than the rank, i.e., the rank without its decimals.

`integer_part = ⌊rank⌋`

;

- The
**fraction part**of the rank - Whatever is left of the rank without its integer part.

`fraction_part = rank - integer_part`

.

With that, we have everything we need for **the percentile formula**. For simplicity of notation, let's denote `m = integer_part`

. Then:

`kth_percentile = aₘ + fraction_part * (aₘ₊₁ - aₘ)`

.

**That wasn't so bad now, was it?** Only a few equations, no need for any quadratic formulas or some pesky roots. A piece of cake if you ask us.

This concludes the explanation on how to find percentiles. We took enough of your time with theory - it's now time to **get down to practice**.

## Example: using the percentile calculator

New year, new you! You decide to **change a few things about your lifestyle** and generally make it healthier and more active. On one hand, you choose to cut down on meat and eat more fruit and vegetables. On the other hand, you want to move around a little more and take up jogging.

The first few weeks were quite promising - **you managed to stick to your plan** and you already feel a lot better! You can even see that in the distance you can run before you feel tired and want to take a nap. It even encouraged you to begin writing down how much you jog each time on a piece of paper and then analyze your improvement.

The distances you wrote in the spreadsheet are:

`1.9 mi`

, `1.7 mi`

, `2 mi`

, `2.3 mi`

, `1.8 mi`

, `2 mi`

, `2.4 mi`

, `2.1 mi`

, `2.4 mi`

.

But then you had a couple of weeks with a difficult project at work with too much overtime, and there was this great show on TV that glued you to the couch for many hours. As a result, **the next few runs were not as impressive** as the previous ones:

`0.9 mi`

, `1.2 mi`

, `1 mi`

, `1.7 mi`

, `1.3 mi`

, `1.3 mi`

.

**But you're not giving up!** You're determined keep up the good work and, as the first step, you decide to reach **the 60th percentile** of the values before they went down. However, just to be realistic about your current stamina, you decided that before you get there, the next few runs should be around **the 60th percentile of all the above values**.

**So, what is the distance you should jog the next time you go to the park? And what is the goal you want to reach in the not-so-distant future?** With our percentile calculator, we'll know how to find the percentiles in no time!

First of all, let's see **how to calculate the percentile of the smaller set** of numbers using the percentile calculator. Observe that there are only eight fields to write the numbers. But don't you worry - once you fill all of them, **more will appear**!

**We input the nine distances** one by one as entries #1 to #9 into the percentile calculator. Lastly, at the bottom, **we need give the percentile** we want to find (in our case, it's `60`

). Once we write that, **the answer will pop up** under the variable fields. That is **the goal distance** that we want to get to soon.

Next, we'd like to see how far we need to run the next time we go jogging. Note, that we can find it by simply **adding the six extra distances** onto what we just input into the percentile calculator. So, we leave what's already there be and simply add entries #10 to #15. **Observe how the percentile changes with every number you give** (as long as you still have the `60`

in the *Percentile* field from before). And, the moment we input the last value, **we get our final answer** at the bottom.

Now, just to be sure, let's see how to calculate the percentiles ourselves to see **how much time the Omni Calculator website can save us on a daily basis**. So sit back at your desk and let's get it over with!

We begin with the calculations **for the smaller data set**. The first thing we need to do when finding percentiles is to **order the entries from smallest to largest**. In our case, this changes the sequence from:

`1.9 mi`

, `1.7 mi`

, `2 mi`

, `2.3 mi`

, `1.8 mi`

, `2 mi`

, `2.4 mi`

, `2.1 mi`

, `2.4 mi`

to:

`1.7 mi`

, `1.8 mi`

, `1.9 mi`

, `2 mi`

, `2 mi`

, `2.1 mi`

, `2.3 mi`

, `2.4 mi`

, `2.4 mi`

.

Now, we follow the steps from the *Percentile formula* section. This means that **we first calculate the rank**:

`rank = (k / 100) * (n + 1) = (60 / 100) * (9 + 1) = 0.6 * 10 = 6`

,

where `k`

is the percentile and `n`

is the number of entries, which, in our case, is `k = 60`

and `n = 9`

.

Next, we find **the integer part** and **the fraction part**:

`integer_part = ⌊rank⌋ = ⌊6⌋ = 6`

;

`fraction_part = rank - integer_part = 6 - 6 = 0`

.

According to the percentile formula:

`kth_percentile = aₘ + fraction_part * (aₘ₊₁ - aₘ)`

,

this gives:

`60th_percentile = a₆ + fraction_part * (a₇ - a₆) = 2.1 mi + 0 * (2.3 mi - 2.1 mi) = 2.1 mi`

.

So our goal distance is `2.1 mi`

. Well, looking at the last few numbers, **we're not really there yet**... Let's check out what we should start with.

Now, we **do the same calculations but for a different, larger data set**. Again, we begin with ordering it, so:

`1.9 mi`

, `1.7 mi`

, `2 mi`

, `2.3 mi`

, `1.8 mi`

, `2 mi`

, `2.4 mi`

, `2.1 mi`

, `2.4 mi`

, `0.9 mi`

, `1.2 mi`

, `1 mi`

, `1.7 mi`

, `1.3 mi`

, `1.3 mi`

becomes:

`0.9 mi`

, `1 mi`

, `1.2 mi`

, `1.3 mi`

, `1.3 mi`

, `1.7 mi`

, `1.7 mi`

, `1.8 mi`

, `1.9 mi`

, `2 mi`

, `2 mi`

, `2.1 mi`

, `2.3 mi`

, `2.4 mi`

, `2.4 mi`

.

Next, we find **the rank**, **the integer part**, and **the fraction part**:

`rank = (60 / 100) * (15 + 1) = 0.6 * 16 = 9.6`

;

`integer_part = ⌊9.6⌋ = 9`

;

`fraction_part = 9.6 - 9 = 0.6`

.

Lastly, we input the numbers into **the percentile formula**:

`60th_percentile = a₉ + fraction_part * (a₁₀ - a₉) = 1.9 mi + 0.6 * (2 mi - 1.9 mi) = `

`= 1.9 mi + 0.6 * 0.1 mi = 1.96 mi`

.

To sum it all up, **we start by running** `1.96`

**miles, but that's just a spring board to get us to** `2.1`

**miles each time**. Fair enough, seems like a good plan.

Encouraged and excited, you put on your running shoes and **go out in the direction of the city park**. You can already **feel the endorphins kicking in** and the calories being burned with each step. Sure enough, you'll have that beach body ready for summer in no time!