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## Quartile Formula

Statistics is an important part of mathematics and is used extensively. The concept of quarters is a fundamental one in statistics. Let us learn more about quarters and the Quartile Formula.

## Quartile Formula

Quartile, as it sounds phonetically, is a statistical term that divides the data into four quarters. It basically divides the data points into a data set in 4 quarters on the number line. One thing we need to keep in mind is that data points can be random and we have to put those numbers in line first on the number line in ascending order and then divide them into quartiles. It is basically an extended version of the median. Median divides the data into two equal parts which quartiles divide it into four parts. Once we divide the data, the four quartiles will be:

- 1
^{st}quartile also known as the lower quartile basically separates the lowest 25% of data from the highest 75%. - 2
^{nd}quartile or the middle quartile also the same as the median it divides numbers into 2 equal parts. - 3
^{rd}quartile or the upper quartile separates the highest 25% of data from the lowest 75%.

Quartile divides a set of observations into 4 equal parts. The first quartile is the value in the middle of the first term and the median. The median is the second quartile. The middle value between the median and the last term is the third quartile. Mathematically, they are represented as follows,

When the set of observations are arranged in ascending order the quartiles are represented as,

- First Quartile(Q1)=((n+1)/4)
^{t}^{h}Term also known as the lower quartile. - The second quartile or the 50th percentile or the Median is given as: Second Quartile(Q2)=((n+1)/2)
^{t}^{h}Term - The third Quartile of the 75th Percentile (Q3) is given as: Third Quartile(Q3)=(3(n+1)/4)
^{t}^{h}Term also known as the upper quartile. - The interquartile range is calculated as: Upper Quartile – Lower Quartile.

## Solved Example for Quartile Formula

**Question:** Find the median, lower quartile, upper quartile and interquartile range of the following data set of values: 19, 21, 23, 20, 23, 27, 25, 24, 31?

**Solution: **Firstly arrange the values in ascending order.

Plugging in the values in the formulas above we get,

Median(Q2)=5^{th }Term=23

Lower Quartile (Q1) = 2.5^{th }Term = 11

Upper Quartile(Q3) = 7.5^{th} Term = 24.5

IQR=Upper Quartile−Lower Quartile

IQR = 24.5 – 11

IQR = 13.5

**Question**: Find the upper quartile for the following set of numbers:

27, 19, 5, 7, 6, 9, 15, 12, 18, 2, 1.

**Solution: **The upper quartile formula is: Q3 = ¾(n + 1)th Term.

The formula doesn’t give you the value for the upper quartile, it gives you the place. For example, 5th place, 8th place etc.

So firstly we put your numbers in order: 1, 2, 5, 6, 7, 9, 12, 15, 18, 19, 27. Note that for very large data sets, you may want to use Excel to place your numbers in order. And then we work the formula. There are 11 numbers in the set, so:

Q3 = ¾(n + 1)th Term.

Q3 = ¾(12)th Term. = 9th Term.

In this set of numbers given, the upper quartile (18) is the 9th term or the 9th place from the left.

## Quartile Formula Definition

A quartile is a statistical term that splits data into quarters or four defined intervals. It basically divides the data points into a data set in 4 quarters on the number line. One thing we need to keep in mind is that data points can be random and we have to put those numbers in line first on the number line in ascending order and then divide them into quartiles. It is basically an extended version of the median. Median divides the data into two equal parts and quartiles divide it into four parts. Once we divide the data, the four quartiles will be:

- 1
^{st}quartile or lower quartile basically separates the lowest 25% of data from the highest 75%. - 2
^{nd}quartile or middle quartile is also the same as the median. It divides numbers into 2 equal parts. - 3
^{rd}quartile or the upper quartile separates the highest 25% of data from the lowest 75%.

Formula For Quartile:

Let’s say that we have a data set with N data points:

X – {X1, X2, X3……….. XN}

The formula for quartiles is given by:

What it basically means is that in a data set with N data points:

((N+1) * 1 / 4)^{th} term is the lower quartile

((N+1) * 2 / 4)^{th} term is the middle quartile

((N+1) * 3 / 4)^{th} term is the upper quartile

**Examples of Quartile Formula (With Excel Template)**

Let’s take an example to understand the calculation of Quartile in a better manner.

#### Quartile Formula – Example #1

**Let’s say we have a data set A which contains 19 data points. Calculate the Quartile for data set A.**

**Data Set:**

First of all, you have to arrange this in an ascending order i.e. from lowest to highest:

Number of data points is calculated as:

Quartile is calculated using the below formula

**Lower Quartile (Q1) = (N+1) * 1 / 4**

- Lower Quartile (Q1) = (19+1) * 1/4
- Lower Quartile (Q1) = 20 / 4 = 5
^{th}data point

So Lower Quartile (Q1) = **29**

**Middle Quartile (Q2) = (N+1) * 2 / 4**

- Middle Quartile (Q2) = (19+1) * 2/4
- Middle Quartile (Q2) = 40 / 4 = 10
^{th}data point

So Middle Quartile (Q2) = **43**

**Upper Quartile (Q3)= (N+1) * 3 / 4**

- Upper Quartile (Q3)= (19+1) * 3/4
- Upper Quartile (Q3)= 60 / 4 = 15
^{th}data point

So Upper Quartile (Q3)= **67**

Interquartile Range is calculated using the formula given below

**Interquartile Range = Q3 – Q1**

- Interquartile Range = 15– 5
- Interquartile Range = 10
^{th}data point

So Interquartile Range = **43**

If you see the data set, the median of this set is: (n+1)/2 = 20/2 = 10^{th} value i.e. 43, this is the same as Q2.

Inference:

- Value 29 divides the data set in such a way that the lowest 25% are above it and the highest 75% are below it
- Value 43 divides the data set into two equal parts
- Value 67 divides the data set in such a way that the highest 25% are below it and the lowest 75% are above it

#### Quartile Formula – Example #2

Let’s see another example of how companies and businesses can use this tool to make an informed decision on which product to produce.

Suppose you are a manufacturer of running shoes and are a well-known brand amongst the athletes who run a marathon, play sports, etc. You have collected the data of the shoe sizes these athletes wear so that in the future you produce more of that size to meet demand.

**You have collected a sample of 15 athletes from different sports.** **Calculate the Quartile.**

The data set is given below:

Arrange the shoe size in ascending order.

Quartile is calculated using below given formula

**Lower Quartile (Q1) = (N+1) * 1 / 4**

- Lower Quartile (Q1)= (15+1) * 1/4
- Lower Quartile (Q1)= 16 / 4 = 4
^{th}data point

So Lower Quartile (Q1)= **9**

**Middle Quartile (Q2) = (N+1) * 2 / 4**

- Middle Quartile (Q2) = (15+1) * 2/4
- Middle Quartile (Q2)= 32 / 4 = 8
^{th}data point

So Middle Quartile (Q2)= **10**

**Upper Quartile (Q3) = (N+1) * 3 / 4**

- Upper Quartile (Q3)= (15+1)*3/4
- Upper Quartile (Q3)= 48 / 4 = 12
^{th}data point

So Upper Quartile (Q3) = **11**

Interquartile Range is calculated using the formula given below

**Interquartile Range = Q3 – Q1**

- Interquartile Range = 12 – 4
- Interquartile Range = 8
^{th}data point

So Interquartile Range = **10**

## What Is Quartile Formula?

The quartile formula helps to divide a set of observations into 4 equal parts. The first quartile lies in the middle of the first term and the median. The median is the second quartile. The middle value lying between the median and the last term is the third quartile. Mathematically, they are represented as follows,

When the set of observations are arranged in ascending order the quartiles are represented as,

- First Quartile(Q1) = ((n + 1)/4)
^{t}^{h}Term - Second Quartile(Q2) = ((n + 1)/2)
^{t}^{h}Term - Third Quartile(Q3) = (3(n + 1)/4)
^{t}^{h}Term

1^{st} quartile is also known as the lower quartile. 2^{nd} quartile is the same as the median dividing data into 2 equal parts. 3^{rd} quartile is also called the upper quartile.

The interquartile range is calculated as Upper Quartile – Lower Quartile.

## Examples Using Quartile Formula

**Example 1: **Calculate the median, lower quartile, upper quartile, and interquartile range of the following data set of values: 20, 19, 21, 22, 23, 24, 25, 27, 26

**Solution:**

Arranging the values in ascending order: 19, 20, 21, 22, 23, 24, 25, 26, 27

Putting the values in the formulas above we get,

Median(Q2) = 5^{th }Term = 23

Lower Quartile (Q1) = Mean of 2^{nd} and 3^{rd }term = (20 + 21)/2 = 20.5

Upper Quartile(Q3) = Mean of 7^{th} and 8^{th }term = (25 + 26)/2 = 25.5

IQR = Upper Quartile−Lower Quartile

IQR = 25.5 – 20.5

IQR = 5

**Answer: IQR = 5**

**Example 2: ****What will be the upper quartile for the following set of numbers?26, 19, 5, 7, 6, 9, 16, 12, 18, 2, 1.**

**Solution:**

The formula for the upper quartile formula is Q3 = ¾(n + 1)^{th} Term.

The formula instead of giving the value for the upper quartile gives us the place. For example, 8^{th} place, 10^{th} place, etc.

So firstly we put your numbers in ascending order: 1, 2, 5, 6, 7, 9, 12, 16, 18, 19, 26. There are a total of 11 numbers, so:

Q3 = ¾(n + 1)^{th} Term.

Q3 = ¾(12)^{th} Term. = 9^{th} Term.

**Solution: The upper quartile (18) is the 9 ^{th} term or on the 9^{th} place from the left.**

**Example 3: Find the 3rd quartile in the following data set: 4, 5, 8, 7, 11, 9, 9**

**Solution: **

Let us first arrange our array in ascending order and it becomes 4, 5, 7, 8, 9, 9, 11

The median of our data is 8.

In order to find the 3rd quartile, we have to deal with the data points that are greater than the median that is 9, 9, 10.

In order to find the 3rd quartile, we have to find the median of the data points that are greater than the median that is 9, 9, 10.

**Answer:** **Hence, the 3rd quartile of our data set is 9.**

## FAQs on Quartile Formula

### What is Quartile Formula Used For?

The quartile formula is used to divide a set of observations into 4 equal parts. The first quartile lies in the middle of the first term and the median. The median is the second quartile. The middle value lying between the median and the last term is the third quartile.

### How to Find Q1, Q2, and Q3 using Quartile Formula?

When the set of observations are arranged in ascending order the quartiles are represented using the quartile formula as,

First Quartile(Q1) = ((n + 1)/4)^{th} Term

Second Quartile(Q2) = ((n + 1)/2)^{th} Term

Third Quartile(Q3) = (3(n + 1)/4)^{th} Term

### What Is Lower Quartile and Upper Quartile Formula?

1^{st} quartile is also known as the lower quartile. 2^{nd} quartile is the same as the median dividing data into 2 equal parts. 3^{rd} quartile is also called the upper quartile. We find the median of the given data points. The median of the set of the data values towards the left of this median forms the lower quartile and the median of the set of data points towards the right of this median forms the upper quartile. Using the quartile formula, the first quartile(Q1) = ((n + 1)/4)^{th} Term and third quartile(Q3) = (3(n + 1)/4)^{th} Term

### How Is Interquartile Range Calculated?

The interquartile range formula is Upper Quartile – Lower Quartile. 1^{st} quartile ((n + 1)/4)^{th} term is also known as the lower quartile and 3^{rd} quartile (3(n + 1)/4)^{th }term is also called the upper quartile. Upper quartile – lower quartile = interquartile range

## Find Quartiles: Examples

**Example:** Divide the following data set into quartiles: 2, 5, 6, 7, 10, 22, 13, 14, 16, 65, 45, 12.

Step 1: Put the numbers in order: 2, 5, 6, 7, 10, 12 13, 14, 16, 22, 45, 65.

Step 2: Count how many numbers there are in your set and then divide by 4 to cut the list of numbers into quarters. There are 12 numbers in this set, so you would have 3 numbers in each quartile.

2, 5, 6, | 7, 10, 12 | 13, 14, 16, | 22, 45, 65

If you have an uneven set of numbers, it’s OK to slice a number down the middle. This *can* get a little tricky (imagine trying to divide 10, 13, 17, 19, 21 into quarters!), so you may want to use an online interquartile range calculator to figure those quartiles out for you. The calculator gives you the 25th Percentile, which is the end of the first quartile, the 50th Percentile which is the end of the second quartile (or the median) and the 75th Percentile, which is the end of the third quartile. For 10, 13, 17, 19 and 21 the results are:

25th Percentile: 11.5

50th Percentile: 17

75th Percentile: 20

Interquartile Range: 8.5.

Why do we need quartiles in statistics? The main reason is to **perform further calculations**, like the interquartile range, which is a measure of how the data is spread out around the mean.

## What is an Upper Quartile?

The upper quartile (sometimes called Q3) is the number dividing the third and fourth quartile. The upper quartile can also be thought of as the median of the upper half of the numbers. The upper quartile is also called the 75th percentile; it splits the lowest 75% of data from the highest 25%.

## Calculating the Upper Quartile

You can find the upper quartile by placing a set of numbers in order and working out Q3 by hand, or you can use the upper quartile formula. If you have a small set of numbers (under about 20), by hand is usually the easiest option. However, the formula works for all sets of numbers, from very small to very large. You may also want to use the formula if you are uncomfortable with finding the median for sets of data with odd or even numbers.

Example question: Find the upper quartile for the following set of numbers:

27, 19, 5, 7, 6, 9, 15, 12, 18, 2, 1.

### By Hand

Step 1: Put your numbers in order: 1, 2, 5, 6, 7, 9, 12, 15, 18, 19, 27

Step 2: Find the median: 1, 2, 5, 6, 7, 9, 12, 15, 18, 19, 27.

Step 3: Place parentheses around the numbers above the median.

1, 2, 5, 6, 7, 9, (12, 15, 18, 19, 27).

Step 4: Find the median of the upper set of numbers. This is the upper quartile:

1, 2, 5, 6, 7, 9, (12, 15, 18 ,19 ,27).

### Using the Formula

The upper quartile formula is:

Q_{3} = ¾(n + 1)^{th} Term.

The formula doesn’t give you the value for the upper quartile, it gives you the *place*. For example, the 5th place, or the 76th place.

Step 1: Put your numbers in order: 1, 2, 5, 6, 7, 9, 12, 15, 18, 19, 27.*Note*: for very large data sets, you may want to use Excel to place your numbers in order. See: Sorting Numbers in Excel.

Step 2: Work the formula. There are 11 numbers in the set, so:

Q_{3} = ¾(n + 1)^{th} Term.

Q_{3} = ¾(11 + 1)^{th} Term.

Q_{3} = ¾(12)^{th} Term.

Q_{3} = 9^{th} Term.

In this set of numbers (1, 2, 5, 6, 7, 9, 12, 15, 18, 19, 27), the upper quartile (18) is the 9th term, or the 9th place from the left.

## Quartile Function Excel

You can also find a quartile in Microsoft Excel using the Excel quartile function.

- Type your data into a single column. For example, type your data into cells A1 to A10.
- Click an empty cell somewhere on the sheet. For example, click cell B1.
- Type “=QUARTILE(A1:A10,
**1**)” and then press “Enter”. This finds the first quartile. To find the third quartile, type “=QUARTILE(A1:A10,**3**)”.**Note**: If your data is in a different cell range other than A1:A10, make sure you change the function to reflect that.

## Difference between a quarter and a quartile

There’s a *slight* difference between a quarter and quartile. A quarter is the whole slice of pizza, but a quartile is the mark the pizza cutter makes at the end of the slice.

A quarter of the pizza is the whole slice; a quartile marks the end of the first quarter and the beginning of the second.

## Quartiles

In statistics, **Quartiles **are the set of values which has three points dividing the data set into four identical parts. We ordinarily deal with a large amount of numerical data, in stats. There are several concepts and formulas, which are extensively applicable in various researches and surveys. One of the best applications of quartiles is defined in box and whisker plot.

Quartiles are the values that divide a list of numerical data into three quarters. The middle part of the three quarters measures the central point of distribution and shows the data which are near to the central point. The lower part of the quarters indicates just half information set which comes under the median and the upper part shows the remaining half, which falls over the median. In all, the quartiles depict the distribution or dispersion of the data set.

## Quartiles Definition

Quartiles divide the entire set into four equal parts. So, there are three quartiles, first, second and third represented by Q_{1}, Q_{2} and Q_{3}, respectively. Q_{2} is nothing but the median, since it indicates the position of the item in the list and thus, is a positional average. To find quartiles of a group of data, we have to arrange the data in ascending order.

In the median, we can measure the distribution with the help of lesser and higher quartile. Apart from mean and median, there are other measures in statistics, which can divide the data into specific equal parts. A median divides a series into two equal parts. We can partition values of a data set mainly into three different ways:

- Quartiles
- Deciles
- Percentiles

### Quartiles Formula

Suppose, Q_{3} is the upper quartile is the median of the upper half of the data set. Whereas, Q_{1} is the lower quartile and median of the lower half of the data set. Q_{2} is the median. Consider, we have n number of items in a data set. Then the quartiles are given by;

Q_{1} = [(n+1)/4]th item

Q_{2} = [(n+1)/2]th item

Q_{3} = [3(n+1)/4]th item

Hence, the formula for quartile can be given by;

Where, Qr is the r^{th} quartile

l_{1} is the lower limit

l_{2} is the upper limit

f is the frequency

c is the cumulative frequency of the class preceding the quartile class.

### Quartiles in Statistics

Similar to the median which divides the data into half so that 50% of the estimation lies below the median and 50% lies above it, the quartile splits the data into quarters so that 25% of the estimation are less than the lower quartile, 50% of estimation are less than the mean, and 75% of estimation are less than the upper quartile. Usually, the data is ordered from smallest to largest:

- First quartile: 25% from smallest to largest of numbers
- Second quartile: between 25.1% and 50% (till median)
- Third quartile: 51% to 75% (above the median)
- Fourth quartile: 25% of largest numbers

### Quartile Deviation

You have learned about standard deviation in statistics**. Quartile deviation** is defined as half of the distance between the third and the first quartile. It is also called **Semi Interquartile range**. If Q_{1} is the first quartile and Q_{3} is the third quartile, then the formula for deviation is given by;

Quartile deviation = (Q_{3}-Q_{1})/2 |

### Interquartile Range

The **interquartile range (IQR)** is the difference between the upper and lower quartile of a given data set and is also called a **midspread**. It is a measure of statistical distribution, which is equal to the difference between the upper and lower quartiles. Also, it is a calculation of variation while dividing a data set into quartiles. If Q_{1} is the first quartile and Q_{3} is the third quartile, then the IQR formula is given by;

IQR = Q_{3 }– Q_{1} |

### Quartiles Examples

**Question 1: Find the quartiles of the following data: 4, 6, 7, 8, 10, 23, 34.**

**Solution:** Here the numbers are arranged in the ascending order and number of items, n = 7

Lower quartile, Q_{1} = [(n+1)/4] th item

Q_{1}= 7+1/4 = 2nd item = 6

Median, Q_{2} = [(n+1)/2]th item

Q_{2}= 7+1/2 item = 4th item = 8

Upper Quartile, Q_{3} = [3(n+1)/4]th item

Q_{3} = 3(7+1)/4 item = 6th item = 23

**Question 2: Find the Quartiles of the following age:-**

**23, 13, 37, 16, 26, 35, 26, 35**

**Solution:**

First, we need to arrange the numbers in increasing order.

Therefore, 13, 16, 23, 26, 26, 35, 35, 37

Number of items, n = 8

Lower quartile, Q_{1 }= [(n+1)/4] th item

Q_{1} = 8+1/4 = 9/4 = 2.25th term

From the quartile formula we can write;

Q_{1} = 2nd term + 0.25(3rd term-2nd term)

Q_{1}= 16+0.25(23-26) = 15.25

Similarly,

Median, Q_{2} = [(n+1)/2]th item

Q_{2} = 8+1/2 = 9/2 = 4.5

Q_{2} = 4th term+0.5 (5th term-4th term)

Q_{2}= 26+0.5(26-26) = 26

And,

Upper Quartile, Q_{3} = [3(n+1)/4]th item

Q_{3} = 3(8+1)/4 = 6.75th term

Q_{3} = 6th term + 0.75(7th term-6th term)

Q_{3} = 35+0.75(35-35) = 35

### Solved example

**Question:** Find the median, lower quartile, upper quartile and inter-quartile range of the following data set of scores: 19, 21, 23, 20, 23, 27, 25, 24, 31 ?

**Solution:**

First, lets arrange of the values in an ascending order: 19, 20, 21, 23, 23, 24, 25, 27, 31

Now let’s calculate the Median,

#### Example

**Teaching private coaching classes is considering rewarding students who are in the top 25% quartile advice to interquartile students lying in that range and retake sessions for the students lying in below Q1.Use the quartile formula to determine what repercussion will student face if he scores an average of 63?**

**Solution :**

Use the following data for the calculation of quartile.

The data is for the 25 students.

The number of observations here is 25, and our first step would be converting the above raw data in ascending order.

Calculation of quartile Q1 can be done as follows,

Q1 = ¼ (n+1)th term

= ¼ (25+1)

= ¼ (26)

Q1 will be –

**Q1 **= **6.5 Term**

The Q1 is 56.00, which is the bottom 25%

Calculation of quartile Q3 can be done as follows,

Q3 = ¾ (n+1)th term

= ¾ (26)

Q3 will be –

**Q3** = **19.50 Term**

Here the average needs to be taken, which is of 19^{th} and 20^{th} terms which are 77 and 77 and the average of same is (77+77)/2 = 77.00

The Q3 is 77, which is the top 25%.

Median or Q2 will be –

Median or Q2=19.50 – 6.5

Median or Q2 will be –

**Median or Q2** = **13 Term**

The Q2 or median is 68.00

Which is 50% of the population.

**The ****R****ange would be:**

56.00 – 68.00

>68.00 – 77.00

77.00

Sometimes a “cut” is between two numbers … the Quartile is the average of the two numbers.

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