## Formula to Calculate Quartile in Statistics

Quartile Formula is a statistical tool to calculate the variance from the given data by dividing the same into 4 defined intervals and then comparing the results with the entire given set of observations and also commenting on the differences if any to the data sets.

It is often used in statistics to measure the variances which describe a division of all the given observations into 4 defined intervals that are based upon the values of the data and to observe as to where they stand when compared with the entire set of the given observations.

It is divided into 3 points –A lower quartile denoted by Q1 which falls between the smallest value and the median of the given data set, median denoted by Q2 which is median, and upper quartile which is denoted by Q3 and is the middle point which lies between the median and the highest number of the given dataset of the distribution.

Quartile Formula in statistics is represented as follows,

The Quartile Formula for Q1= ¼(n+1)^{th }term The Quartile Formula for Q3= ¾(n+1)^{th }term The Quartile Formula for Q2= Q3–Q1(Equivalent to Median)

### Explanation

The quartiles will divide the set of measurements of the given data set or the given sample into 4 similar or say equal parts. 25% of the measurements of the given dataset (that are represented by Q1) are not greater than the lower quartile, then the 50% of the measurements are not greater than the median i.e. Q2 and lastly, 75% of the measurements will be less than the upper quartile which is denoted by Q3. So, one can say that 50% of the measurements of the given dataset are in between the Q1 which the lower quartile is and Q2 which is the upper quartile.

### Examples

Let’s see some simple to advanced examples of a quartile in excel to understand it better.

#### Example #1

**Consider a data set of following numbers: 10, 2, 4, 7, 8, 5, 11, 3, 12. You are required to calculate all the 3 quartiles.**

**Solution:**

Use the following data for the calculation of quartile.

Calculation of Median or Q2 can be done as follows,

Median or Q2 = Sum(2+3+4+5+7+8+10+11+12)/9

Median or Q2 will be –

**Median or Q2** = **7**

Now since the number of observations is odd which is 9, the median would lie on 5^{th} position which is 7 and same will be Q2 for this example.

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Calculation of Q1 can be done as follows,

Q1= ¼ (9 + 1)

= ¼ (10)

Q1 will be –

**Q1** = **2.5**

This means that Q1 is average of 2^{nd} and 3^{rd} position of the observations which is 3 & 4 here and the average of same is (3+4)/2 = 3.5

Calculation of Q3 can be done as follows,

Q3 = ¾ (9 + 1)

= ¾ (10)

Q3 will be –

**Q3** = **7.5 Term**

This means that Q3 is average of 8^{th} and 9^{th} position of the observations which is 10 & 11 here and the average of same is (10+11)/2 = 10.5

#### Example #2

**Simple ltd. is a clothing manufacturer and is working upon a scheme to please their employees for their efforts. The management is in discussion to start a new initiative which states they want to divide their employees as per the following:**

- Top 25% lying above Q3- $25 per cloth
- Greater than Middle one but less than Q3 – $20 per cloth
- Greater than Q1 but less than Q2 – $18 per cloth
- The management has collected their average daily production data for the last 10 days per (average) employee.
- 55, 69, 88, 50, 77, 45, 40, 90, 75, 56.
- Use the quartile formula to build the reward structure.
- What rewards would an employee get if he has produced 76 clothes ready?

**Solution:**

Use the following data for the calculation of quartile.

The number of observations here is 10 and our first step would be converting above raw data in ascending order.

** **40, 45, 50, 55, 56, 69, 75, 77, 88, 90

Calculation of quartile Q1 can be done as follows,

Q1 = ¼ (n+1)th term

= ¼ (10+1)

= ¼ (11)

Q1 will be –

**Q1** = **2.75 Term**

Here the average needs to be taken which is of 2^{nd} and 3^{rd} terms which are 45 and 50 and the average formula of same is (45+50)/2 = 47.50

The Q1 is 47.50 which is bottom 25%

Calculation of quartile Q3 can be done as follows,

Q3 = ¾ (n+1)th term

= ¾ (11)

Q3 will be –

**Q3 **= **8.25 Term**

Here the average needs to be taken which is of 8^{th} and 9^{th} terms which are 88 and 90 and the average of same is (88+90)/2 = 89.00

The Q3 is 89 which is top 25%

Calculation of Median or Q2 can be done as follows,

The Median Value (Q2) = 8.25 – 2.75

Median or Q2 will be –

**Median or Q2**= **5.5 Term**

Here the average needs to be taken which is of 5^{th} and 6^{th} 56 and 69 and average of same is (56+69)/2 = 62.5

The Q2 or median is 62.5

Which is 50% of the population.

**The Reward Range would be:**

47.50 – 62.50 will get $18 per cloth

>62.50 – 89 will get $20 per cloth

>89.00 will get $25 per cloth

** **If an employee produces 76 then he would lie above Q1 and hence would be eligible for a $20 bonus.

#### Example #3

**Teaching private coaching classes is considering rewarding students who are in top 25% quartile advise 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 average 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 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 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 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

### Relevance and Use of Quartile Formula

Quartiles let one quickly divide a given dataset or given sample into 4 major groups, making it simple as well easy for the user to evaluate which of the 4 groups a data point in. is. While the median which measures the central point of the dataset is a robust estimator of the location, but it does not say anything about how much the data of the observations lie on either side or how widely it is dispersed or spread. The quartile measures the spread or dispersion of values that are above and below the arithmetic mean or arithmetic average by dividing the distribution into 4 major groups which are already discussed above.

### Recommended Articles

This has been a guide to Quartile Formula. Here we learn how to calculate quartile in statistics using its formula along with practical examples and downloadable excel template. You can learn more about excel modeling from the following articles –

- Point Estimators
- Quartile Deviation
- Coefficient of Variation Formula
- Calculate the Correlation Coefficient

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