## What is Quartile Deviation?

Quartile deviationis based on the difference between the first quartile and the third quartile in the frequency distribution and the difference is also known as the interquartile range, the difference divided by two is known as quartile deviation or semi interquartile range.

When one takes half of the difference or variance between the 3^{rd} quartile and the 1^{st} quartile of a simple distribution or frequency distribution is the quartile deviation.

### Formula

A Quartile Deviation (Q.D.) formula is used in statistics to measure spread or in other words to measure dispersion. This can also be called a Semi Inter-Quartile Range.

**Q.D. = Q3 – Q1 / 2**

- The formula includes Q3 and Q1 in the calculation which is top 25% and lowers 25% data respectively and when the difference is taken between these two and when this number is halved then it gives measures of spread or dispersion.
- So, to calculate Quartile deviation, you need to first find out Q1 then the second step is to find Q3 and then take a difference of both and the final step is to divide by 2.
- This is one of the best methods of dispersion for open-ended data.

### Examples

#### Example #1

**Consider a data set of following numbers: 22, 12, 14, 7, 18, 16, 11, 15, 12. You are required to calculate the Quartile Deviation.**

**Solution:**

First, we need to arrange data in ascending order to find Q3 and Q1 and avoid any duplicates.

7, 11, 12, 13, 14, 15, 16, 18, 22

Calculation of Q1 can be done as follows,

Q1 = ¼ (9 + 1)

=¼ (10)

**Q1**=**2.5 Term**

Calculation of Q3 can be done as follows,

Q3=¾ (9 + 1)

=¾ (10)

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

Calculation of quartile deviation can be done as follows,

- Q1 is an average of 2
^{nd}which is11 and adds the product of the difference between 3^{rd}& 4^{th}and 0.5 which is (12-11)*0.5 = 11.50. - Q3 is 7
^{th}term and product of 0.5 and the difference between 8^{th}and 7^{th}term which is (18-16)*0.5 and the result is 16 + 1 = 17.

**Q.D. = Q3 – Q1 / 2**

Using quartile deviation formula , we have (17-11.50) / 2

=5.5/2

**Q.D.**=**2.75.**

#### Example #2

**Harry ltd. is a textile manufacturer and is working upon a reward structure. The management is in discussion to start a new initiative, but they first want to know how much their production spread is.**

**The management has collected its average daily production data for the last 10 days per (average) employee.**

**155, 169, 188, 150, 177, 145, 140, 190, 175, 156.**

**Use the Quartile Deviation formula to help management find dispersion.**

**Solution:**

The number of observations here is 10 and our first step would be to arrange data n ascending order.

140, 145, 150, 155, 156, 169, 175, 177, 188, 190

Calculation of Q1 can be done as follows,

Q1= ¼ (n+1)th term

=¼ (10+1)

=¼ (11)

**Q1**= **2.75 ^{th} Term**

Calculation of Q3 can be done as follows,

Q3= ¾ (n+1)th term

=¾ (11)

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

Calculation of quartile deviation can be done as follows,

- 2
^{nd}term is 145 and now adding to this 0.75 * (150 – 145) which is 3.75 and the result is 148.75 - 8
^{th}term is 177 and now adding to this 0.25 * (188 – 177) which is 2.75 and the result is 179.75

**Q.D. = Q3 – Q1 / 2**

Using the quartile deviation formula, we have (179.75-148.75) / 2

=31/2

**Q.D.**=**15.50.**

#### Example #3

**Ryan’s international academy wants to analyze how much percentage score marks of their students are spread out.**

**The data is for the 25 students.**

**Use the Quartile Deviation formula to find out the dispersion in % marks.**

**Solution:**

The number of observations here is 25 and our first step would be arranging data in ascending order.

Calculation of Q1 can be done as follows,

Q1= ¼ (n+1)th term

=¼ (25+1)

=¼ (26)

**Q1**= **6.5 ^{th} Term**

Calculation of Q3 can be done as follows,

Q3=¾ (n+1)th term

=¾ (26)

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

Calculation of quartile deviation or semi interquartile range can be done as follows,

- 6
^{th}term is 154 and now adding to this 0.50 * (156 – 154) which is 1 and the result is 155.00 - 19
^{th}term is 177 and now adding to this 0.50 * (177 – 177) which is 0 and the result is 177

**Q.D. = Q3 – Q1 / 2**

Using the quartile deviation formula, we have (177-155) / 2

=22/2

**Q.D.**= **11.**

#### Example #4

**Let us now determine the value through an excel template for Practical example I.**

**Solution:**

Use the following data for the calculation of quartile deviation.

Calculation of Q1 can be done as follows,

**Q1**=**148.75**

Calculation of Q3 can be done as follows,

**Q3**= **179.75**

Calculation of quartile deviation can be done as follows,

Using the quartile deviation formula, we have (179.75-148.75 )/ 2

Q.D. will be –

**QD** = **15.50**

### Relevance and Uses

Quartile deviation which is also renowned as a semi interquartile range. Again, the difference of the variance between the 3^{rd} and 1^{st} quartiles is termed as the interquartile range. The interquartile range depicts the extent to which the observations or the values of the given dataset are spread out from the mean or their average. The Quartile deviation or semi interquartile range is the majority used in a case where one wants to learn or say a study about the dispersion of the observations or the samples of the given data sets that lie in the main or middle body of the given series. This case would usually happen in a distribution where the data or the observations tends to lie intensely in the main body or middle of the given set of data or the series and the distribution or the values do not lie towards the extremes and if they lie then they are not of much significance for the calculation.

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

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