Quartile Deviation

A quartile is a value that divides a dataset into four equal parts, which helps in understanding how data is distributed. There are three quartiles:

First Quartile (Q₁): The value below which 25% of the data falls (also called the lower below and 50% above).

Second Quartile (Q₂): The median of the dataset, which divides the data into two equal halves (50% below and 50% above).

Third Quartile (Q₃): The value below which 75% of the data falls (also called the upper quartile).

A quartile deviation is a measure of dispersion that indicates the spread of the middle 50% of a dataset. In order to find the deviation, generally we find the difference between the third quartile and the first quartile. 

    Quartile Deviation (Q.D) = (Q₃– Q₁) / 2

Quartile deviation represents the spread of the middle 50% of the data. It is derived from the interquartile range by measuring the deviation around the median. This quartile deviation, which is unaffected by extreme values, measures the absolute degree of dispersion. The coefficient of quartile deviation is the relative measure in relation to quartile deviation.

    Coefficient of Quartile Deviation =  (Q₃ – Q₁) / (Q₃ +  Q₁)

Quartile deviation can be calculated for grouped and ungrouped data. And it can be measured using two different methods according to the type of data provided. Let’s analyze the steps in order to find the quartile deviation,

Step 1: In the given data, for example, you got the data (class mark) as 20, 12, 28, 15, 50, 40, 30, you need to first rearrange this data in ascending order (small to big).

Step 2: The rearranged data is 12, 15, 20, 28, 30, 40, 50. Next, find the first quartile and third quartile value using the formulas. For ungrouped data, the formula is 

Q₁ = (n + 1) / 4 

Where, 

Q₁ = First Quartile Value

n = Number of terms (class mark)

Step 3: Insert the values from the data 12, 15, 20, 28, 30, 40, 50. 

        Q₁ = (7 + 1) / 4

        Q₁ = 8/4 = 2

Step 4: Since you got 2 as Q1, you should find the second number from the data. That number will be the first quartile. In this case, which is 15. 

∴  Q₁ = 15

Step 5: Next find the third quartile, which is the same step but multiply it thrice.

        
        Q₁ = 3 × (n + 1) / 4 

        Q₁ = 3 × (7 + 1) / 4 

        Q₁ = 3 × 2 = 6

Here you got 6 as the answer, so the 6th number in the data is 40.

∴  Q₂ = 40

Step 6: Now using the Quartile Deviation formula 

        Q.D = (Q₃– Q₁) / 2

        Q.D = 40 - 15 / 2

        Q.D = 25 / 2 = 12.5

So the quartile deviation of 12, 15, 20, 28, 30, 40, 50 is 12.5
 

The coefficient of quartile deviation can be found first by finding the quartile deviation using the formula for grouped data and then applying it to the coefficient formula. Finding the quartile deviation of grouped data is a long process as compared to ungrouped data, but with simple step-by-step process, you can find it easily.

    Q₁ = L + ((n/4) -cf / f) × i

Where, 

Q₁ =  First Quartile Value

L = Lower limit

n = Number of Frequencies 

cf = Cumulative Frequency 

f = Frequency

i = Difference Between Upper Limit and Lower Limit
 

Step 1: In ungrouped data we find Q₁ using the formula Q₁ = (n + 1) / 4. But in ungrouped data, we find Q₁ using the formula Q₁ = n / 4.

    Q₁ = n / 4 

    Q₁ =  60 / 4 = 15

Step 2: Find the value 15 from the cumulative frequency of the table. 

    15 comes under 37 in the cumulative frequency, 15 falls under 30 - 35 (class mark).

Step 3: Substitute the values in the formula of ungrouped data.

    Q₁ = L + ((n/4) -cf / f) × i

    Q₁ = 30 + ((15/4) - 37 / 25) × 5

    Q₁ = 30 + 32 / 5 × 5

    Q₁ = 30 + 3 /5 

    Q₁ = 30 + 0.6
    
    Q₁ = 30.6

Step 4: Next, find the third quartile.

    Q₃ = 3 × L + ((n/4) -cf / f) × i

        
    Q₃ = 3 × 30.6

    Q₃ = 91.8

Step 5: Using the coefficient of quartile deviation equation,  (Q₃ – Q₁) / (Q₃ +  Q₁).

    Coefficient =  (91.8 – 30.6) / (91.8 + 30.6)

            =  61.2 / 122.4

            = 0.5

Thus, the coefficient of quartile deviation is 0.5
 

Quartile deviation is widely applied in real-life scenarios such as education, economics, sports, healthcare, income distribution, or data stability. By focusing on quartiles rather than the entire range, quartile deviation provides a clearer picture of trends and patterns in various fields.

Student Performance Analysis:

A school wants to analyze student scores in a math test. Instead of using the full range (which might include extreme high or low scores), they use quartile deviation to measure the spread of middle-performing students. 

Income Distribution in Economics:

Economists use quartile deviation to analyze income inequality. It helps understand how middle-income groups are distributed without being affected by extreme rich or poor values. 

Weather Data Analysis:

Meteorologists study temperature variations using quartile deviation. For example, in a city, daily high temperatures might vary greatly, but the quartile deviation helps focus in the middle 50% of temperatures, ignoring extreme heat waves or cold spells.

Real Estate:

Real estate agents use quartile deviation to study house price variations in a neighborhood. It helps buyers and sellers understand typical price ranges, without extreme expensive or cheap outliers affecting the results.