Quartile Deviation

Quartile deviation is a measure that shows how spread out the middle 50% of a dataset is. Instead of considering all values, it focuses only on the central portion, which helps avoid the effects of extreme or outlier values.

Quartiles divide the data into four equal parts:
 

• Q1: The median of the lower half
   
• Q2: The median of the entire dataset
   
• Q3: The median of the upper half
   

The difference between Q3 and Q1 is called the interquartile range (IQR). The IQR is equal to half the interquartile range, also known as the quartile deviation or semi-interquartile range.

Formula for quartile deviation:

\(Quartile Deviation = \frac{Q_3 - Q_1}{2}\)
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It is a simple way to understand how tightly or widely the central part of the data is spread, and it is not affected by extreme values.

Here is an example for Quartile Deviation:
The test scores of eight students are:
10, 12, 14, 16, 18, 20, 22, 24

a) Find Q₁, Q₂, and Q₃.
b) Find the Interquartile Range (IQR).
c) Find the Quartile Deviation.

Answer

Step 1: Find Q₁, Q₂, and Q₃

The data is already arranged in ascending order.

Q₂ (Median): The middle two values are 16 and 18

 \(Q_2 = \frac{16 + 18}{2} = 17\) 

Lower half: 10, 12, 14, 16

Median of lower half:

 \(Q_1 = \frac{12 + 14}{2} = 13\) 

Upper half: 18, 20, 22, 24

Median of upper half:

 \(Q_3 = \frac{20 + 22}{2} = 21\)

Step 2: Interquartile Range (IQR)

\(IQR =Q_3 − Q_1 = 21 - 13 = 8\)

Step 3: Quartile Deviation

\(\text{Quartile Deviation}= \frac{Q_3 - Q_1}{2} = \frac{8}{2} = 4\) 

Final Answers

Q1 = 13
Q2 = 17
Q3 = 21
IQR = 8
Quartile Deviation = 4

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 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 values using the formulas. For ungrouped data, the formula is 

 \(Q_1 =\frac{(n + 1)}{4}\)

Where, 

Q₁ = First Quartile Value

n = Number of terms (class mark)

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

         \(Q_1 = \frac{7 + 1}{4}\)

         \(Q_1 = \frac{8}{4} = 2\)

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.

  \(Q_1 =3 \times \frac{ (n + 1)}{4}\)
 

  \(Q_1 =3 \times \frac{ (7 + 1)}{4}\)
 

  \( Q_1 = 3 × 2 = 6\)

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

∴  Q₃ = 40

Step 6: Now using the Quartile Deviation formula 

\(Q.D = \frac{Q_3 - Q_1}{2}\)

\(Q.D =\frac{40 + 15}{2}\)

\(Q.D = \frac{25}{2} = 12.5\)

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

Quartile deviation shows us how much the middle 50% of the data values are spread out, but it is an absolute measure. This means we cannot compare two datasets if they are used in different units, for example, one measured in kilograms and the other in grams. To make a fair comparison, we use the Coefficient of Quartile Deviation, which is a relative measure. It allows us to compare the variation between datasets even when their units are different.

The coefficient of quartile deviation formula:

\(\text{Coefficient of Quartile Deviation} = \frac{Q_3 - Q_1}{Q_3 + Q_1}\)

Both ungrouped and grouped data can be used to calculate quartile deviation. In both cases, it helps us to understand how widely the middle 50% of the values are spread around the center.

Quartile Deviation for Grouped Data 

In grouped data, the values are given as the class intervals, so we don’t know the exact individual data points. So, quartiles must be estimated using the class boundaries, frequencies, and cumulative frequencies.

Formula to find Q1 and Q3

\(Q_r = l_1 + \frac{rN/4 - c_f}{f} \times (l_2 - l_1)\)

Where:

Qᵣ = the r-th quartile, either Q₁ (first quartile) or Q₃ (third quartile)

l₁ = lower limit of the quartile class

l₂ = upper limit of the quartile class

f = frequency of the quartile class

c = cumulative frequency of the class before the quartile class

N = total number of observations

This formula helps to locate the exact quartile values within a class interval.

Quartile Deviation for Ungrouped Data

In ungrouped data, the values are individual, not in class intervals. To find the quartiles, we can arrange the data in ascending order and use their positions. For ungrouped data, quartiles can be found using the following positions:

\(Q_1 = \frac{n + 1}{4} th\ item\)

\(Q_2 = \frac{n + 1}{2}th\ item\)

\(Q_3 = \frac{n + 1}{3}th\ item\)

Here, n is the total number of observations.

The data must be arranged in ascending order before calculating the quartiles. 
 

• Q₂ is the middle value when all the data is arranged in ascending order.
   
• Q₁ is the middle value of the lower 50% of the data.
   
• Q₃ is the middle value of the upper 50% of the data.

If n is even, the method for finding the median remains the same.

Mastering quartile deviation helps you measure data spread while minimizing the effect of outliers. These tips guide you to calculate and interpret it accurately in real-world scenarios.

• Understand the concept of quartiles and how they divide data into four equal parts.
   
• Learn the formula for quartile deviation.
   
• Practice identifying Q1 and Q3 from both raw and grouped data.
   
• Use quartile deviation to measure spread without being affected by extreme values.
   
• Solve real-life examples to strengthen your understanding and application of the concept.
   
• Use real-life data, such as students’ test scores or daily temperatures, to illustrate how quartiles divide the data into four equal parts.
   
• Explain Q1 and Q3 visually, using the number lines or simple charts, so children can see how the middle 50% of data behaves.
   
• Compare the quartile deviation with the standard deviation, so learners can see how the quartile deviation is less affected by extreme values.
   
• Introduce the online tools, such as a quartile deviation calculator, to help parents and teachers verify answers and boost learners’ confidence.
   
• Discuss how quartile deviation behaves in a normal distribution, which may help the learners to understand when to use quartile vs. standard deviation.

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.
 

Sports Performance Analysis: Coaches use quartile deviation to evaluate athletes’ performance by focusing on the middle 50% of scores or timings. This helps identify consistent performers without being skewed by exceptionally high or low results.