Quartile deviation, also known as the semi-interquartile range or half the interquartile range (IQR), is a measure of statistical dispersion or variability that quantifies the spread of data around the median. The interquartile range (IQR) represents the range within which the middle 50% of the data values lie, and the quartile deviation is half of this range.

Formula for Quartile Deviation:

$\text{Quartile Deviation (QD)}=\frac{\text{IQR}}{2}$

Quartile Deviation (QD)=2IQR​

Where:

• 
  
  $\text{IQR}=<span\; class="mord\; mathnormal"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"></span>$

   

  
  $<span\; class="mord\; mathnormal"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;">Q3-Q1</span><br>$

   

  IQR=Q3−Q1

• 
  
  $\mathrm{<br>}1$

   

  Q1 represents the first quartile (25th percentile)

• 
  
  $\mathrm{<br>}3$

   

  Q3 represents the third quartile (75th percentile)

Characteristics and Considerations:

1. Interquartile Range (IQR): Quartile deviation is closely related to the interquartile range, which provides a measure of the spread of the central portion (middle 50%) of the data values.
2. Robustness: Similar to the median and IQR, quartile deviation is less sensitive to extreme values or outliers compared to the mean and standard deviation, making it a robust measure of dispersion for skewed or asymmetric distributions.
3. Simplicity: Quartile deviation offers a simple and interpretable measure of variability that complements other descriptive statistics, such as the median and quartiles, in summarizing the distribution of data.
4. Relative Measure: Quartile deviation is a relative measure of dispersion that focuses on the central portion of the data values, providing insights into the variability within the range where the majority of the data points lie.
5. Comparison with Other Measures: While quartile deviation provides a measure of variability around the median, the standard deviation or variance quantifies the average variability around the mean. The choice between these measures depends on the nature of the data and the specific objectives of the analysis.

Example:

Consider the following dataset representing the ages (in years) of a sample of individuals:

$20,25,30,35,40,45,50,55,60$

20,25,30,35,40,45,50,55,60

To calculate the quartile deviation:

1. Calculate the first quartile (Q1):
   
   $\mathrm{<br>}1=30\text{ years}$

    

   Q1=30 years

2. Calculate the third quartile (Q3):
   
   $\mathrm{<br>}3=50\text{ years}$

    

   Q3=50 years

3. Calculate the interquartile range (IQR):
   
   $\text{IQR}=\mathrm{<span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal">Q</span><span\; class="mord">3</span><span\; class="mspace"></span><span\; class="mbin">-</span><span\; class="mspace"></span></span><span\; class="base"><span\; class="strut"></span><span\; class="mord\; mathnormal">Q</span><span\; class="mord">1</span></span></span>}=50-30=20\text{ years}$

    

   IQR=Q3−Q1=50−30=20 years

4. Calculate the quartile deviation:
   
   $\text{Quartile Deviation (QD)}=\frac{\text{IQR}}{2}=\frac{20}{2}=10\text{ years}$

    

   Quartile Deviation (QD)=2IQR​=220​=10 years

The quartile deviation of the ages in the sample is 10 years, indicating the average spread of ages around the median age of 40 years within the middle 50% of the data values.

Quartile deviation is a measure of statistical dispersion that quantifies the spread of data around the median by considering half of the interquartile range. As a robust and interpretable measure, quartile deviation provides valuable insights into the variability of data values within the central portion of a distribution, complementing other descriptive statistics and facilitating a comprehensive understanding of data characteristics and patterns