Quartiles

1. Introduction

Quartiles are statistical measures that divide a dataset into four equal parts, each containing 25% of the data. They help in understanding the distribution, variability, and central tendency of data, making them particularly useful in business analytics, income distribution analysis, and quality control.

Quartiles are essential in detecting skewness, outliers, and dispersion in a dataset. They provide a better understanding of data spread than simple averages, allowing businesses and analysts to make more informed decisions.

2. Definition of Quartiles

A dataset is divided into four quartiles:

• Q1 (First Quartile or Lower Quartile) → The 25th percentile (the value below which 25% of data lies).
• Q2 (Second Quartile or Median) → The 50th percentile (the middle value when data is arranged in order).
• Q3 (Third Quartile or Upper Quartile) → The 75th percentile (the value below which 75% of data lies).
• Q4 (Fourth Quartile or Maximum Value) → The highest value in the dataset.

These quartiles split the dataset into four equal parts, providing a detailed look at data distribution.

3. Formula to Calculate Quartiles

For a dataset arranged in ascending order, quartiles are calculated as follows:

Where:

• n = Total number of observations
• If the quartile falls between two values, take the average of the two nearest values.

4. Example Calculation

A company records the monthly sales (in ₹’000) for 12 months:

Step 1: Arrange the Data in Ascending Order

40, 42, 45, 47, 50, 52, 55, 58, 60, 63, 65, 68

Step 2: Calculate Q1, Q2, and Q3

• Q1 (First Quartile) = (n+1)/4 = (12+1)/4 = 3.25th value
  • The 3rd value = 45, 4th value = 47
  • Q1 = 45 + 0.25(47 – 45) = 45.5
• Q2 (Second Quartile or Median) = (n+1)/2 = (12+1)/2 = 6.5th value
  • The 6th value = 52, 7th value = 55
  • Q2 = 52 + 0.5(55 – 52) = 53.5
• Q3 (Third Quartile) = 3(n+1)/4 = 3(12+1)/4 = 9.75th value
  • The 9th value = 60, 10th value = 63
  • Q3 = 60 + 0.75(63 – 60) = 62.25

Thus, the quartiles for the sales dataset are:

• Q1 = 45.5
• Q2 (Median) = 53.5
• Q3 = 62.25

5. Uses of Quartiles in Business Analysis

(i) Income and Salary Distribution

Businesses use quartiles to analyze salary distributions within an organization. If Q3 is significantly higher than Q1, it indicates income inequality among employees.

(ii) Customer Spending Analysis

Retail businesses segment customers into low-spending (Q1), average-spending (Q2), and high-spending (Q3) groups to create targeted marketing strategies.

(iii) Financial Risk Assessment

Investment analysts use quartiles to compare stock performance and assess risk-return distributions in portfolio analysis.

(iv) Quality Control in Manufacturing

Manufacturing companies use quartiles to analyze product quality variations. If the difference between Q3 and Q1 is too large, it may indicate inconsistent production quality.

(v) Performance Benchmarking

Organizations compare employee performance, sales trends, and customer satisfaction ratings using quartiles. This helps identify top and bottom performers in different areas.

6. Interquartile Range (IQR) and Outliers

Interquartile Range (IQR)

The Interquartile Range (IQR) measures data spread by calculating the difference between Q3 and Q1.

IQR = Q3−Q1

A larger IQR indicates high variability in data, whereas a smaller IQR suggests consistent data.

Detecting Outliers Using Quartiles

Outliers are extreme values that lie far outside the normal range of data. They are detected using the formula:

• Lower Bound: Q1 − 1.5 × IQR
• Upper Bound: Q3 + 1.5 × IQR

Any data point outside these bounds is considered an outlier.

7. Advantages of Quartiles

• Resistant to Outliers – Unlike the mean, quartiles are not affected by extreme values.
• Useful for Skewed Data – Median and quartiles provide a better representation of central tendency when data is skewed.
• Effective in Comparisons – Helps compare different groups within a dataset, such as income levels or customer segments.

8. Limitations of Quartiles

• Ignores Extreme Values – Quartiles do not account for the impact of maximum and minimum values.
• Does Not Provide a Single Central Value – Unlike the mean, quartiles do not give an exact representative number for the dataset.
• Not Suitable for Small Datasets – In small samples, quartiles may not provide meaningful insights.

9. Conclusion

Quartiles are a powerful tool in business statistics that help in understanding data distribution, detecting outliers, and making comparisons across different segments. They provide insights into income distribution, customer spending behaviour, and performance benchmarking. Businesses rely on quartiles for risk assessment, decision-making, and financial analysis.