1. Introduction
Measures of dispersion describe the spread, variability, or distribution of data in a dataset. While measures of central tendency (mean, median, and mode) give a central value, measures of dispersion show how much the data deviates from the center.
Two fundamental measures of dispersion are:
- Range – The simplest measure of dispersion that represents the difference between the highest and lowest values in a dataset.
- Interquartile Range (IQR) – A more refined measure that shows the spread of the middle 50% of data, helping to eliminate the effect of outliers.
These measures help businesses and analysts understand the variability and consistency in financial, operational, and market data.
2. Range
Definition
The range is the simplest measure of dispersion, calculated as the difference between the maximum and minimum values in a dataset. It shows the total spread of data but does not consider how data is distributed within the range.
Formula
Range = Maximum Value − Minimum Value
Example Calculation
A company records the daily sales (₹) for one week:
| Day | Sales (₹ ‘000) |
|---|---|
| Monday | 50 |
| Tuesday | 42 |
| Wednesday | 60 |
| Thursday | 45 |
| Friday | 70 |
| Saturday | 55 |
| Sunday | 65 |
- Maximum Value (Highest Sales) = 70
- Minimum Value (Lowest Sales) = 42
Range = 70−42 = 28
Thus, the range is ₹28,000, showing the total spread in sales data over the week.
Advantages of Range
✔ Easy to calculate and interpret.
✔ Gives a quick measure of data spread.
✔ Useful in comparing variability between different datasets.
Disadvantages of Range
❌ Highly sensitive to outliers – A single extreme value can distort the range.
❌ Does not show distribution – It only gives the spread, not how values are distributed.
3. Interquartile Range (IQR)
Definition
The Interquartile Range (IQR) measures the spread of the middle 50% of data by calculating the difference between the third quartile (Q3) and first quartile (Q1). It is a robust measure of dispersion because it is not affected by extreme values (outliers).
Formula
IQR = Q3−Q1
Where:
- Q1 (First Quartile) = 25th percentile (value below which 25% of data lies).
- Q3 (Third Quartile) = 75th percentile (value below which 75% of data lies).
Example Calculation
A company records monthly sales (₹) for one year:
| Month | Sales (₹ ‘000) |
|---|---|
| Jan | 40 |
| Feb | 42 |
| Mar | 45 |
| Apr | 47 |
| May | 50 |
| Jun | 52 |
| Jul | 55 |
| Aug | 58 |
| Sep | 60 |
| Oct | 63 |
| Nov | 65 |
| Dec | 68 |
Step 1: Arrange the Data in Ascending Order
40, 42, 45, 47, 50, 52, 55, 58, 60, 63, 65, 68
Step 2: Find Q1 and Q3
- Q1 (First Quartile) = (n+1)/4 = (12+1)/4 = 3.25th value
- 3rd value = 45, 4th value = 47
- Q1 = 45 + 0.25(47 – 45) = 45.5
- Q3 (Third Quartile) = 3(n+1)/4 = 3(12+1)/4 = 9.75th value
- 9th value = 60, 10th value = 63
- Q3 = 60 + 0.75(63 – 60) = 62.25
IQR = Q3−Q1=62.25−45.5 = 16.75
Thus, the IQR is ₹16,750, meaning the middle 50% of the data is spread over ₹16,750.
Advantages of IQR
✔ Not affected by outliers, making it more reliable than range.
✔ Focuses on the middle 50% of data, reducing bias from extreme values.
✔ Useful for skewed distributions where mean and standard deviation may not be reliable.
Disadvantages of IQR
❌ Ignores the extremes – Does not consider maximum and minimum values.
❌ More complex to calculate compared to range.
4. Detecting Outliers Using IQR
The Interquartile Range (IQR) Rule helps in detecting outliers, which are values that lie significantly outside the typical range.
Formula for Identifying Outliers
- Lower Bound (Minimum Acceptable Value): Q1− (1.5 × IQR)
- Upper Bound (Maximum Acceptable Value): Q3+ (1.5 × IQR)
Any data point outside this range is considered an outlier.
Example: Outlier Detection
Using our previous sales dataset where:
- Q1 = 45.5,
- Q3 = 62.25,
- IQR = 16.75,
We calculate:
- Lower Bound = 45.5 – (1.5 × 16.75) = 20.375
- Upper Bound = 62.25 + (1.5 × 16.75) = 87.375
If any sales value is below ₹20,375 or above ₹87,375, it is considered an outlier.
5. Comparison of Range and IQR
| Measure | Formula | Best Used For | Sensitivity to Outliers |
|---|---|---|---|
| Range | Max – Min | Quick measure of spread | Highly sensitive |
| Interquartile Range (IQR) | Q3 – Q1 | Robust measure for skewed data | Not sensitive |
6. Applications of Range and IQR in Business Statistics
(i) Risk Assessment in Investments
Investors use range and IQR to measure the volatility of stock prices and assess financial risk. If a stock has a large range, it may be more volatile, whereas a low IQR suggests stable returns.
(ii) Quality Control in Manufacturing
Companies use IQR to monitor product quality and defect rates. If production data has high variability, it indicates inconsistent product quality.
(iii) Salary and Income Analysis
Organizations use IQR to analyze employee salaries. If the IQR is small, salaries are consistent, whereas a large IQR suggests wage disparity.
(iv) Customer Purchase Behavior
Retail businesses use range and IQR to analyze spending patterns. If IQR is narrow, most customers spend around the same amount; if it’s wide, customer spending habits vary significantly.
7. Conclusion
Range and Interquartile Range (IQR) are crucial in understanding the spread and variability of data. While the range is easy to calculate and provides a quick measure, it is highly affected by outliers. On the other hand, IQR is more reliable for business analysis as it focuses on the middle 50% of data, eliminating extreme values.