How to Find Interquartile Range Formula Steps and Solved Examples
The concept of interquartile range (IQR) plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. It measures how spread out the central 50% of your data is, helping you understand variation and identify outliers in a set of values.
What Is Interquartile Range?
The interquartile range is a measure of statistical dispersion, or how spread out values in a data set are. Specifically, the IQR is the difference between the upper quartile (Q3) and the lower quartile (Q1). This shows the range covered by the middle 50% of the data. You’ll find this concept applied in box and whisker plots, outlier detection, and comparing variability between different data sets.
Key Formula for Interquartile Range
Here’s the standard formula: \( \text{Interquartile Range (IQR)} = Q_3 - Q_1 \)
Where:
- Q1 (Lower Quartile): 25% of data lies below this value
- Q3 (Upper Quartile): 75% of data lies below this value
Cross-Disciplinary Usage
Interquartile range is not only useful in Maths but also plays an important role in Physics, Computer Science, Biology, and data science. It is a reliable measure to understand variability and spot outliers. Students preparing for CBSE, ICSE, JEE, or NEET will see IQR questions in both theoretical and practical contexts.
Step-by-Step Illustration
- Arrange all data values in ascending order.
- Identify the positions for Q1 and Q3.
For an odd number of values, Q1 is the median of the lower half and Q3 is the median of the upper half.
For an even number, use the average of two middle values. - Find the values for Q1 and Q3.
- Subtract Q1 from Q3.
IQR = Q3 − Q1
Example: Find the interquartile range of these 10 scores: 56, 62, 63, 64, 64, 70, 72, 76, 77, 81.
- Data in order: 56, 62, 63, 64, 64, 70, 72, 76, 77, 81
- Lower half (first 5): 56, 62, 63, 64, 64 (Q1 is 63)
- Upper half (last 5): 70, 72, 76, 77, 81 (Q3 is 76)
- IQR = Q3 – Q1 = 76 – 63 = 13
Speed Trick or Vedic Shortcut
Here’s a quick shortcut that helps solve problems faster when working with interquartile range. If your data set is already in order and has a small, even number of values, just split it in half, find the medians of each half (Q1 and Q3), and subtract.
Example Trick:
For 8 numbers: 3, 4, 7, 8, 10, 12, 15, 18
- Split into lower half: 3, 4, 7, 8 (Q1 = (4 + 7)/2 = 5.5)
- Upper half: 10, 12, 15, 18 (Q3 = (12 + 15)/2 = 13.5)
- IQR = Q3 – Q1 = 13.5 – 5.5 = 8
Students often use this for class 10 board math problems and quick competitive exams! Vedantu’s live classes discuss more such easy calculation tips.
Try These Yourself
- Calculate the IQR for the data set: 5, 8, 10, 12, 13, 14, 15, 17
- If Q1 = 20 and Q3 = 40, what is the IQR?
- Identify any outliers in: 14, 16, 17, 17, 20, 22, 70
- Find IQR when the data set is 25, 26, 27, 28, 29, 29, 30, 32, 33, 40
Frequent Errors and Misunderstandings
- Mixing up range and interquartile range—they are different!
- Forgetting to re-arrange data in ascending order before finding Q1 and Q3.
- Using wrong formula (subtracting Q1 from Q3, not the other way around).
- Confusing quartiles with percentiles.
Relation to Other Concepts
The idea of interquartile range connects closely with topics such as Mean, Standard Deviation, and Box Plot. Mastering this will make you stronger at understanding data spread and advanced statistics in higher grades.
Classroom Tip
A quick way to remember interquartile range: IQR = Q3 – Q1, or “upper minus lower quartile.” Vedantu’s teachers often show this on a box plot diagram, highlighting the ‘box’ width as the IQR for easy recall.
We explored interquartile range—from its definition, formula, stepwise examples, mistakes to avoid, and how it links with other Maths concepts. Continue practicing with Vedantu to become confident in solving IQR questions for school, board, and competitive exams!
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FAQs on Interquartile Range in Statistics and Data Analysis
1. What is the interquartile range (IQR) in statistics?
The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1), measuring the spread of the middle 50% of data. It shows how dispersed the central values are and is calculated using the formula IQR = Q3 − Q1.
- Q1 = 25th percentile
- Q3 = 75th percentile
- The IQR ignores extreme values (outliers)
2. How do you calculate the interquartile range step by step?
To calculate the interquartile range (IQR), first find Q1 and Q3, then subtract Q1 from Q3. Follow these steps:
- Arrange the data in ascending order.
- Find the median (Q2).
- Find Q1 (median of the lower half).
- Find Q3 (median of the upper half).
- Compute IQR = Q3 − Q1.
3. What is the formula for the interquartile range?
The formula for the interquartile range is IQR = Q3 − Q1. Here:
- Q1 is the first quartile (25th percentile).
- Q3 is the third quartile (75th percentile).
4. What does the interquartile range tell you?
The interquartile range tells you how spread out the middle 50% of the data values are. A larger IQR means greater variability, while a smaller IQR means the data are more closely clustered.
- It is resistant to outliers.
- It shows central data dispersion.
- It is used in box-and-whisker plots.
5. How do you find Q1 and Q3 when calculating IQR?
To find Q1 and Q3, divide the ordered data set into two halves and find the median of each half. Steps:
- Order the data from smallest to largest.
- Find the median (Q2).
- Q1 = median of the lower half (excluding Q2 if odd number of values).
- Q3 = median of the upper half.
6. Can you give an example of calculating the interquartile range?
Yes, the interquartile range is calculated by subtracting Q1 from Q3 after finding the quartiles. Example data: 1, 3, 5, 7, 9, 11, 13.
- Median (Q2) = 7
- Lower half: 1, 3, 5 → Q1 = 3
- Upper half: 9, 11, 13 → Q3 = 11
- IQR = 11 − 3 = 8
7. What is the difference between range and interquartile range?
The range measures total spread (maximum − minimum), while the interquartile range measures the spread of the middle 50% (Q3 − Q1).
- Range = Maximum − Minimum
- IQR = Q3 − Q1
- Range is affected by outliers.
- IQR is resistant to extreme values.
8. How is the interquartile range used to identify outliers?
The interquartile range identifies outliers using the 1.5 × IQR rule. A value is an outlier if it is:
- Less than Q1 − 1.5 × IQR
- Greater than Q3 + 1.5 × IQR
9. Why is the interquartile range considered a robust measure of spread?
The interquartile range is considered robust because it is not affected by extreme values or outliers. Since it only uses Q1 and Q3, it focuses on the central 50% of the data.
- Ignores minimum and maximum values.
- Works well for skewed distributions.
- Provides stable variability measurement.
10. What does a small or large interquartile range mean?
A small IQR means the middle 50% of values are close together, while a large IQR means they are more spread out. Specifically:
- Small IQR → low variability in central data.
- Large IQR → high variability in central data.
