How to Calculate Interquartile Range Step by Step
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 (IQR): Meaning, Formula, Calculation & Examples
1. What is the interquartile range (IQR) in Maths?
The interquartile range (IQR) is a measure of statistical dispersion, describing the spread of the middle 50% of a data set. It's calculated as the difference between the upper quartile (Q3) and the lower quartile (Q1). The IQR is useful because it's less sensitive to outliers than the range.
2. How do you calculate the interquartile range step by step?
Calculating the IQR involves these steps:
1. Arrange the data in ascending order.
2. Find the median (Q2): This is the middle value if the data set has an odd number of values; otherwise, it's the average of the two middle values.
3. Identify Q1: The median of the lower half of the data (values below Q2).
4. Identify Q3: The median of the upper half of the data (values above Q2).
5. Calculate the IQR: IQR = Q3 - Q1.
3. What are Q1 and Q3 in a data set?
Q1 (Lower Quartile) is the value that separates the bottom 25% of the data from the top 75%. Q3 (Upper Quartile) is the value separating the bottom 75% from the top 25%. These quartiles, along with the median (Q2), divide the data into four equal parts.
4. How is IQR different from the range?
The range is the difference between the highest and lowest values in a data set. It's highly sensitive to outliers. The IQR, however, focuses on the middle 50%, making it less affected by extreme values. The IQR provides a more robust measure of spread in datasets with outliers.
5. Why is the IQR important for outlier detection?
The IQR helps identify outliers because data points significantly outside the range of Q1 - 1.5 * IQR and Q3 + 1.5 * IQR are considered potential outliers. These are points that lie far from the central mass of the data.
6. Can I calculate IQR for grouped data?
Yes, you can. For grouped data, you need to estimate Q1 and Q3 using interpolation within the relevant class intervals. Formulas exist for this calculation, but it's more complex than for ungrouped data.
7. How can interquartile range be used to compare data variability between two groups?
Comparing the IQRs of two groups provides a way to assess which group exhibits more variability. A larger IQR suggests greater dispersion among the data points within the group. This comparison should be done along with other relevant descriptive statistics.
8. What happens to the IQR if all data values are multiplied by a constant?
If all data values are multiplied by a constant 'k', then the IQR will also be multiplied by 'k'. This is because the quartiles (Q1 and Q3) are directly affected by the scaling of the data, resulting in a proportionally scaled IQR.
9. How do you estimate IQR from an incomplete box and whisker plot?
If a box plot is incomplete, you can estimate the IQR by using the visible boundaries of the box. The width of the box represents the IQR. In some cases, you might need to use additional information to fill the gaps.
10. Why might IQR be more reliable than standard deviation for non-normal data?
The standard deviation is highly sensitive to outliers and assumes a roughly normal distribution. The IQR is less sensitive to outliers and doesn't require any distributional assumptions. For non-normal data, the IQR often provides a more robust measure of spread.
11. Can IQR ever be zero, and what does that mean?
Yes, the IQR can be zero. This happens only when all data points in the dataset are identical. It indicates that there is absolutely no variability in the data.
