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What is the Interquartile Range?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
The Interquartile Range (IQR) is a measure of how spread out the middle 50% of your data is. It tells you the range between the 25th percentile (first quartile, Q1) and the 75th percentile (third quartile, Q3) of a dataset. A smaller IQR means the middle data points are closer together.
Simple Example
Quick Example
Imagine you have the marks of 10 students in a Maths test. The IQR would tell you how much the marks vary for the middle half of the students. If the IQR is small, most students in the middle scored similar marks, like between 60 and 70. If it's large, their marks are more spread out, like between 40 and 90.
Worked Example
Step-by-Step
Let's find the IQR for the following daily chai prices (in Rupees) at 9 different stalls: 10, 12, 15, 15, 18, 20, 22, 25, 30.
STEP 1: Arrange the data in ascending order. (It's already done here: 10, 12, 15, 15, 18, 20, 22, 25, 30)
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STEP 2: Find the Median (Q2). For 9 data points, the median is the (9+1)/2 = 5th value. The 5th value is 18.
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STEP 3: Find Q1 (First Quartile). This is the median of the lower half of the data (excluding the overall median if n is odd). Lower half: 10, 12, 15, 15. The median of these 4 values is (12+15)/2 = 13.5. So, Q1 = 13.5.
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STEP 4: Find Q3 (Third Quartile). This is the median of the upper half of the data. Upper half: 20, 22, 25, 30. The median of these 4 values is (22+25)/2 = 23.5. So, Q3 = 23.5.
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STEP 5: Calculate the IQR. IQR = Q3 - Q1.
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IQR = 23.5 - 13.5 = 10.
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ANSWER: The Interquartile Range (IQR) is 10 Rupees.
Why It Matters
The IQR is super useful in fields like AI/ML to understand data spread and find unusual data points (outliers). In FinTech, it helps analyze stock price volatility, while in medicine, it can show the range of patient responses to a new drug. Data scientists and financial analysts use it daily to make informed decisions.
Common Mistakes
MISTAKE: Not arranging the data in order before finding quartiles. | CORRECTION: Always sort your data from smallest to largest first. This is the crucial first step.
MISTAKE: Including the median in both the lower and upper halves when finding Q1 and Q3 for an odd number of data points. | CORRECTION: If the total number of data points (n) is odd, exclude the overall median when splitting the data into lower and upper halves to find Q1 and Q3.
MISTAKE: Confusing IQR with the full range (max - min). | CORRECTION: The IQR only measures the spread of the middle 50% of the data, ignoring extreme values, making it less affected by outliers.
Practice Questions
Try It Yourself
QUESTION: Find the IQR for the following number of push-ups done by 7 friends: 10, 18, 12, 25, 15, 20, 13. | ANSWER: Q1 = 12.5, Q3 = 21.5, IQR = 9
QUESTION: A class of 10 students scored these marks in a science quiz: 50, 65, 70, 72, 75, 80, 82, 85, 90, 95. Calculate the Interquartile Range. | ANSWER: Q1 = 70, Q3 = 85, IQR = 15
QUESTION: The daily mobile data usage (in GB) for a week is: 1.5, 2.0, 1.2, 3.0, 1.8, 2.5, 1.0. If the usage on the 8th day was 4.0 GB, how does the IQR change if you include this new data point? | ANSWER: Original IQR: Q1=1.2, Q3=2.5, IQR=1.3. New data (sorted): 1.0, 1.2, 1.5, 1.8, 2.0, 2.5, 3.0, 4.0. New Q1=(1.2+1.5)/2=1.35. New Q3=(2.5+3.0)/2=2.75. New IQR=2.75-1.35=1.4. The IQR increases from 1.3 to 1.4.
MCQ
Quick Quiz
What does a small Interquartile Range (IQR) indicate about a dataset?
The data is very spread out.
The middle 50% of the data points are close to each other.
There are many outliers in the data.
The average of the data is very high.
The Correct Answer Is:
B
A small IQR means the difference between the 25th and 75th percentiles is small, indicating that the middle 50% of the data points are tightly clustered together. Options A, C, and D describe different aspects of data or misinterpret the IQR's meaning.
Real World Connection
In the Real World
In cricket, analysts use IQR to compare the consistency of batsmen. If Virat Kohli's scores have a small IQR, it means his middle 50% of scores are very consistent. Similarly, in e-commerce, companies like Flipkart might use IQR to understand the typical price range of a product category, helping them set competitive prices and identify abnormally priced items.
Key Vocabulary
Key Terms
QUARTILE: A value that divides a dataset into four equal parts | MEDIAN: The middle value of a dataset when arranged in order (Q2) | OUTLIER: A data point that is significantly different from other data points | DATA SPREAD: How varied or dispersed the data values are | PERCENTILE: A measure indicating the value below which a given percentage of observations in a group of observations falls
What's Next
What to Learn Next
Now that you understand IQR, you can explore other measures of data spread like Standard Deviation and Variance. These concepts will help you analyze data more deeply and are essential for understanding advanced topics in statistics and machine learning.
