What is Location of Quartiles?

S3-SA3-0305

Grade Level:

Class 8

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The location of quartiles tells us where to find the three points (Q1, Q2, Q3) that divide a sorted list of data into four equal parts. Think of it like cutting a long ribbon into four equal pieces. These locations help us understand the spread of data.

Simple Example

Quick Example

Imagine you have the marks of 9 students in a science test: 30, 45, 50, 55, 60, 65, 70, 75, 80. To find the location of the median (Q2), you'd find the middle mark. For Q1 and Q3, you'd find the middle of the first half and second half of the marks, respectively.

Worked Example

Step-by-Step

Let's find the location of quartiles for the number of runs scored by a cricketer in 11 matches: 10, 25, 30, 40, 50, 60, 70, 80, 90, 100, 110.

STEP 1: Arrange the data in ascending order. (It's already sorted here: 10, 25, 30, 40, 50, 60, 70, 80, 90, 100, 110). The total number of data points (n) is 11.

STEP 2: Find the location of the Median (Q2). The formula for the position of Q2 is (n+1)/2. So, (11+1)/2 = 12/2 = 6. The 6th data point is Q2.

STEP 3: Identify Q2. The 6th data point is 60. So, Q2 = 60.

STEP 4: Find the location of the First Quartile (Q1). Q1 is the median of the lower half of the data (before Q2). The lower half is: 10, 25, 30, 40, 50. Here, n=5. The position of Q1 is (5+1)/2 = 3. The 3rd data point in the lower half is Q1.

STEP 5: Identify Q1. The 3rd data point in the lower half is 30. So, Q1 = 30.

STEP 6: Find the location of the Third Quartile (Q3). Q3 is the median of the upper half of the data (after Q2). The upper half is: 70, 80, 90, 100, 110. Here, n=5. The position of Q3 is (5+1)/2 = 3. The 3rd data point in the upper half is Q3.

STEP 7: Identify Q3. The 3rd data point in the upper half is 90. So, Q3 = 90.

Answer: The location of Q1 is the 3rd position (value 30), Q2 is the 6th position (value 60), and Q3 is the 9th position (value 90).

Why It Matters

Understanding quartile locations is crucial for data scientists and economists who analyze large datasets to find trends. It helps engineers understand the performance range of products and even helps in AI/ML to identify outliers in data. These skills are vital for careers in data analytics and financial planning.

Common Mistakes

MISTAKE: Not sorting the data first before finding quartile locations. | CORRECTION: Always arrange the data in ascending (smallest to largest) order before calculating any quartile positions.

MISTAKE: Including the median (Q2) in both the lower and upper halves when finding Q1 and Q3, especially for an odd number of data points. | CORRECTION: If 'n' is odd, exclude the median (Q2) from both the lower and upper halves when calculating Q1 and Q3. If 'n' is even, Q2 is the average of two middle values, so the data naturally splits into two halves without Q2 being a single data point.

MISTAKE: Using the wrong formula for position, especially confusing (n+1)/2 with n/2. | CORRECTION: For the position of Q1, Q2, and Q3, use (n+1)/2 for the median of a given set. Remember that Q1 is the median of the lower half and Q3 is the median of the upper half.

Practice Questions

Try It Yourself

QUESTION: Find the location of Q2 for the following number of passengers in 7 auto-rickshaws: 2, 4, 1, 3, 5, 2, 3. | ANSWER: Sorted data: 1, 2, 2, 3, 3, 4, 5. n=7. Location of Q2 = (7+1)/2 = 4. So, Q2 is the 4th position.

QUESTION: For the data set: 10, 12, 15, 18, 20, 22, 25, 28, 30, 32. Find the location of Q1 and Q3. | ANSWER: n=10. Q2 is between 5th and 6th (10/2=5, 5+1=6). Q2 = (20+22)/2 = 21. Lower half: 10, 12, 15, 18, 20. n=5. Location of Q1 = (5+1)/2 = 3. Q1 is the 3rd position (value 15). Upper half: 22, 25, 28, 30, 32. n=5. Location of Q3 = (5+1)/2 = 3. Q3 is the 3rd position (value 28).

QUESTION: A small shop sold these many samosas each day for 8 days: 45, 50, 40, 60, 55, 70, 65, 35. Find the values of Q1, Q2, and Q3. | ANSWER: Sorted data: 35, 40, 45, 50, 55, 60, 65, 70. n=8. Location of Q2 = (8+1)/2 = 4.5. Q2 is the average of 4th and 5th values = (50+55)/2 = 52.5. Lower half: 35, 40, 45, 50. n=4. Location of Q1 = (4+1)/2 = 2.5. Q1 is the average of 2nd and 3rd values = (40+45)/2 = 42.5. Upper half: 55, 60, 65, 70. n=4. Location of Q3 = (4+1)/2 = 2.5. Q3 is the average of 2nd and 3rd values = (60+65)/2 = 62.5.

MCQ

Quick Quiz

What is the first step to find the location of quartiles for any given data set?

Calculate the mean of the data

Arrange the data in ascending order

Identify the largest and smallest values

Count the number of data points

The Correct Answer Is:

Before finding any quartile, the data must always be arranged from smallest to largest. This ensures the correct middle points are identified.

Real World Connection

In the Real World

In cricket analytics, statisticians use quartiles to understand a batsman's scoring pattern. They might look at the Q1, Q2 (median), and Q3 of a player's runs over a season to see if they consistently score high or have many low scores, helping coaches make strategic decisions.

Key Vocabulary

Key Terms

QUARTILE: A value that divides a data set into four equal parts | MEDIAN (Q2): The middle value of a sorted data set | ASCENDING ORDER: Arranging numbers from smallest to largest | DATA SET: A collection of numbers or values | LOCATION: The position of a quartile within a sorted data set

What's Next

What to Learn Next

Now that you understand where quartiles are located, you can learn about the Interquartile Range (IQR). IQR uses the values of Q1 and Q3 to measure how spread out the middle 50% of your data is, which is a very powerful tool in statistics!