How to Find Quartiles Formula Steps and Solved Examples
The concept of quartiles plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Quartiles help us organize large sets of numbers into manageable pieces, making it easier to see patterns, identify outliers, and compare data. Students often encounter quartiles in statistics, competitive exams, and even in daily problem-solving involving data.
What Is Quartile?
A quartile is defined as a statistical value that divides a dataset into four equal sections, each containing a quarter of the data. You’ll find this concept applied in areas such as data analysis, exam score breakdowns, and practical statistics for sciences. The three main quartiles are:
- Lower Quartile (Q1): The value below which 25% of the data falls.
- Median or Second Quartile (Q2): The 50th percentile, dividing the data into two halves.
- Upper Quartile (Q3): The value below which 75% of the data falls.
Key Formula for Quartiles
Here’s the standard formula for quartiles in an ordered dataset with ‘n’ items:
| Quartile | Formula | Position |
|---|---|---|
| Q1 (Lower Quartile) | \( Q_1 = \dfrac{(n+1)}{4} \)th item | 25% |
| Q2 (Median) | \( Q_2 = \dfrac{(n+1)}{2} \)th item | 50% |
| Q3 (Upper Quartile) | \( Q_3 = \dfrac{3(n+1)}{4} \)th item | 75% |
Cross-Disciplinary Usage
Quartiles are not only useful in Maths but also play an important role in Physics, Computer Science, Economics, data science, and daily logical reasoning. For example, students preparing for JEE, NEET, or board exams frequently use quartiles to analyze score distributions, understand box-and-whisker plots, and evaluate surveys in social sciences. Quartiles also help in identifying outliers in big datasets.
Step-by-Step Illustration
Let’s calculate quartiles with a quick example: Find the quartiles for the data set: 5, 7, 9, 12, 15, 18, 21, 24.
2. Count the number of terms: n = 8.
3. Use the formulas:
2nd value + 0.25 × (3rd value – 2nd value): 7 + 0.25×(9–7) = 7.5
- Q2 position (Median): \( (8+1)/2 = 4.5 \)th item
4th value + 0.5 × (5th value – 4th value): 12 + 0.5×(15–12) = 13.5
- Q3 position: \( 3(8+1)/4 = 6.75 \)th item
6th value + 0.75 × (7th value – 6th value): 18 + 0.75×(21–18) = 20.25
Thus, Q1 = 7.5, Q2 = 13.5, Q3 = 20.25.
Speed Trick or Vedic Shortcut
Here’s a quick shortcut for quartiles in a small dataset: If your list is short, simply split the data into four sections after sorting. The number at each split gives you Q1, Q2, and Q3. For odd-sized lists, interpolate (find the average between two numbers) at the split points. This method is handy for quick MCQs and revision under time pressure.
Example Trick: For a list of 9 numbers, Q1 is the 3rd number, Median is the 5th, and Q3 is the 7th value in the ordered list. Memorizing this can save precious seconds in exams!
Tricks like this are frequently taught in Vedantu’s online Maths classes to help you solve more questions in less time.
Try These Yourself
- Find the quartiles for the dataset: 10, 12, 14, 18, 20, 22, 25, 28.
- Which quartile equals the median?
- If Q1=21 in a test, what does it mean for the top 25% of students?
- Calculate the interquartile range for: 3, 7, 9, 10, 15, 18, 19, 22.
Frequent Errors and Misunderstandings
- Forgetting to order the data before applying formulas.
- Mixing up quartiles with percentiles, deciles, or the mean.
- Incorrectly handling datasets with even or odd numbers of terms.
- Assuming Q3 is always the highest number (it’s not unless the dataset is uniform).
Relation to Other Concepts
The idea of quartiles connects closely with other topics in statistics such as mean, median, interquartile range, and variance. Learning quartiles helps in visualizing box plots, understanding data spread, and mastering advanced concepts in probability and data handling.
Classroom Tip
A quick way to remember quartiles: “Q1 cuts below, Q2 is the middle, Q3 cuts above.” Students can draw a number line or graph to visually ‘chop’ the dataset into four. Vedantu’s teachers often use color-coding or block diagrams for quartiles during live sessions, making the concept clear and memorable.
We explored quartiles—from definition, formula, examples, mistakes, and connections to other subjects. Continue practicing with Vedantu to become confident in solving problems using this concept. Don’t forget to check related topics like median and box and whiskers plots to build a complete understanding!
FAQs on Quartiles in Statistics Explained Simply
1. What are quartiles in statistics?
Quartiles are the three values that divide an ordered data set into four equal parts. The three quartiles are:
- Q1 (First Quartile): separates the lowest 25% of the data.
- Q2 (Second Quartile): the median, separates the lowest 50%.
- Q3 (Third Quartile): separates the lowest 75% of the data.
2. How do you calculate quartiles step by step?
To calculate quartiles, first arrange the data in ascending order and then find Q1, Q2, and Q3 using median positions. Follow these steps:
- Step 1: Arrange data from smallest to largest.
- Step 2: Find Q2 (median).
- Step 3: Find Q1 as the median of the lower half (excluding Q2 if odd).
- Step 4: Find Q3 as the median of the upper half.
3. What is the formula for quartiles?
The position of quartiles in ungrouped data can be found using formulas based on (n + 1). The formulas are:
- Q1 position = (n + 1) / 4
- Q2 position = (n + 1) / 2
- Q3 position = 3(n + 1) / 4
4. What is the difference between quartiles and percentiles?
Quartiles divide data into four equal parts, while percentiles divide data into 100 equal parts. Specifically:
- Quartiles: Q1 (25%), Q2 (50%), Q3 (75%).
- Percentiles: P1 to P99 divide data into 100 parts.
5. What is the interquartile range (IQR)?
The interquartile range (IQR) is the difference between the third and first quartiles. The formula is IQR = Q3 − Q1. It measures the spread of the middle 50% of data and helps identify variability. For example, if Q3 = 18 and Q1 = 10, then IQR = 18 − 10 = 8.
6. How do you find quartiles for grouped data?
Quartiles for grouped data are calculated using the quartile class and a specific formula. The formula is:
- Qk = L + [(kN/4 − cf) / f] × h
- L = lower boundary of quartile class
- N = total frequency
- cf = cumulative frequency before quartile class
- f = frequency of quartile class
- h = class width
7. Why are quartiles important in statistics?
Quartiles are important because they measure data spread and help detect outliers. They are useful for:
- Summarizing distributions
- Constructing box and whisker plots
- Calculating interquartile range (IQR)
- Identifying skewness and variability
8. How do you find Q1, Q2, and Q3 in an odd-numbered data set?
In an odd-numbered data set, Q2 is the middle value, and Q1 and Q3 are medians of the lower and upper halves excluding Q2. Example: For 1, 3, 5, 7, 9:
- Q2 = 5
- Lower half: 1, 3 → Q1 = 2nd value = 3
- Upper half: 7, 9 → Q3 = 2nd value = 7
9. How do quartiles relate to a box and whisker plot?
Quartiles form the key components of a box and whisker plot. In a box plot:
- The left edge of the box is Q1.
- The line inside the box is Q2 (median).
- The right edge of the box is Q3.
- The box length represents the IQR.
10. Can you give a simple example of finding quartiles?
Yes, quartiles can be found by ordering the data and calculating medians of halves. Example: Data = 4, 8, 10, 12, 14, 18, 20.
- Step 1: Median (Q2) = 12.
- Step 2: Lower half = 4, 8, 10 → Q1 = 8.
- Step 3: Upper half = 14, 18, 20 → Q3 = 18.
