# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/10/mathematics/statistics/quartile-data-single
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/statistics/quartile-data-single/en.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "Quartiles for Ungrouped Data",
  description: "Calculate Q1, Q2, Q3 quartiles for ungrouped data with simple position formulas. Learn to divide sorted data into four equal parts step-by-step.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/22/2025",
  subject: "Statistics",
};

## What Are Quartiles?

The median is like a ruler that divides data into two equal parts, right in the middle (50%). Well, there's another friend of the median, called **quartiles**.

If the median divides data into two, quartiles are even better; they divide the sorted data into **four** equal parts! Imagine you have a chocolate bar, and you break it into four equal pieces. Quartiles are the breaking points.

There are three quartile breaking points:

1.  **Lower Quartile (<InlineMath math="Q_1" />):** This is the first break. It separates the smallest 25% of the data from the rest. Like the first quarter of the chocolate.
2.  **Middle Quartile (<InlineMath math="Q_2" />):** This is the **median**! It's exactly in the middle, dividing the data in half (50% left, 50% right). Like the break in the middle of the chocolate.
3.  **Upper Quartile (<InlineMath math="Q_3" />):** This is the last break. It separates the smallest 75% of the data from the largest 25%. Like the boundary after three-quarters of the chocolate.

So, <InlineMath math="Q_1" />, <InlineMath math="Q_2" />, and <InlineMath math="Q_3" /> divide our data into four small groups with the same number of data points (25% each).

## How to Find the Position of Quartiles

Okay, now how do we know the position (rank) of <InlineMath math="Q_1" />, <InlineMath math="Q_2" />, and <InlineMath math="Q_3" /> in our ordered data?

Assume we have <InlineMath math="n" /> data points that we have sorted from smallest to largest.

### Q1 (Lower Quartile)

The formula is simple:

<BlockMath math="\text{Position of } Q_1 = \text{Data point at }\frac{1}{4}(n+1)" />

- **If the result is a whole number**, for example 5, then <InlineMath math="Q_1" /> is the value of the 5th data point.
- **If the result has a decimal**, for example <InlineMath math="5.25" />, then <InlineMath math="Q_1" /> lies between the 5th and 6th data points. (There's a way to calculate its value later, but for now, we're just finding the position).

**Simple Example:**

Suppose we have 20 data points (<InlineMath math="n=20" />).

Position of <InlineMath math="Q_1" /> = Data point at <InlineMath math="\frac{1}{4}(20+1)" /> = Data point at <InlineMath math="\frac{21}{4}" /> = Data point at 5.25.

This means <InlineMath math="Q_1" /> is between the 5th and 6th data points.

### Q2 (Median or Middle Quartile)

This is the median, so the formula is:

<BlockMath math="\text{Position of } Q_2 = \text{Data point at }\frac{1}{2}(n+1)" />

The rules are the same as for <InlineMath math="Q_1" />:

- **If the result is a whole number**, say 10, <InlineMath math="Q_2" /> is the value of the 10th data point.
- **If the result has a decimal**, say 10.5, <InlineMath math="Q_2" /> is between the 10th and 11th data points.

**Simple Example (<InlineMath math="n=20" />):**

Position of <InlineMath math="Q_2" /> = Data point at <InlineMath math="\frac{1}{2}(20+1)" /> = Data point at <InlineMath math="\frac{21}{2}" /> = Data point at 10.5.

This means <InlineMath math="Q_2" /> (the median) is between the 10th and 11th data points.

### Q3 (Upper Quartile)

The formula is similar again:

<BlockMath math="\text{Position of } Q_3 = \text{Data point at }\frac{3}{4}(n+1)" />

The rules are exactly the same:

- **If the result is a whole number**, say 15, <InlineMath math="Q_3" /> is the value of the 15th data point.
- **If the result has a decimal**, say 15.75, <InlineMath math="Q_3" /> is between the 15th and 16th data points.

**Simple Example (<InlineMath math="n=20" />):**

Position of <InlineMath math="Q_3" /> = Data point at <InlineMath math="\frac{3}{4}(20+1)" /> = Data point at <InlineMath math="\frac{63}{4}" /> = Data point at 15.75.

This means <InlineMath math="Q_3" /> is between the 15th and 16th data points.

## Exercise

Try to find the position of <InlineMath math="Q_1" />, <InlineMath math="Q_2" />, and <InlineMath math="Q_3" /> from the math test scores of these 7 children:

**Scores:** 7, 5, 8, 6, 9, 7, 10

**Step 1: Sort the data first!**

Sorted data: 5, 6, 7, 7, 8, 9, 10

Number of data points (<InlineMath math="n" />) = 7

**Step 2: Find the quartile positions using the formulas**

- **Position of <InlineMath math="Q_1" />:**

  <InlineMath math="\text{Data point at }\frac{1}{4}(n+1) = \text{Data point at }\frac{1}{4}(7+1) = \text{Data point at }\frac{8}{4} = \text{Data point at }2" />

  The result is a whole number (2), so <InlineMath math="Q_1" /> is the 2nd data point.

- **Position of <InlineMath math="Q_2" /> (Median):**

  <InlineMath math="\text{Data point at }\frac{1}{2}(n+1) = \text{Data point at }\frac{1}{2}(7+1) = \text{Data point at }\frac{8}{2} = \text{Data point at }4" />

  The result is a whole number (4), so <InlineMath math="Q_2" /> is the 4th data point.

- **Position of <InlineMath math="Q_3" />:**

  <InlineMath math="\text{Data point at }\frac{3}{4}(n+1) = \text{Data point at }\frac{3}{4}(7+1) = \text{Data point at }\frac{24}{4} = \text{Data point at }6" />

  The result is a whole number (6), so <InlineMath math="Q_3" /> is the 6th data point.

**Step 3: Determine the quartile values**

Look at the sorted data: 5, 6, 7, 7, 8, 9, 10

- <InlineMath math="Q_1" /> = 2nd data point = **6**
- <InlineMath math="Q_2" /> = 4th data point = **7**
- <InlineMath math="Q_3" /> = 6th data point = **9**

## The Fourth Quartile (Q4)

You might be wondering, "If there's <InlineMath math="Q_1" />, <InlineMath math="Q_2" />, and <InlineMath math="Q_3" />, is there a <InlineMath math="Q_4" />?"

Technically, the concept of quartiles divides the data into four parts. <InlineMath math="Q_1" /> is the boundary for the first 25%, <InlineMath math="Q_2" /> (the median) is the 50% boundary, and <InlineMath math="Q_3" /> is the 75% boundary. The final boundary, which encompasses 100% of the data, is actually the **maximum value** of the dataset.

So, while we could refer to the maximum value as "<InlineMath math="Q_4" />", in statistical analysis, we don't typically use the term <InlineMath math="Q_4" /> explicitly. The main focus is on <InlineMath math="Q_1" />, <InlineMath math="Q_2" />, and <InlineMath math="Q_3" /> because they provide important information about the spread and center of the data in the lower, middle, and upper sections. The minimum value is sometimes called "<InlineMath math="Q_0" />", but like <InlineMath math="Q_4" />, it's less commonly used than <InlineMath math="Q_1" />, <InlineMath math="Q_2" />, and <InlineMath math="Q_3" />.