3.2 Quartiles, Percentiles, Range and Interquartile Range We are learning to... - Understand how to find the range of a data set - Understand the purpose of interquartile range as a measure of spread - Understand quartiles as specific percentiles - Use statistical measures to compare two data sets I know I have succeeded if I can... Find the range of a set of data Find the interquartile range of a set of data - Find the percentile value of a set of data Solve reverse problems involving percentile of a set of data Apply statistical measures to solve problems Introduction: In a set of data, range = the largest value - smallest value. The interquartile range of a set of data is the difference between the upper quartile $$ \left(\mathrm{Q}_{3}\right) $$ an the lower quartile $$ \left(\mathrm{Q}_{1}\right) $$. Interquartile range $$ =\mathrm{Q}_{3}-\mathrm{Q}_{1} $$

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To help you understand how to find the range, interquartile range, and percentiles of a data set, let's break down the concepts step by step.

### 1. Finding the Range of a Data Set
The range of a data set is calculated using the formula:
$$
\text{Range} = \text{Largest Value} - \text{Smallest Value}
$$
**Example:**
Consider the data set: $$ [3, 7, 8, 5, 12] $$

- Largest Value = 12
- Smallest Value = 3

Calculating the range:
$$
\text{Range} = 12 - 3 = 9
$$

### 2. Understanding Quartiles and Interquartile Range
Quartiles divide a data set into four equal parts. The lower quartile $$ (Q_1) $$ is the median of the first half of the data, and the upper quartile $$ (Q_3) $$ is the median of the second half. The interquartile range (IQR) is the difference between these two quartiles:
$$
\text{Interquartile Range} = Q_3 - Q_1
$$

**Example:**
Using the same data set $$ [3, 5, 7, 8, 12] $$:
1. **Order the data** (already ordered).
2. **Find $$ Q_1 $$**: The median of the first half $$ [3, 5] $$ is $$ 4 $$.
3. **Find $$ Q_3 $$**: The median of the second half $$ [7, 8, 12] $$ is $$ 8 $$.

Calculating the interquartile range:
$$
\text{IQR} = Q_3 - Q_1 = 8 - 4 = 4
$$

### 3. Finding Percentiles
Percentiles indicate the relative standing of a value within a data set. The $$ k $$-th percentile is the value below which $$ k \% $$ of the data falls.

**Example:**
To find the 50th percentile (median) of the data set $$ [3, 5, 7, 8, 12] $$:
- The median is $$ 7 $$ (as it is the middle value).

### 4. Reverse Problems Involving Percentiles
To find the value corresponding to a specific percentile, you can use the formula:
$$
P_k = \frac{k}{100} \times (n + 1)
$$
where $$ P_k $$ is the position of the $$ k $$-th percentile and $$ n $$ is the number of data points.

**Example:**
To find the 25th percentile in the data set $$ [3, 5, 7, 8, 12] $$:
- $$ n = 5 $$
- $$ P_{25} = \frac{25}{100} \times (5 + 1) = 1.5 $$

This means the 25th percentile is between the 1st and 2nd values in the ordered data set, which are $$ 3 $$ and $$ 5 $$. Thus, the 25th percentile is $$ 4 $$.

### 5. Applying Statistical Measures to Compare Two Data Sets
When comparing two data sets, you can calculate the range, IQR, and percentiles for each set and analyze how they differ in terms of spread and central tendency.

**Example:**
Data Set A: $$ [3, 5, 7, 8, 12] $$  
Data Set B: $$ [2, 4, 6, 10, 14] $$

- Calculate the range and IQR for both sets.
- Compare the results to understand which data set has a greater spread.

By following these steps, you can effectively analyze and compare data sets using statistical measures. If you have specific data sets you'd like to analyze, please provide them, and I can help you calculate the range, IQR, and percentiles.
To find the range, interquartile range, and percentiles of a data set, follow these steps: 1. **Range**: Subtract the smallest value from the largest value. 2. **Interquartile Range (IQR)**: Find the first quartile (Q₁) and third quartile (Q₃), then subtract Q₁ from Q₃. 3. **Percentiles**: Determine the value below which a certain percentage of data falls. By calculating these measures, you can understand the spread and central tendency of your data.

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