A set of observations is divided into four equal parts by quartiles.
Quartiles are essential statistical measures that segment a dataset into specific proportions, offering valuable insights into its distribution. As defined, "The quartile is defined as the middle number between the smallest number and the median of the data set. They are those values which divide the total set of data into four equal parts."
Understanding Quartiles
While the provided definition specifically describes the first quartile (Q1), it's crucial to understand that there are three main quartiles that facilitate this division:
- First Quartile (Q1) or Lower Quartile: This is the median of the lower half of the dataset, excluding the median itself if the total number of data points is odd. It represents the 25th percentile, meaning 25% of the data falls below this value. This aligns with the reference: "the middle number between the smallest number and the median of the data set."
- Second Quartile (Q2) or Median: This is the middle value of the entire dataset. It represents the 50th percentile, with 50% of the data falling below it.
- Third Quartile (Q3) or Upper Quartile: This is the median of the upper half of the dataset, excluding the median itself if the total number of data points is odd. It represents the 75th percentile, meaning 75% of the data falls below this value (or 25% of the data falls above it).
The Four Equal Parts Created by Quartiles
These three quartiles (Q1, Q2, and Q3) act as dividing points, splitting the ordered dataset into four segments, each containing 25% of the data.
Here's how the four parts are formed:
| Part Number | Description | Data Range | Percentage of Data |
|---|---|---|---|
| Part 1 | From the Minimum to Q1 | Smallest Value to Q1 | 25% |
| Part 2 | From Q1 to Q2 (Median) | Q1 to Median | 25% |
| Part 3 | From Q2 (Median) to Q3 | Median to Q3 | 25% |
| Part 4 | From Q3 to the Maximum | Q3 to Largest Value | 25% |
Practical Example
Let's consider a dataset of 12 numbers representing monthly sales figures:
[10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65]
- Order the data: The data is already ordered.
- Find Q2 (Median): With 12 data points, the median is the average of the 6th and 7th values:
(35 + 40) / 2 = 37.5. So, Q2 = 37.5. - Find Q1 (Lower Quartile): This is the median of the lower half of the data (
[10, 15, 20, 25, 30, 35]). The median of these 6 values is(25 + 30) / 2 = 27.5. So, Q1 = 27.5. - Find Q3 (Upper Quartile): This is the median of the upper half of the data (
[40, 45, 50, 55, 60, 65]). The median of these 6 values is(50 + 55) / 2 = 52.5. So, Q3 = 52.5.
Now, let's see the four parts:
- Part 1: From
10(Min) to27.5(Q1) - Contains values: 10, 15, 20, 25 - Part 2: From
27.5(Q1) to37.5(Q2) - Contains values: 30, 35 - Part 3: From
37.5(Q2) to52.5(Q3) - Contains values: 40, 45, 50 - Part 4: From
52.5(Q3) to65(Max) - Contains values: 55, 60, 65
Each of these parts contains 3 data points, which is 25% of the total 12 data points, demonstrating the division into four equal parts.
Why Quartiles Matter
Quartiles provide a robust way to understand the spread and central tendency of data, especially when dealing with skewed distributions or outliers. They are crucial for:
- Identifying Spread: The Interquartile Range (IQR), the difference between Q3 and Q1 (Q3 - Q1), measures the spread of the middle 50% of the data, making it less sensitive to extreme values than the full range.
- Detecting Outliers: Data points significantly outside the range of (Q1 - 1.5 IQR) and (Q3 + 1.5 IQR) are often considered outliers.
- Data Visualization: Quartiles are fundamental to creating box plots, which visually represent the distribution of a dataset by showing its median, quartiles, and potential outliers.
In summary, quartiles divide a dataset into four distinct, equally sized segments, each representing 25% of the total data points, enabling a comprehensive understanding of data distribution.
