Identifying and understanding quartiles and outliers: A guide with box plot examples

Understanding Quartiles: First Quartile (Q1) and Third Quartile (Q3) Explained with Examples

The first quartile (Q1) and the third quartile (Q3) are measures from descriptive statistics that divide a dataset into four equal parts. They help to describe the spread and central tendency of the data.

First Quartile (Q1)

The first quartile, also known as the lower quartile, is the median of the lower half of the dataset. It is the value that separates the lowest 25% of the data from the rest. In other words, 25% of the data points lie below Q1.

Third Quartile (Q3)

The third quartile, also known as the upper quartile, is the median of the upper half of the dataset. It is the value that separates the lowest 75% of the data from the highest 25%. In other words, 75% of the data points lie below Q3, and 25% lie above Q3.

Example

Consider the following dataset:
[ 4, 8, 15, 16, 23, 42, 50 ]

1. Find the median (Q2): The median is the middle value of the dataset. For this dataset, the median is 16.
2. Divide the data into halves:

• Lower half (excluding the median): 4, 8, 15
• Upper half (excluding the median): 23, 42, 50

1. Find Q1 (median of the lower half):

• The lower half is 4, 8, 15. The median of these values is 8. So, Q1 = 8.

1. Find Q3 (median of the upper half):

• The upper half is 23, 42, 50. The median of these values is 42. So, Q3 = 42.

Summary

• Q1 (First Quartile): 8
• Q3 (Third Quartile): 42

These quartiles help in understanding the distribution of the data, identifying outliers, and analyzing the spread of the dataset.

Identifying outliers in box plots using the interquartile range (IQR) method

In statistics, an outlier is typically defined as a data point that lies significantly outside the range of the rest of the data. For a box plot, the most common criterion used to declare a value as an outlier is the interquartile range (IQR) method. Here’s how it works:

1. Calculate the IQR: The IQR is the difference between the first quartile (Q1) and the third quartile (Q3):
   [
   \text{IQR} = Q3 – Q1
   ]
2. Determine the outlier boundaries:

• The lower boundary is calculated as:
  [
  \text{Lower Bound} = Q1 – 1.5 \times \text{IQR}
  ]
• The upper boundary is calculated as:
  [
  \text{Upper Bound} = Q3 + 1.5 \times \text{IQR}
  ]

1. Identify outliers:

• Any data point below the lower boundary is considered a lower outlier.
• Any data point above the upper boundary is considered an upper outlier.

These boundaries are sometimes called the “fences.” Values that fall outside these fences are typically marked as outliers and are represented as individual points on a box plot. The whiskers of the box plot extend to the smallest and largest values within the lower and upper boundaries, respectively.

Example

Suppose you have the following dataset: [5, 7, 8, 9, 10, 11, 12, 13, 15, 16, 25]

Let’s correct the calculation step-by-step:

Dataset:

[ [5, 7, 8, 9, 10, 11, 12, 13, 15, 16, 25] ]

Calculate Q1 and Q3:

1. Arrange the data in ascending order:
   (Already in ascending order)
   [ [5, 7, 8, 9, 10, 11, 12, 13, 15, 16, 25] ]
2. Calculate Q1 (the first quartile):
   The first quartile (Q1) is the median of the first half of the data (excluding the median if the number of data points is odd).
   [ [5, 7, 8, 9, 10] ]
   The median of this subset is 8.
   [ Q1 = 8 ]
3. Calculate Q3 (the third quartile):
   The third quartile (Q3) is the median of the second half of the data (excluding the median if the number of data points is odd).
   [ [12, 13, 15, 16, 25] ]
   The median of this subset is 15.
   [ Q3 = 15 ]

Calculate the IQR:

[ \text{IQR} = Q3 – Q1 = 15 – 8 = 7 ]

Determine the outlier boundaries:

• Lower Bound:
  [ Q1 – 1.5 \times \text{IQR} = 8 – 1.5 \times 7 = 8 – 10.5 = -2.5 ]
• Upper Bound:
  [ Q3 + 1.5 \times \text{IQR} = 15 + 1.5 \times 7 = 15 + 10.5 = 25.5 ]

Box plot: Easiest way to detect anomalies?
byu/DigitalSplendid inAskStatistics

Identify outliers:

• Any data point below -2.5 is a lower outlier.
• Any data point above 25.5 is an upper outlier.

In this dataset, no value is below -2.5, and no value is above 25.5. Therefore, there are no outliers according to the IQR method.

Comment
byu/DigitalSplendid from discussion
inAskStatistics

Using this method ensures that values significantly deviating from the central range are identified and treated as outliers.

Comment
byu/DigitalSplendid from discussion
inAskStatistics

Finding Q1, median, Q3 through formula on Excel
byu/DigitalSplendid inexcel

How to make formula applicable for the whole column
byu/DigitalSplendid inexcel

Disclaimer: This article was generated with the assistance of large language models, including ChatGPT and Google Gemini. While I (the author) provided the direction and topic, these AI tools helped with research, content creation, and phrasing.