The Interquartile Range (IQR) helps us understand the spread of data which focuses on the middle 50% of a dataset. Unlike other measures of spread such as the range which can be heavily influenced by extreme values or outliers, the IQR isolates the central portion of the data. By calculating the difference between the first and third quartiles (Q1 and Q3) it gives us a clearer picture of how data points are distributed around the median. In this article, we'll see how to calculate the IQR and how it can be applied in real-world scenarios.
Some Key Concepts
Before moving into the calculation and applications of the IQR in statistics lets understand some basic concepts in better way:
1. Quartiles:
Quartiles are defined as a statistical measure which divides the given dataset into four equal parts:
- Q1 (First Quartile): This value separates the lowest 25% of the data. It is the median of the lower half of the dataset.
- Q2 (Second Quartile or median of the data): This is the middle value of the dataset which splitts it into two equal parts.
- Q3 (Third Quartile): This value separates the top 25% of the data. It is the median of the upper half of the dataset.
These quartiles allow us to divide the data into four equal parts and the IQR focuses on the range between Q1 and Q3 which captures the middle 50% of the data.
2. Interquartile Range (IQR):
IQR in Statistics is used to measure variability by dividing a data set into quartiles. The data is sorted in ascending order and split into 4 equal parts: Q1, Q2, Q3 called first, second and third quartiles respectively in the given data.
The IQR is simply the difference between the third quartile (Q3) and the first quartile (Q1) which is calculated as:
Interquartile range = Upper Quartile (Q3)– Lower Quartile(Q1)
It tells us how spread out the central 50% of the data is which helps to gauge the data's variability without being influenced by outliers.
Read More about Percentile.
To calculate the IQR, follow these step-by-step instructions:
Step 1: Sort the Data: Arrange the dataset in ascending order.
Step 2: Find the Median (Q2): The median is the middle value of the dataset. If the number of data points is odd, the median is the middle value. If the count is even the median is the average of the two middle values.
Step 3: Finding Q1 and Q3:
- Split the dataset into two halves based on the median.
- Q1 is the median of the lower half (not including the median of the entire dataset) and Q3 is the median of the upper half.
Step 4: Calculate the IQR: Subtract Q1 from Q3:
IQR=Q3−Q1
This gives the spread of the middle 50% of the data.
Example Calculation
We will see how to Calculate the Interquartile Range using an example:
Consider the following dataset of exam scores for a class tenth:
77, 85, 92, 64, 78, 95, 82
1. Sort the data:
64, 77, 78, 82, 85, 92, 95
2. Find the Median (Q2):
Q2=82
3. Divide the data into two halves:
- Lower half: 64, 77, 78
- Upper half: 85, 92, 95
4. Find Q1 and Q3:
- Q1 is the median of the lower half (64, 77, 78) which is 77.
- Q3 is the median of the upper half (85, 92, 95) which is 92.
5. Calculate the IQR:
IQR=Q3−Q1=92−77=15
So the IQR for this dataset is 15.
Understanding Semi Interquartile Range (SIQR)
Semi interquartile range is also known as the Quartile deviation is a measure of how spread out the middle 50% of the data is. It is useful for datasets with skewed distributions and is not affected much by extreme values or outliers.
Key Characteristics:
- SIQR is half of the Interquartile Range (IQR).
- It provides insight into how data is distributed around a central point (the median).
- Extreme values have little impact on the SIQR helps in making it ideal for datasets with outliers.
How to Find Semi Interquartile Range?
The semi interquartile range is calculated by the following steps:-
Step 1: Find Q1: Identify the first quartile (Q1) from the data.
Step 2: Find Q3: Identify the third quartile (Q3) from the data.
Step 3: Subtract Q1 from Q3:
IQR=Q3−Q1
Step 4: Divide by 2: SIQR is half of the IQR:
\text{SIQR} = \frac{Q_3 - Q_1}{2}
Formula:
\text{SIQR} = \frac{1}{2} \times (Q_3 - Q_1)
The IQR Median is the median of the interquartile range which provides a measure of the central tendency for the middle 50% of our data. It minimizes the impact of extreme values helps in providing a more accurate reflection of the data's central distribution.
Relationship between Median and IQR:
- The Median (Q2) is the middle value of the dataset helps in splitting it into two equal parts.
- The IQR represents the range of values that lie between Q1 (25th percentile) and Q3 (75th percentile) capturing the middle 50% of the data.
When dealing with skewed distributions it’s better to use the median (Q2) for central tendency and IQR for variability as these are less affected by extreme outliers.
Applications of the Interquartile Range (IQR)
The Interquartile Range (IQR) has a variety of applications across different fields which includes:
- Outlier Detection: IQR is used in finance, healthcare and quality control to detect outliers. Data points that fall outside the range Q_1 - 1.5 \times IQR \quad \text{to} \quad Q_3 + 1.5 \times IQR
are considered outliers.
- Measure of Variability for Skewed Distributions: Unlike the range, IQR is not sensitive to extreme values or outliers. It is useful for measuring variability in skewed datasets helps in providing a better representation of spread.
- Data Summary and Comparison: It acts as a tool for summarizing data when the dataset is non-normally distributed. It provides a focused view of the data's central 50% which offers valuable insights into data spread and central tendency.
- Predictive Data Analysis: IQR can be applied in predictive analytics where understanding the distribution of data plays an important role in model accuracy and prediction reliability.
- Central Tendency: While the mean can be skewed by extreme values, it focuses on the central 50% of the data which provides a clearer understanding of its true distribution.
You can refer to more related articles:
Solved Examples on Interquartile Range
Example 1: You are given a dataset of the ages of students in a classroom:
18, 19, 20, 21, 22, 35, 13, 23,find the Interquartile Range ?
Solution:
- Arrange in ascending order: 13, 18, 19, 20, 21, 22, 23, 35
- Count the given values i.e is 8 so median is average of two numbers median =20+21/2 = 20.5
- Lower half is 13, 18, 19, 20
- Median of the lower half (Q1)= (18+ 19) / 2 = 18.5
- Upper half is 21, 22, 23, 35
- Median of the Upper half (Q3)= (22+ 23) / 2 = 22.5
- Finally
IQR = Q3 - Q1 = 22.5-18.5 = 4
Example 2: The age of a group of young gymnasts are 4, 5, 6, 3, 12, 14, 15, 13 Find the interquartile range and the semi-interquartile range?
Solution:
- Arrange in ascending order:3,4,5,6,12,13,14,15
- Count the given values i.e is 8 so median is average of two numbers median =6+12/2= 9
- Lower half is 3,4,5
- Median of the lower half (Q1)= 4
- Upper half is 13,14,15
- Median of the Upper half (Q3)= 14
- Finally
IQR = Q3 - Q1 =14-4 = 10
Semi Interquartile Range = IQR/2 = 10/2 = 5.
Practice Questions On Interquartile Range
Q1. Calculate the Interquartile Range for the following dataset: 12, 15, 20, 25, 30, 35, 40, 45?
Q2. A dataset of temperatures in degrees Celsius for a week is given as follows: 18, 22, 20, 25, 19, 28, 17. Find the Interquartile Range?
Q3. You have a dataset of the heights (in inches) of a group of individuals: 62, 67, 71, 68, 70, 75, 61, 66, 69, 70. Determine the Interquartile Range of heights?
By understanding and applying the Interquartile Range (IQR) we can gain deeper insights into the data distribution, manage outliers effectively and make more informed decisions based on the central tendencies of our dataset.
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