To calculate the interquartile range (IQR) and identify outliers for the provided dataset, we can follow these steps.
Given data:
\[ 20, 25, 25, 30, 30, 35, 35, 38, 40, 60 \]
Step 1: Sort the data
The data is already sorted: \[ 20, 25, 25, 30, 30, 35, 35, 38, 40, 60 \]
Step 2: Calculate Q1 and Q3
- Q1 (First quartile) is the median of the first half of the dataset.
First half: \[ 20, 25, 25, 30, 30 \] (5 values)
Median of this half (the third element): \[ Q1 = 25 \]
- Q3 (Third quartile) is the median of the second half of the dataset.
Second half: \[ 35, 35, 38, 40, 60 \] (5 values)
Median of this half (the third element): \[ Q3 = 38 \]
Step 3: Calculate IQR
\[ \text{IQR} = Q3 - Q1 = 38 - 25 = 13 \]
Step 4: Identify outliers
Outliers are usually identified using the following formulas:
- Lower Bound: \( Q1 - 1.5 \cdot \text{IQR} \)
- Upper Bound: \( Q3 + 1.5 \cdot \text{IQR} \)
Calculating the bounds:
-
Lower Bound: \[ Q1 - 1.5 \cdot \text{IQR} = 25 - 1.5 \cdot 13 = 25 - 19.5 = 5.5 \]
-
Upper Bound: \[ Q3 + 1.5 \cdot \text{IQR} = 38 + 1.5 \cdot 13 = 38 + 19.5 = 57.5 \]
Outliers
- Any value lower than 5.5 or higher than 57.5 is considered an outlier.
- In the dataset, the only outlier is:
- \( 60 \) (since it is greater than 57.5)
Summary
- Q1 = 25
- Q3 = 38
- IQR = 13
- Values lower than 5.5 and higher than 57.5 are considered outliers.
- Outliers in the dataset: 60
Let me know if you need further assistance!