What is the interquartile range of this set of data? 15, 19, 20, 25, 31, 38, 41 6 13 19 26
The Correct Answer and Explanation is :
To find the interquartile range (IQR) of the given data set, we first need to organize the data in ascending order and then determine the quartiles.
Step 1: Organize the Data
The provided data set is:
- 15, 19, 20, 25, 31, 38, 41, 6, 13, 19, 26
First, let’s arrange the numbers in ascending order:
- Ordered Data: 6, 13, 15, 19, 19, 20, 25, 26, 31, 38, 41
Step 2: Determine Quartiles
The quartiles divide the data set into four equal parts:
- Q1 (First Quartile): The median of the first half of the data.
- Q2 (Second Quartile): The median of the entire data set.
- Q3 (Third Quartile): The median of the second half of the data.
Finding Q2 (Median):
- The data set has 11 values. The median (Q2) is the middle value, which is the 6th value:
- Q2 = 20
Finding Q1:
- The first half of the data (before the median) is:
- 6, 13, 15, 19, 19
- Q1 is the median of this half. Since there are 5 values, the median is the 3rd value:
- Q1 = 15
Finding Q3:
- The second half of the data (after the median) is:
- 25, 26, 31, 38, 41
- Q3 is the median of this half. With 5 values, the median is again the 3rd value:
- Q3 = 31
Step 3: Calculate the Interquartile Range (IQR)
The IQR is calculated as:
[
\text{IQR} = Q3 – Q1
]
Substituting in our values:
[
\text{IQR} = 31 – 15 = 16
]
Conclusion
The interquartile range (IQR) of the given data set is 16. The IQR is a measure of statistical dispersion and represents the range within which the central 50% of the data falls, effectively providing insight into the spread of the middle portion of the data while reducing the influence of outliers. In this case, the IQR indicates a moderate spread of the values between the lower and upper quartiles, which can be useful for understanding the variability in the dataset.