Given the data set of $X = \{3, 5, 5, 6, 9, 11\}$, find the IQR.

Sort the data set: X = { 3 , 5 , 5 , 6 , 9 , 11 } .
Find the median: 2 5 + 6 ​ = 5.5 .
Find Q1 (median of lower half { 3 , 5 , 5 } ): Q 1 = 5 .
Find Q3 (median of upper half { 6 , 9 , 11 } ): Q 3 = 9 .
Calculate IQR: I QR = Q 3 − Q 1 = 9 − 5 = 4 ​ .

Explanation

Understand the problem and provided data We are given the data set X = { 3 , 5 , 5 , 6 , 9 , 11 } . Our goal is to find the Interquartile Range (IQR). The IQR is a measure of statistical dispersion and is calculated as the difference between the third quartile (Q3) and the first quartile (Q1).

Sort the data set First, we need to sort the data set in ascending order. In this case, the data is already sorted: X = { 3 , 5 , 5 , 6 , 9 , 11 } .

Find the median Next, we find the median of the data set. Since there are 6 data points (an even number), the median is the average of the 3rd and 4th values. The 3rd value is 5 and the 4th value is 6. So, the median is 2 5 + 6 ​ = 2 11 ​ = 5.5 .

Find the first quartile (Q1) Now, we find the first quartile (Q1), which is the median of the lower half of the data. The lower half is { 3 , 5 , 5 } . The median of this lower half is 5. Therefore, Q1 = 5.

Find the third quartile (Q3) Then, we find the third quartile (Q3), which is the median of the upper half of the data. The upper half is { 6 , 9 , 11 } . The median of this upper half is 9. Therefore, Q3 = 9.

Calculate the IQR Finally, we calculate the IQR as Q3 - Q1. IQR = 9 - 5 = 4.

State the final answer The Interquartile Range (IQR) for the given data set is 4.


Examples
Understanding the IQR can help in many real-world scenarios. For example, in analyzing student test scores, the IQR shows the spread of the middle 50% of the scores, indicating how consistent the performance is. In finance, the IQR can be used to assess the volatility of stock prices, helping investors understand the risk involved. In environmental science, the IQR can help analyze the range of pollution levels in a region, providing insights into the consistency of environmental quality. These applications demonstrate how understanding the IQR can provide valuable insights in various fields.

Answered by GinnyAnswer | 2025-07-08

The Interquartile Range (IQR) for the dataset X = { 3 , 5 , 5 , 6 , 9 , 11 } is calculated by first finding the first (Q1) and third (Q3) quartiles, which are 5 and 9, respectively. The IQR is then found by subtracting Q1 from Q3, resulting in an IQR of 4. Therefore, the final answer is that the IQR is 4.
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Answered by Anonymous | 2025-07-16