What is the interquartile range of this data set?
3, 4, 8, 11, 14, 15, 16
The Correct Answer and Explanation is :
To find the interquartile range (IQR) of the data set (3, 4, 8, 11, 14, 15, 16), we need to first arrange the data in ascending order, which is already done in this case. Next, we identify the quartiles of the data.
Steps to Calculate the Interquartile Range:
- Identify the Quartiles:
- The median (Q2) divides the data set into two halves. For the seven data points, the median is the fourth number in the ordered set:
[
Q2 = 11
] - Now, we find the first quartile (Q1) and third quartile (Q3):
- Q1 is the median of the lower half of the data (the first three numbers): (3, 4, 8). Since there are three numbers, the median is the second number:
[
Q1 = 4
] - Q3 is the median of the upper half of the data (the last three numbers): (14, 15, 16). Similarly, the median is the second number:
[
Q3 = 15
]
- Q1 is the median of the lower half of the data (the first three numbers): (3, 4, 8). Since there are three numbers, the median is the second number:
- Calculate the Interquartile Range (IQR):
The interquartile range is found by subtracting Q1 from Q3:
[
IQR = Q3 – Q1 = 15 – 4 = 11
]
Explanation of the Interquartile Range:
The interquartile range (IQR) is a measure of statistical dispersion that represents the range within which the central 50% of the data points lie. By focusing on the middle half of the data, the IQR is less affected by outliers or extreme values, making it a robust statistic for understanding variability.
In our data set, the IQR of 11 indicates that the spread of the middle 50% of the values is fairly wide. This is particularly useful in contexts where understanding the core of the data distribution is critical, as it provides a clearer picture of the data’s variability without the influence of extreme values. For example, in fields such as finance, biology, or quality control, understanding the central tendency and spread of data can help inform decision-making and analyses.
Thus, the interquartile range provides insight into the distribution of data, allowing for better comprehension of underlying trends and patterns.