Ezra counts the number of passengers who are already on the bus when he gets on each morning. His data for the last 15 days are shown below. What is the interquartile range of the data? 8, 9, 9, 10, 12, 12, 13, 14, 15, 15, 15, 16, 18, 19, 20
The Correct Answer and Explanation is:
To find the interquartile range (IQR) of a data set, follow these steps:
Step 1: Order the Data
The data is already in ascending order:
8, 9, 9, 10, 12, 12, 13, 14, 15, 15, 15, 16, 18, 19, 20
Step 2: Identify the Median (Q2)
There are 15 numbers, so the median is the 8th number:
- Q2 = 14
Step 3: Find the Lower Quartile (Q1)
The lower half of the data includes the numbers before the median:
8, 9, 9, 10, 12, 12, 13
This is 7 numbers. The median of these is the 4th number:
- Q1 = 10
Step 4: Find the Upper Quartile (Q3)
The upper half includes the numbers after the median:
15, 15, 15, 16, 18, 19, 20
This is also 7 numbers. The median of these is the 4th number:
- Q3 = 16
Step 5: Calculate the Interquartile Range (IQR)
IQR=Q3−Q1=16−10=6\text{IQR} = Q3 – Q1 = 16 – 10 = \boxed{6}
Explanation:
The interquartile range (IQR) is a measure of statistical dispersion, or how spread out the middle 50% of a data set is. It is particularly useful because it ignores outliers and extreme values, focusing only on the central portion of the data. To calculate the IQR, we use the first quartile (Q1) and the third quartile (Q3). Q1 marks the 25th percentile, while Q3 marks the 75th percentile.
In Ezra’s case, he recorded the number of passengers on the bus over 15 days. His data is already sorted, making analysis straightforward. Since there are 15 values, the median (or second quartile, Q2) is the 8th number, which is 14. The lower half of the data (first 7 numbers) is used to find Q1, and the upper half (last 7 numbers) is used for Q3. The middle of the lower half is 10, so Q1 = 10. The middle of the upper half is 16, so Q3 = 16.
Subtracting Q1 from Q3 gives the IQR: IQR=Q3−Q1=16−10=6\text{IQR} = Q3 – Q1 = 16 – 10 = 6
This tells us that the middle 50% of Ezra’s observations range over 6 passengers. The IQR is a robust way to understand variability in data because it isn’t affected by unusually high or low values. It helps in identifying how consistently the number of passengers remains near the middle range.
Final Answer: 6\boxed{6}.
