In a study of the lizard Sceloporus occidentalis, biologists measured the distance (m) run in 2 minutes for each of 15 animals. The results (listed in increasing order) were as follows:42 18.4 22.2 24.5 26.4 27.5 28.7 30.6 32.9 32.9 34.0 34.8 37.5 42.1 45.5 45.5 (a) Determine the quartiles and the interquartile range. (b) Determine the range.
The Correct Answer and Explanation is:
To solve this problem, let’s break it down step by step.
Part (a) Determine the Quartiles and Interquartile Range (IQR)
Step 1: Organize the Data
The data set is already ordered in increasing order:
[ 18.4, 22.2, 24.5, 26.4, 27.5, 28.7, 30.6, 32.9, 32.9, 34.0, 34.8, 37.5, 42.1, 42.0, 45.5, 45.5 ]
There are 16 data points, so:
- Median (Q2): The median is the middle value. Since there are an even number of data points (16), we take the average of the two middle values (the 8th and 9th values):
[
Q2 = \frac{32.9 + 32.9}{2} = 32.9
]
Step 2: Find Q1 and Q3
- Q1 (First Quartile): This is the median of the lower half of the data (excluding the median). The lower half of the data set is:
[
18.4, 22.2, 24.5, 26.4, 27.5, 28.7, 30.6, 32.9
]
The median of this group is the average of the 4th and 5th values:
[
Q1 = \frac{26.4 + 27.5}{2} = 26.95
] - Q3 (Third Quartile): This is the median of the upper half of the data (excluding the median). The upper half of the data set is:
[
32.9, 34.0, 34.8, 37.5, 42.1, 42.0, 45.5, 45.5
]
The median of this group is the average of the 4th and 5th values:
[
Q3 = \frac{37.5 + 42.1}{2} = 39.8
]
Step 3: Calculate the Interquartile Range (IQR)
The IQR is the difference between the third quartile (Q3) and the first quartile (Q1):
[
IQR = Q3 – Q1 = 39.8 – 26.95 = 12.85
]
Part (b) Determine the Range
The range is the difference between the largest and smallest values in the data set:
[
\text{Range} = 45.5 – 18.4 = 27.1
]
Summary
- Quartiles:
- Q1 = 26.95
- Q2 (Median) = 32.9
- Q3 = 39.8
- Interquartile Range (IQR) = 12.85
- Range = 27.1
Explanation
To find the quartiles, we divided the data into four parts. The first quartile (Q1) represents the value below which 25% of the data falls, and the third quartile (Q3) represents the value below which 75% of the data falls. The median (Q2) splits the data in half. The interquartile range (IQR) measures the spread of the middle 50% of the data, which is the difference between Q3 and Q1. The range gives the overall spread of the data by subtracting the smallest value from the largest value.
The quartiles and IQR are useful for understanding the distribution of data, as they provide insight into the central tendency and variability, excluding outliers. The range is a simpler measure but can be influenced by extreme values.