Question

In a study of the lizard Sceloporus occidentalis, biologists measured the distance \((\mathrm{m})\) run in 2 minutes for each of 15 animals. The results (listed in increasing order) were as follows: \(^{42}\) \(\begin{array}{llllllll}18.4 & 22.2 & 24.5 & 26.4 & 27.5 & 28.7 & 30.6 & 32.9 \\\ 32.9 & 34.0 & 34.8 & 37.5 & 42.1 & 45.5 & 45.5 & \end{array}\) (a) Determine the quartiles and the interquartile range. (b) Determine the range.

Short answer

The quartiles are Q1 = 26.4, Q2 = 32.9, Q3 = 37.5. The interquartile range (IQR) is Q3 - Q1 = 37.5 - 26.4 = 11.1 m. The range is 45.5 - 18.4 = 27.1 m.

Step-by-step solution

- Organizing the Data

List the given data in increasing order (as it's already provided).

- Calculating the Quartiles

The first quartile (Q1) is the median of the first half of the data, the second quartile (Q2) is the median of the entire data set, and the third quartile (Q3) is the median of the second half of the data.

- Finding the First Quartile (Q1)

Q1 is the median of the first 7 numbers. So it will be the 4th value in the ordered list.

- Finding the Second Quartile (Q2) or Median

Q2 is the median of all the data points. Since there are 15 numbers, Q2 will be the 8th value in the ordered list.

- Finding the Third Quartile (Q3)

Q3 is the median of the last 7 numbers. So it will be the 12th value in the ordered list.

- Calculating the Interquartile Range (IQR)

Subtract Q1 from Q3 to determine the interquartile range (IQR).

- Determining the Range

Subtract the smallest value in the data set from the largest value to calculate the range.

Key concepts

Data Organization

Organizing data is the first crucial step in statistical analysis, setting the stage for computation of various measures, including quartiles. When dealing with a dataset, like the measurements of lizard running distances, the data should be listed in ascending order. This not only simplifies the identification of various statistical parameters but also aids in visualizing the distribution of data points.

For instance, in an ordered list, it becomes immediately apparent if the data is skewed towards higher or lower values, or if there are any outliers that might influence subsequent calculations. Correctly organizing data can prevent errors during analysis and ensures accuracy when determining central tendencies and other statistics. As seen in our lizard speed study, with the data neatly aligned, it's easier to jump to the next step, which is finding the median, or the second quartile.

Median Calculation

The median is a measure of central tendency that divides a dataset into two equal parts. To calculate the median in a dataset with an odd number of observations, like the 15 lizards in our study, you locate the middle value. This is done simply by counting to the data point exactly at the mid-position: the 8th value in this case.When the number of data points is even, the median is found by averaging the two central values. The median is particularly useful as it is not influenced by extreme values or outliers in the data. It can be seen as the true middle, providing a sense of the 'central' tendency beyond the average, which might be skewed by non-representative, extreme datapoints. Knowing how to find the median is essential for determining the quartiles, which are metrics that divide the data into quarters.

Range in Statistics

The range is one of the simplest measures of variability in a dataset. It is calculated by subtracting the smallest value from the largest value. This gives a quick sense of the spread of the data, showing us the extent to which the values differ from each other. In our lizard example, the range informs us of the difference between the fastest and the slowest recorded runs. However, the range doesn't give us information about the distribution of values between those two extremes.This is where the interquartile range (IQR) provides more nuance. The IQR is determined by subtracting the first quartile (the median of the lower half of the data) from the third quartile (the median of the upper half). It tells us about the spread of the middle 50% of the data, which is often more representative of the typical conditions than the full range.