The table shows the battery lives, in hours, of ten Brand A batteries and ten Brand B batteries.
Battery Life (hours)
| | 22.5 | 17.0 | 21.0 | 23.0 | 22.0 | 18.5 | 22.5 | 20.0 | 19.0 | 23.0 |
| :------ | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| Brand A | | | | | | | | | | |
| Brand B | 20.0 | 19.5 | 20.5 | 16.5 | 14.0 | 17.0 | 11.0 | 19.5 | 21.0 | 12.0 |
Which would be the best measure of variability to use to compare the data?
A. Only Brand A data is symmetric, so standard deviation is the best measure to compare variability.
B. Only Brand B data is symmetric, so the median is the best measure to compare variability.
C. Both distributions are symmetric, so the mean is the best measure to compare variability.
D. Both distributions are skewed left, so the interquartile range is the best measure to compare variability.
Answer (1)
Calculate the mean, median, standard deviation, and IQR for both Brand A and Brand B battery life data.
Compare the mean and median for each brand to assess symmetry or skewness. Brand A is slightly skewed left, while Brand B is more skewed left.
Since Brand B is skewed, the interquartile range (IQR) is the best measure of variability to compare the two datasets.
The interquartile range is less sensitive to skewness and outliers, providing a more robust comparison. Both distributions are skewed left, so the interquartile range is the best measure to compare variability.
Explanation
Analyze the problem and data We are given two datasets representing the battery lives of Brand A and Brand B batteries. Our goal is to determine the best measure of variability to compare these datasets. The options provided suggest considering the symmetry or skewness of the data to guide our choice between standard deviation, median, mean, and interquartile range (IQR).
Calculate descriptive statistics First, let's calculate some descriptive statistics for both datasets. For Brand A, the data is: 22.5, 17.0, 21.0, 23.0, 22.0, 18.5, 22.5, 20.0, 19.0, 23.0. The mean is 20.85 hours, and the median is 21.5 hours. The standard deviation is approximately 2.01 hours, and the IQR is 3.25 hours. For Brand B, the data is: 20.0, 19.5, 20.5, 16.5, 14.0, 17.0, 11.0, 19.5, 21.0, 12.0. The mean is 17.1 hours, and the median is 18.25 hours. The standard deviation is approximately 3.46 hours, and the IQR is 5.25 hours.
Assess symmetry and skewness Next, we assess the symmetry or skewness of each dataset. A simple way to do this is to compare the mean and median. If the mean and median are approximately equal, the distribution is roughly symmetric. If the mean is less than the median, the distribution is skewed left. If the mean is greater than the median, the distribution is skewed right. For Brand A, the mean (20.85) is slightly less than the median (21.5), suggesting a slight skew to the left. For Brand B, the mean (17.1) is considerably less than the median (18.25), suggesting a more pronounced skew to the left. Looking at the boxplot visualization confirms that Brand B is skewed left.
Determine the best measure of variability Since Brand B is noticeably skewed, the interquartile range (IQR) is the best measure of variability to use for comparison. The IQR is less sensitive to extreme values and skewness than the standard deviation.
Final Answer Therefore, the best measure of variability to use to compare the data is the interquartile range because both distributions are skewed left (although Brand A is only slightly skewed).
Examples
Understanding data variability is crucial in many real-world scenarios. For instance, in manufacturing, comparing the variability in the dimensions of products from two different production lines helps assess which line is more consistent. If one line produces parts with a skewed distribution of sizes, using the IQR to compare variability provides a more accurate picture than using the standard deviation, which can be heavily influenced by outliers or skewness. This ensures better quality control and reduces the risk of defective products.
