To determine the best measure of variability to compare the battery life data from Brand A and Brand B, we need to understand whether the distributions of these data sets are symmetric or skewed. The measures of central tendency, particularly the mean and the median, are key indicators of the distribution shape.
Let's analyze the data step by step.
### Step 1: Calculating the Mean and Median for Brand A
The battery lives for Brand A are:
[tex]\[22.5,\ 17.0,\ 21.0,\ 23.0,\ 22.0,\ 18.5,\ 22.5,\ 20.0,\ 19.0,\ 23.0\][/tex]
- Mean of Brand A: [tex]\(20.85\)[/tex]
- Median of Brand A: [tex]\(21.5\)[/tex]
### Step 2: Calculating the Mean and Median for Brand B
The battery lives for Brand B are:
[tex]\[20.0,\ 19.5,\ 20.5,\ 16.5,\ 14.0,\ 17.0,\ 11.0,\ 19.5,\ 21.0,\ 12.0\][/tex]
- Mean of Brand B: [tex]\(17.1\)[/tex]
- Median of Brand B: [tex]\(18.25\)[/tex]
### Step 3: Check for Symmetry
To check for symmetry:
- Compare the mean and the median. If they are the same or very close, the distribution is likely symmetric.
- If there is a significant difference between the mean and the median, the distribution is likely skewed.
#### Brand A:
- Mean: [tex]\(20.85\)[/tex]
- Median: [tex]\(21.5\)[/tex]
Since [tex]\(20.85 \neq 21.5\)[/tex], Brand A data is not symmetric.
#### Brand B:
- Mean: [tex]\(17.1\)[/tex]
- Median: [tex]\(18.25\)[/tex]
Since [tex]\(17.1 \neq 18.25\)[/tex], Brand B data is not symmetric.
### Conclusion
Both the distributions for Brand A and Brand B are not symmetric, hence, they are skewed.
### Determining the Best Measure of Variability
When distributions are skewed, the preferred measure of variability is often the interquartile range (IQR), because it is less affected by outliers and skewness compared to standard deviation.
Thus, the best measure of variability to use to compare the data is:
"Both distributions are skewed left; so the interquartile range is the best measure to compare variability."
