To measure the spread of the radii of 12 randomly selected tennis balls, you can use the following formulas based on whether your data represents a sample or the entire population.
1. Sample Variance (s²):
[tex]\[ s^2 = \frac{\sum_{i=1}^n (a_i - \bar{x})^2}{n - 1} \][/tex]
This formula is used to measure the spread within a sample. Here, [tex]\(a_i\)[/tex] represents each individual measurement, [tex]\(\bar{x}\)[/tex] is the sample mean, and [tex]\(n\)[/tex] is the number of observations (in this case, 12).
2. Sample Standard Deviation (s):
[tex]\[ s = \sqrt{ \frac{\sum_{i=1}^n (a_i - \bar{x})^2}{n - 1} } \][/tex]
This is the square root of the sample variance and provides a measure of spread in the same units as the data itself.
3. Population Variance (σ²):
[tex]\[ \sigma^2 = \frac{\sum_{i=1}^n (a_i - \mu)^2}{n} \][/tex]
This formula is used if you have data for the entire population. Here, [tex]\(a_i\)[/tex] represents each individual measurement, [tex]\(\mu\)[/tex] is the population mean, and [tex]\(n\)[/tex] is the total number of population observations.
4. Population Standard Deviation (σ):
[tex]\[ \sigma = \sqrt{ \frac{\sum_{i=1}^n (a_i - \mu)^2}{n} } \][/tex]
This is the square root of the population variance and provides a measure of spread in the same units as the data itself.
Since you have data from a randomly selected sample of 12 tennis balls, you should typically use the sample variance and sample standard deviation formulas (1) and (2) to measure the spread:
- Sample Variance: [tex]\[ s^2=\sum_{i=1}^n \frac{a_i-\bar{x}}{n-1} \][/tex]
- Sample Standard Deviation: [tex]\[ s=\sqrt{\sum_{i=1}^n \frac{a_i-\bar{x}}{n-1}} \][/tex]
These formulas will help you understand how much the radii of the tennis balls vary from the average radius within your sample.
