Answer :
To find the sample standard deviation for the given data set, we'll follow these steps:
Calculate the Mean (Average):
First, add up all the numbers in the data set and then divide by the total number of observations.
[tex]\text{Mean} = \frac{98.5 + 99.9 + 100.3 + 99.2 + 99.8 + 99.8 + 98.6 + 97.6 + 99.1 + 98.4 + 97.5 + 96.5 + 99.8 + 100.5 + 96.8 + 100.4 + 96.5 + 98.8 + 97.8 + 97.2}{20}[/tex]
[tex]\text{Mean} = \frac{1981.7}{20} = 99.085[/tex]
Calculate the Variance:
For each data point, subtract the mean and square the result (these are called squared deviations). Then, sum up all these squared deviations and divide by the number of data points minus one.
[tex]\text{Variance} = \frac{(98.5 - 99.085)^2 + (99.9 - 99.085)^2 + ... + (97.2 - 99.085)^2}{19}[/tex]
Computing each squared deviation and summing them:
[tex](98.5 - 99.085)^2 + (99.9 - 99.085)^2 + \ldots + (97.2 - 99.085)^2 = 18.846[/tex]
Divide by 19:
[tex]\text{Variance} = \frac{18.846}{19} = 0.9929[/tex]
Calculate the Standard Deviation:
The standard deviation is the square root of the variance.
[tex]\text{Standard Deviation} = \sqrt{0.9929} \approx 0.9964[/tex]
So, the sample standard deviation for the given data set is approximately [tex]0.996[/tex] when rounded to three decimal places, which is one more decimal place than the data set.
In summary, the sample standard deviation provides a measure of the amount of variation or dispersion in a set of values. It's particularly useful in statistics for understanding how data is spread out in relation to the mean.