Final answer:
To calculate the standard deviation for the provided data, we find the midpoints of each class, calculate the mean using these midpoints and frequencies, calculate the variance, and then take the square of the variance to determine the standard deviation. A statistical calculator or software can be used to simplify the process.
Step-by-step explanation:
To calculate the standard deviation for the given frequency table, we must first find the midpoint for each class. The midpoints for the classes are 1, 3, 5, 7, 9, 11, and 13, respectively. Once we have the midpoints, we can then use them alongside the frequencies to calculate the mean or expected value, then use the mean to find the variance, and finally, take the square root of the variance to obtain the standard deviation.
The formula for the mean is:
\(\bar{x} = \frac{\sum (f \times m)}{\sum f}\)
Where f represents the frequency and m represents the midpoint for each class. Once we calculate the mean, the variance is found using:
\(\sigma^2 = \frac{\sum (f \times (m - \bar{x})^2)}{N}\)
Where \(\sigma^2\) is the variance and N is the sum of all frequencies. Finally, the standard deviation is the square root of the variance. This calculation can be facilitated with statistical tools such as a TI-83/84 calculator or statistical software.
If we were given a specific value, such as the enrollment of a school, we could determine how many standard deviations this value is from the mean by subtracting the mean from the value and dividing the result by the standard deviation.
This process encapsulates the formula and method for calculating both mean and standard deviation and how to interpret those values in context.