Final answer:
The standard deviation of a set of observations measures how spread out the values are from the mean. It can be calculated using a formula that involves finding the mean, calculating the squared differences between each observation and the mean, and taking the square root of the variance. In this case, the standard deviation of the 18 observations is approximately 2.
Step-by-step explanation:
The standard deviation of a set of observations is a measure of how spread out the values are from the mean. To calculate the standard deviation, use the formula:
Step 1:
Calculate the mean (average) of the observations by dividing the sum of all the observations by the number of observations:
mean = Σxi / n = 72 / 18 = 4
Step 2:
Calculate the difference between each observation and the mean:
xi - mean = (x1 - 4), (x2 - 4), ..., (x18 - 4)
Step 3:
Square each difference:
(xi - mean)^2 = (x1 - 4)^2, (x2 - 4)^2, ..., (x18 - 4)^2
Step 4:
Calculate the sum of all the squared differences:
Σ(xi - mean)^2 = (x1 - 4)^2 + (x2 - 4)^2 + ... + (x18 - 4)^2
Step 5:
Divide the sum of squared differences by the number of observations minus 1 (n-1) to find the variance:
variance = Σ(xi - mean)^2 / (n - 1)
Step 6:
Finally, take the square root of the variance to find the standard deviation:
standard deviation = sqrt(variance)
Using the given data, Σxi = 72 and there are n = 18 observations. Plugging these values into the formulas, we can calculate the standard deviation:
mean = 72 / 18 = 4
variance = Σ(xi - mean)^2 / (n - 1) = (x1 - 4)^2 + (x2 - 4)^2 + ... + (x18 - 4)^2 / (18 - 1)
standard deviation = sqrt(variance)
Therefore, the standard deviation of the 18 observations is approximately 2.