Extending Concepts

A...

Question

Extending Concepts

Alternative Formula You used SSₓ = Σ(x − x̄)² when calculating variance and standard deviation. An alternative formula that is sometimes more convenient for hand calculations is

SSₓ = Σ x² − (Σ x)² / n.

You can find the sample variance by dividing the sum of squares by n − 1 and the sample standard deviation by finding the square root of the sample variance.

b. Use the alternative formula to calculate the sample standard deviation for the data set in Exercise 15.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says the number of hours 5 employees worked in a week is as follows 32, 36, 40, 44, and 48. What is the sample standard deviation of the hours work? Use the formula S is equal to the square root of the quantity of the sum of X2 minus the quantity of the sum of X in quantity squared divided by N in quantity, divided by the quantity of N minus 1 in quantity. And we're given 4 possible choices as our answers. For choice A, we have 7.32 hours. For choice B, we have 6.23 hours. For choice C, we have 5.32 hours, and for choice D, we have 6.32 hours. So, here we just need to calculate our sample standard deviation, and we're actually given the formula and the problem. So, we need to identify a couple quantities to be used in this formula. The first one is going to be N, and in here is the number of data points that we have. And so here we actually have 5 points in our data set. So that means N is going to be equal to 5. Next, we'll need to calculate the sum of X. And here we'll just add up all of our values in our data set. So, the sum of X is going to be equal to 32 plus 36. Plus 40 + 44 plus 48. And when we add up all of these values, that means that the sum is going to be equal to 200. Next, we'll need to calculate the sum of X2. So here we're just going to square each of our values in our data set, and then add them together. So, we'll have 32 squared plus 36. Squared Plus 40 squared. Plus 44 squared. And then finally plus 48 squared. And when we set up all these values, that means that this is going to be equal to. 8160. So now we can actually use our formula for our sample standard deviation. So that means that S is going to be equal to the square root. Of the quantity, and in the numerator here, we'll have the sum of X, which is 8160, then we'll have minus. In the next term we have the quantity of the sum of X and quantity squared, so we'll have 200. Squared, and that's going to be divided by N, which is 5. And then that whole quantity is going to be divided by the quantity of N minus 1, which is going to be 5 minus 1. And so, evaluating this expression, we see that S is going to be approximately equal to 6.32. And since we're dealing with our here, That means that this is going to be 6.32 hours. So that is going to be the answer for this problem. And if we compare our answer here to our answer choices, we see that the correct answer choice actually corresponds to answer choice D. So, I hope that this explanation has been useful, and I'll see you in the next video.

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Key Concepts

Sum of Squares (SS)

Sample Variance

Standard Deviation

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