Calculate the standard deviation of the values 45, 35, 42, 49, 39, and 34. If necessary, give your answer to three decimal places.
So what is a standard deviation? The standard deviation is a measure of how spread out numbers are. To mathematically find it, we have to take the square root of the variance. Well, what is the variance? Variance is the average of the squared differences from the mean. So let’s first begin by finding the mean.
To find the mean, we add all of the numbers together and then divide by the total. So we divide by six because there are six total numbers. Adding the values on the numerator together, we get 244. Now I need to take that number and divide by six. Rounding three decimal places, the mean is equal to 40.666.
Now to find the variance, we need to take the average of the squared differences from the mean. So the difference from the mean means we need to take the mean and subtract each of these values from it. That way we know how far apart each of these values are from the mean. Here we can see that we took each value and subtracted it from the mean. Then we will square each of these values, add them together, and then divide by six. That’s how you find the variance. So let’s first begin by subtracting each of these numbers from the mean inside these parentheses.
Now that we’ve subtracted inside each parenthesis, the next step would be to square each of these numbers. Now notice, some of the numbers are negative. However, when you square a negative, it turns positive. So all of our values will be positive on the numerator. Now that we have squared each number on the numerator, the next step would be to add each of these numbers together equaling 169.336. Now divide by six, and we get that the variance is 28.223. Now in the beginning, we said that the standard deviation is equal to the square root of the variance. So our last step would be to square root this number, square root the variance which equals 5.3125. Now it says to give our answer to three decimal places. So we have to decide if we keep the two at two or round it up to three. So we look at the number to the right, the five. So since five is five or larger, we will take the two and round it up to a three.
Therefore, the standard deviation is equal to 5.313.