Video: Calculating Standard Deviation

If βˆ‘(π‘₯ βˆ’ the mean)Β² for a set of 6 values equals 25, find the standard deviation of the set, and round the result to the nearest thousandth.

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Video Transcript

If the sum of all the π‘₯-values minus the mean squared for a set of six values equals 25, find the standard deviation of the set and round the result to the nearest thousandth.

Before we go any further, I want us to think about and examine the formula for standard deviation. Standard deviation is given by the formula 𝜎 equals the square root of the summation of all π‘₯-values minus the mean of the π‘₯-values squared divided by 𝑛, the number of π‘₯-values.

How can we use the information we’re given to solve for the standard deviation? If we look closely, we’ll see that we’re already given the summation we need. The sum of all the π‘₯-values minus the mean of the π‘₯-values squared equals 25.

We now know that the numerator inside the square root is 25. We also know that the 𝑛 value is the number of π‘₯-values. In this case, we have six π‘₯-values. So we plug in six for 𝑛. We can simplify this by taking the square root of the numerator and the square root of the denominator. The square root of 25 equals five and we can’t simplify the square root of six any further.

The standard deviation for these six values will be equal to five over the square root of six. However, we want the value rounded to the nearest thousandth. To do that would mean we would need to divide five by the square root of six. We can use a calculator for this process by plugging in five divided by six. And the calculator returns an irrational number to us: 2.041241452.

We want to round to the thousandth place. The thousandth place is three spaces to the right of the decimal. The digit to the right of the thousandth place is a two. Therefore, we round down. The one in the thousandth place stays the same. And we bring down everything to the left of the thousandth place: 2.041.

The standard deviation of the set of six values is equal to 2.041.

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