Question Video: Calculating the Standard Deviation of a Data Set | Nagwa Question Video: Calculating the Standard Deviation of a Data Set | Nagwa

Question Video: Calculating the Standard Deviation of a Data Set Mathematics • Third Year of Preparatory School

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Calculate the standard deviation of the values 45, 35, 42, 49, 39, and 34. Give your answer to three decimal places.

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Video Transcript

Calculate the standard deviation of the values 45, 35, 42, 49, 39, and 34. Give your answer to three decimal places.

Let’s begin by recalling what we mean by the standard deviation. The standard deviation is a measure of how spread out a set of numbers are. And to find it mathematically, we have to take the square root of the variance. We might see the standard deviation written as the Greek letter 𝜎 and the variance as 𝜎 squared.

Next, we recall what we mean by the variance. This is the average of the squared differences from the mean. We will therefore begin by calculating the mean. To calculate the mean of a set of numbers, we add them all together and then divide by how many there are. In this question, we need to find the sum of 45, 35, 42, 49, 39, and 34 and then divide this total by six. The six numbers sum to 244, and dividing this by six gives us 40.6 recurring. However, for accuracy, we will leave our answer as the fraction 122 over three. The mean of the six values is 122 over three.

We are now in a position to find the variance. And as already mentioned, this is the average of the squared differences from the mean. We begin by subtracting each of our six values from the mean. We then square each of these values, add them together, and divide by how many values we have. The variance in this question is therefore equal to the following equation. Subtracting 45 from 122 over three and then squaring the answer gives us 169 over nine. Squaring 122 over three minus 35 gives us 289 over nine. And repeating this for the four other terms on the numerator, we have 16 over nine, 625 over nine, 25 over nine, and 400 over nine.

The variance is the sum of these six values all divided by six. The numerator is equal to 1524 over nine. And dividing this by six, we have 254 over nine. Once again, we will leave this as a fraction. The variance of our six numbers is 254 over nine. We are now in a position to calculate the standard deviation. It is equal to the square root of 254 over nine. Typing this into our calculator gives us 5.31245 and so on.

We are asked to give our answer to three decimal places. Since the fourth digit after the decimal point is a four, we round down. And we can therefore conclude that the standard deviation of the six values, correct to three decimal places, is 5.312. Whilst it is not covered in this video, it is worth noting that some scientific calculators will calculate the mean, variance, and standard deviation for us by simply typing in the values.

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