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

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Calculate the standard deviation of the data. Round your answer to two decimal places.

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Video Transcript

Calculate the standard deviation of the following data. Round your answer to two decimal places.

For the data, we have scores of one, two, three, four, and five and frequencies of three, nine, 12, five, and four, respectively. To find the standard deviation from a frequency table, there are a couple of methods available to us. One formulation we might be familiar with is the square root of the sum of each score π‘₯ 𝑖 minus the mean πœ‡ all squared times each frequency 𝑓 𝑖 and then divided by the sum of the frequencies. Equivalently, this is equal to the square root of the sum of each frequency times its score squared divided by the sum of the frequencies minus πœ‡ squared. Although we won’t prove it here, these two formulations are exactly the same. But the second version is often slightly easier to use, so let’s use it.

Incidentally, it’s also simpler to just write this expression without the 𝑖’s as long as we remember that we just want to multiply entries in each column together. So the first values we will need to find are the squares of the scores, that is, π‘₯ squared, which we can get by just squaring the scores in the corresponding columns above. So the first entry is one squared, which is just one. The second entry is two squared, which is four, and we can continue doing this to get nine, 16, and 25.

Now, we’re also going to need to find this πœ‡ squared term. And we can recall that πœ‡ is the mean, which for a frequency table is the sum of the frequencies times the scores divided by the sum of the frequencies. So, in particular, we will need to find the product of each frequency with its corresponding score.

For the first value, we multiply one by three, giving us three. And we can do this for all the other entries, giving us two times nine is 18, three times 12 is 36, and four times five or five times four are both 20. So now we can actually work out the mean if we sum up these 𝑓π‘₯ values and divide by the sum of the 𝑓 values. The sum of the 𝑓π‘₯ values is 97, and the sum of the frequencies is 33, which means the mean is 97 over 33. We don’t need to simplify this, but this is approximately 2.93.

Now, we have the mean and the π‘₯ squared values, but we will also need 𝑓 times π‘₯ squared. We can find these values by multiplying the π‘₯ squared values we previously found by the 𝑓-values. So the first entry is three times one, which is three. Then, we have nine times four, which is 36, and we continue this process, giving us 108, 80, and 100.

Now, we can finally calculate the standard deviation. We take the sum of the 𝑓π‘₯ squared values, which is equal to 327. And we already found that the sum of the 𝑓’s is 33. Therefore, the standard deviation will be the square root of 327 over 33 minus 97 over 33 squared. And putting this all into a calculator gives us 1.1265 et cetera, which we can round to two decimal places as specified.

Thus, our final answer is that the standard deviation of our data set to two decimal places is 1.13.

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