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saurabh9gupta
Joined: 10 Jan 2013
Last visit: 24 May 2026
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Location: India
Concentration: General Management, Strategy
Schools: Kenan-Flagler '24 (D)
GRE 1: Q163 V155
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19 Aug 2018, 00:08
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19 Aug 2018, 21:28
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saurabh9gupta
The first bit is just saying that if you add a number (positive or negative) to every element in the set, the standard deviation remains the same. For example:
1. For the set \({1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\) we have \(σ=2.87\) (approximate value) and mean \(μ=5.5\)
2. For the set \({11, 12, 13, 14, 15, 16, 17, 18, 19, 20}\) (adding 10 to each element) we have standard deviation \(σ=2.87\) (approximate value) and mean \(μ=15.5\)
What matters for the standard deviation is how far the elements are from the mean. The actual values of the elements do not matter here.
The second portion seems to be talking about the inclusion of one more element in the set, but as far as I know, it is not very easy to predict the impact of such an inclusion on the standard deviation, so I'll leave that to someone else. What we usually discuss is the inclusion of a pair of values that are symmetric about the mean.
Let's say the original set is \({1, 3, 5}\). Here \(σ=1.633\) and \(μ=3\). If we now put a pair of elements \(x\) and \(y\) into the set, the new set will be \({1, 3, 5, x, y}\). If \(x\) and \(y\) are very far from the mean, the standard deviation will go up, and if \(x\) and \(y\) are very close to the mean, the standard deviation will go down.
1. \(x=(3-4)=-1\) and \(y=(3+4)=7\): The new set is \({-1, 1, 3, 5, 7}\) and its standard deviation is \(2.828\), which is greater than the original standard deviation.
2. \(x=(3-1.633)=1.367\) and \(y=3+1.633=4.633\): The new set is \({1, 1.367, 3, 4.633, 5}\) and its standard deviation is \(1.633\), which is the same as the original standard deviation.
3. \(x=(3-1)=2\) and \(y=3+1=4\): The new set is \({1, 2, 3, 4, 5}\) and its standard deviation is \(1.414\), which is less than the original standard deviation.
4. \(x=(3-0)=3\) and \(y=3+0=3\): The new set is \({1, 3, 3, 3, 5}\) and its standard deviation is \(1.265\), which is the lowest the standard deviation can go with a pair of new elements added to the set (the difference between these values and the mean is \(0\)).
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