# How many standard deviations around the mean do I need to guarantee covering at least one sample

I am sampling readings $$X = \{ x \}_i$$, $$i=1..N$$ from an unknown distribution. I am estimating the mean $$\mu$$ and the standard deviation $$\sigma$$ from the sample.

My question is: how many $$\sigma$$-s do I need so that I am guaranteed that, given any distribution independently of how "pathological" it is, I will always have at least one sample within $$[ \mu - n \sigma; \mu + n \sigma ]$$ ?

My "intuition" tells me that the worst distribution I can have is the one where "all samples are furthest from the mean at the same time", ie for example $$X = \{ -1.0, 1.0 \}$$, in which case $$\mu = 0$$, $$\sigma = 1$$, so I would guess that $$n=1$$ would be the value I am looking for. But is this correct?

## 1 Answer

You should use different notation for the sample mean and standard deviation, for example $$\bar x$$ and $$s$$. There are no guarantees for the location of sample observations compared to the population mean and standard deviation, just probabilities.

There is a slight issue if your calculation of the sample standard deviation $$s$$ uses the $$\frac1{n-1}$$ expression of $$s= \sqrt{\frac{1}{n-1}\sum(x_i-\bar x)^2}$$ rather than $$s= \sqrt{\frac{1}{n}\sum(x_i-\bar x)^2}$$, as for $$n=1$$ the $$\frac1{n-1}$$ expression will give $$\frac00$$. When $$n=1$$, you have the observation equal to the sample mean.

Ignoring that issue and using either expression for $$s$$, you can say that for all $$n>0$$, at least one sample observations will be in the interval $$[\bar x -s, \bar x +s]$$. If none were then you would have $$\sum(x_i-\bar x)^2 > \sum_i s^2 = n s^2$$, leading to the contradictory $$s>s$$.

All the sample observations will be in the interval $$[\bar x -\sqrt{n}s, \bar x +\sqrt{n}s]$$. If any observation $$x_j$$ was outside that interval you would have $$\sum(x_i-\bar x)^2 > (x_j-\bar x)^2 > ns^2$$ again leading to the contradictory $$s>s$$.

Both $$[\bar x -s, \bar x +s]$$ and $$[\bar x -\sqrt{n}s, \bar x +\sqrt{n}s]$$ are narrower than $$[\bar x -ns, \bar x +ns]$$ for $$n>0$$, with equality only when $$n=1$$.

• I can't quite remember, but there's a named theorem asserting that all sample data points are in some interval involving the sample mean and sample standard deviation. Do you perhaps know the name? Is the interval the $\sqrt{n}$-interval you provided (or is it narrower)? – user257566 May 20 at 0:54
• Ah, this is it: en.wikipedia.org/wiki/Samuelson%27s_inequality . I'm not so forgetful after all! – user257566 May 20 at 0:55