[Math] Mean equal to standard deviation, with positive values

standard deviationstatistics

I read the question and the answers regarding the idea of the mean being equal to the standard deviation, in which the mean can always be adjusted to equal anything. But, if the data is restricted to only positive values, then it seems the normal distribution would not work, while the exponential distribution and Poisson would work. Any ideas?

Best Answer

There are many other examples.

The simplest non-degenerate one is $1$ or $0$ with equal probabilities which has mean and standard deviation $\frac12$.

Another is to take the value $\sqrt{p(1-p)} - p$ with probability $1-p$ and the value $\sqrt{p(1-p)} - p+1$ with probability $p$, for $p \le \frac12$.

You could devise many more. A $\chi^2$ distribution with two degrees of freedom. Or if $X$ has a $\chi^2$ distribution with $1$ degree of freedom then $X+\sqrt{2}-1$.

You can multiply any example by any positive number $k$ and retain the property.

Related Solutions

[Math] Calculating mean and standard deviation of very large sample sizes

Posting as an answer in response to comments.

Here's a way to compute the mean and standard deviation in one pass over the file. (Pseudocode.)

n = r1 = r2 = 0;
while (more_samples()) {
    s = next_sample();
    n += 1;
    r1 += s;
    r2 += s*s;
}
mean = r1 / n;
stddev = sqrt(r2/n - (mean * mean));

Essentially, you keep a running total of the sum of the samples and the sum of their squares. This lets you easily compute the standard deviation at the end.

[Math] Simple question on standard deviation and mean.

$\def\implies{\quad\Rightarrow\quad}$If the mean is 5, then you know that

$$\frac{1}{10} \sum_i a_i = 5 \implies \sum_ia_i = 50$$

and if the standard deviation is 1 (and hence the variance is 1) then

$$\frac{1}{10} \sum_i(a_i-5)^2 = 1 \implies \sum_i(a_i-5)^2 = 10$$

To make the range as big as possible, we choose two values $a^+$ and $a^-$ to be as far from the mean as possible (symmetrically) and set the rest of the values to the mean, i.e. $a^+=5+x$ and $a^-=5-x$. What does that tell you about $x$?

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Related Solutions

[Math] Calculating mean and standard deviation of very large sample sizes

[Math] Simple question on standard deviation and mean.

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