# Alternative Standard Deviation Calculations

I recently inherited some code that does bounds checking from a coworker no longer with the company. It has the following comment

//Currently, we use 3 sigma(standard deviation) as upper bound, where
// sigma = SQRT(mean * (1 - mean) / (sample size)).

And the following implementation is supposed to represent the upper bound to check against

MEAN + 3 * SQRT(MEAN * (1 - MEAN) / COUNT)

Does either the formula or the implementation seem correct? I'm not that familiar with Standard Deviation, but the Wiki article on the topic mentions a much different formula.

• This works if the mean represents a proportion – Kamster Feb 6 '15 at 22:20
• Could you go into further detail? – Kobaj Feb 6 '15 at 22:22

An analysis of this would require an undergraduate course in calculus-based probability.

Specifically, let $$\hat{p} = \dfrac{X}{n}$$ where $X$ follows a binomal distribution with the usual parameters $n$ (being the sample size), $p$. In general, if we have a statistic $\hat{\theta}$ which follows a normal distribution, $$\mathbb{E}[\hat{\theta}] \pm z_{\alpha /2} \cdot \sigma_{\hat{\theta}}$$ - where $\mathbb{E}$ is expectation (or mean) - is a $1-\alpha$ confidence interval centered around $\hat{\theta}$ for the parameter $\theta$. The upper bound would be $$\mathbb{E}[\hat{\theta}] + z_{\alpha / 2} \cdot \sigma_{\hat{\theta}}\text{.}$$ Now our task is to show that:

1) $\hat{p}$ follows a normal distribution

2) $\sigma_{\hat{p}} = \sqrt{\dfrac{\mathbb{E}[\hat{p}]\left(1-\mathbb{E}[\hat{p}]\right)}{n}}$

In the elementary statistics courses I was a teaching assistant in, 1) is usually tested by making sure that $$np(1-p) \geq 10\text{.}$$ Assuming 1) is satisfied, we have \begin{align} &\mathbb{E}[\hat{p}] = \dfrac{\mathbb{E}[X]}{n} = \dfrac{np}{n} = p \\ &\text{Var}[\hat{p}] = \dfrac{\text{Var}[X]}{n^2} = \dfrac{np(1-p)}{n^2} = \dfrac{n\cdot \mathbb{E}[\hat{p}](1-\mathbb{E}[\hat{p}])}{n^2} = \dfrac{\mathbb{E}[\hat{p}](1-\mathbb{E}[\hat{p}])}{n}\text{,}\end{align} or, $$\sigma_{\hat{p}} = \sqrt{\text{Var}[\hat{p}]} = \sqrt{\dfrac{\mathbb{E}[\hat{p}](1-\mathbb{E}[\hat{p}])}{n}}$$ so that our upper bound is $$\mathbb{E}[\hat{p}] + z_{\alpha/2}\sqrt{\dfrac{\mathbb{E}[\hat{p}](1-\mathbb{E}[\hat{p}])}{n}}\text{.}$$ Take the above formula, and set $z_{\alpha/2} = 3$.

ETA: If I have time, I will try to write a more intuitive explanation.

I'll just explain the topic in a little bit more depth than needed but just so you understand fully.

First off for independent random variables, $X_1,...,X_n$ that are from same distribution (you can think of this as sample of data) where $\mu_X=E(X_i)$ and $Var(X_i)=\sigma_X^2$. Now, for the arithmetic average/mean $\overline{X}=\frac{1}{n}\sum_{i=1}^{n}X_i$. Now, one important theorem you should know is the Centeral Limit Theorem (CLT) (this says that sum of iid random variables has an approximate normal distribution for large $n$ ie sample size). Thus CLT asserts that $\overline{X}$ has a normal distribution where $\mu=\mu_X$ and $\sigma^2=\frac{\sigma_X^2}{n}$.

Now, I think the notion of upper bound that you are using is a little misleading. The bound you are using is a the upper bound of a 99% confidence interval (which should not be interpreted as an upper bound for $\overline{X}$. This bound is $$\mu+3\sigma=\mu_X+3\frac{\sigma_X}{\sqrt{n}}$$

Now this is were your problem lies. $\frac{\sigma_X}{\sqrt{n}}\approx \sqrt{\frac{(1-\overline{X})\overline{X}}{n}}$ is only true when $\overline{X}$ represents a proportion (this standard deviation for a Bernoulli random variable