Probability Of 2/3 observations being >= 2 standard deviations in a set of observations

I'm using statistical process control charts where the values of the first n observations provide an average and standard deviation, and the position of the subsequent observations relative to those values highlights any variation that is not considered normal in the process.

One of those markers is where 2 of 3 observations are equal or above 2 standard deviations. My question is how would I calculate the probability of that occurring given that the observations are independent (e.g the number of people buying cars, number of people buying ice creams, number of people admitted to a hospital)?

Here's my workings:

The normal distribution table shows zscore of 2 and -2 as 0.022750

The probability that the absolute zscore is >= 2 is 2(0.022750) = 0.0455

There are 4 ways that 2/3 observations above 2 standard deviations can occur: I then multiply the probability of each obs together for each combination and sum the results to get the probability of 2/3 observations being above 2 standard deviations.

Any help or corrections on this would be greatly appreciated.

Many Thanks

Nick

• If the mean $\mu$ and standard deviation $\sigma$ are fixed (not updated to account for each new observation), then the probability of an observation from a normally distributed process 'in control' is above $\mu + 2\sigma$ is about 0.023. [In R statistical software: 1 - pnorm(2) returns 0.02275013.] Then the probability of $\ge 2$ out of the next 3 above $\mu +2\sigma$ is found from the binomial distribution to be about 0.0016 (assuming the process remains in control). [In R: sum(dbinom(2:3, 3, .023)) returns 0.00156267 .] If you need more, please say what & someone may be able to help. – BruceET Jul 9 '18 at 18:10
• Your method seems OK, except you are looking for observations outside $\mu \pm 2\sigma$ instead of observations above $\mu + 2\sigma.$ sum(dbinom(2:3, 3, .0455)) returns 0.006022357, as you say. Do you want more than $2\sigma$ above $\mu$ or more than $2\sigma$ away from $\mu?$ – BruceET Jul 9 '18 at 18:31