margin of error formula

[robhelps]

Hi,

I have a team of electricians and i want to work out their response rates to jobs within a certain time. The objective is to get to 98% within 24 hours.

I have taken a sample of each persons jobs to work out the percentage that were done in 24 hours, over 24 hours and over 48 hours.

Eeach person has done about 1200 jobs but I have taken a sample of roughly 25 (varies on each person)

My main issue is, when I work out the percentage for person C for example, the one call he did not make within 24 hours has altered his percentage to 95%.

I know that mathematically this is correct, but is there a formula that can take into consideration a margin of error for just the odd job gone wrong? Increasing the sample size is obviously one answer but ideally I want to work with the data I have.

I've attached the worksheet.

Many thanks for reading.

[AndyPS]

Without getting into heavy statistics where the relationship of sample size to population matters, a very simple (could say simplistic) measure of the success rate against the standard would be to estimate the standard error. Say that in the sample, 97% of jobs took place within 24 hours - thus 3% did not. The (not quite) standard deviation is estimated as SQRT(f*(1-f)/s) where f is the percentage expressed as a fraction of the sample - e.g. 96% = 0.96) and s equals the sample size (25?).
The next thing to do is decide what confidence level is needed. Let's assume it is 95% - i.e. 19 times out of 20 samples, the real rate falls within our estimates. This can be expressed as the number of standard deviations = 2 for 95%.
Putting this together for a sample of 25, a result of 96% and 95% confidence, we calculate 2*SQRT(0.96*0.04/25) = 7.8%.
Thus the true percentage success lies between 88% and 100% (can't have more!) for the given level of confidence.
As you will realise, with sampling you can also incorrectly estimate that an operative achieved the target even when that was not the case.
The only way of getting tighter limits is to increase the sample.
Hope this helps.

[robhelps]

Thanks AndyPS