If 50% of all the people in a population of 20000 people drink coffee in the morning,
and if you were repeat the survey of 377 people ("Did you drink coffee this morning?")
many times, then 95% of the time, your survey would find that between 45% and 55% of
the people in your sample answered "Yes".
The remaining 5% of the time, or for 1 in 20 survey questions, you would expect the
survey response to more than the margin of error away from the true answer.
When you
survey a sample of the population, you don't know that you've found the correct
answer, but you do know that there's a 95% chance that you're within the margin of
error of the correct answer.
Try changing your sample size and watch what happens to the * alternate scenarios* .
That tells you what happens if you don't use the recommended sample size, and how and confidence level (that 95%) are related.
To learn more if you're a beginner, read Basic
Statistics: A Modern Approach and
The Cartoon Guide to Statistics . Otherwise, look at the
more advanced books .

In terms of the numbers you selected above, the sample size * n* and margin of error
* E* are given by
* x* = * Z* ( * c* / 100 ) 2 * r* (100-* r* )
* n* =
* N x* / ((* N* -1)* E* 2 + * x* )
* E* = Sqrt[ (* N* - * n* )* x* / * n* (* N* -1) ]
where * N* is the population size, * r* is the fraction of
responses that you are interested in, and * Z* (* c* /100) is
the critical
value for the confidence level * c* .

If you'd like to see how we perform the calculation, view the page
source. This calculation is based on the Normal
distribution , and assumes you have more than about 30 samples.

About Response distribution : If you ask a random sample of
10 people if they like donuts, and 9 of them say, "Yes", then the
prediction that you make about the general population is different than it
would be if 5 had said, "Yes", and 5 had said, "No". Setting the response
distribution to 50% is the most conservative assumption. So just leave it
at 50% unless you know what you're doing.
The sample size calculator computes the critical value for the normal
distribution. Wikipedia has good articles on statistics.
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