Worked
Example
If
a retailer would like to estimate the
proportion of their customers who bought an
item after viewing their website on a
certain day with a 95% confidence level and
5% margin of error, how many customers do
they have to monitor? Given that
their website has on average 10,000 views
per day and they are uncertain of their
current conversion rate, then they would
need to sample 370 customers. If,
however they know from previous studies that
they would expect a conversion rate of 5%,
then a sample size of 73 would be
sufficient.
Formula
This
calculator uses the following formula for
the sample size n:
n
= N*X / (X + N  1),
where,
X
= Z_{α/2}^{2} *p*(1p) /
MOE^{2},
and
Z_{α/2} is the critical value of the
Normal distribution at α/2 (e.g. for a
confidence level of 95%, α is 0.05 and the
critical value is 1.96), MOE is the margin
of error, p is the sample proportion, and N
is the population size. Note that a
Finite Population Correction has been
applied to the sample size formula.
Discussion
The
above sample size calculator provides you
with the recommended number of samples
required to estimate the true proportion
mean with the required margin of error and
confidence level.
You
can use the Alternative Scenarios to see how
changing the four inputs (the margin of
error, confidence level, population size and
sample proportion) affect the sample
size. By watching what happens to the
alternative scenarios you can see how each
input is related to the sample size and what
would happen if you didn't use the
recommended sample size. The larger the
sample size, the more certain you can be
that the estimates reflect the population,
so the narrower the confidence interval.
However, the relationship is not linear,
e.g., doubling the sample size does not
halve the confidence interval.
Definitions
Margin of
error
The
margin of error is the the level of
precision you require. This is the plus or
minus number that is often reported with an
estimated proportion and is also called the
confidence interval. It is the range in
which the true population proportion is
estimated to be and is often expressed in
percentage points (e.g., ±2%). Note
that the actual precision achieved after you
collect your data will be more or less than
this target amount, because it will be based
on the proportion estimated from the data
and not your expected sample proportion.
Confidence
level
The
level of confidence indicates the likelihood
that the true proportion is within the
margin of error. It is reasonable to assume
that the true value will fall within this
range on 95% of the occasions if the study
is repeated and the range is calculated each
time. You can be more confident that the
interval contains the true proportion if
your confidence level is higher.
Population
size
This
is the total number of distinct individuals
in your population. To account for
sampling from small populations, we use a
finite population correction in this
formula. If you're not sure how big your
population is, you can use 100,000 as a
conservative estimate. If a population is
over 100,000 people, the sample size doesn't
change all that much.
Sample
proportion
The
proportion of the population that you expect
to see in the results is known as the sample
proportion. The results of a previous survey
or a small pilot study can often be used to
make this determination. Whenever you aren't
sure, go with a conservative estimate like
50%, which will give you the largest
possible sample size. Note that the Normal
approximation to the Binomial distribution
is used to calculate the sample size. In
cases where the proportion of the sample is
close to 0 or 1, this approximation is not
valid and you should use a different method
for calculating sample size.
Sample size
With
the required margin of error and level of
confidence, this is the bare minimum sample
size you require. Be aware that if some
respondents choose not to respond, they will
be excluded from your sample. If this is a
concern, you will need to increase your
sample size. Nonresponse will often lead to
biases in your estimate, so the higher the
response rate, the better your estimate will
be.
