Here is number of points in the sample. Thus, as the size of the sample increases, the uncertainty of the estimate decreases like one over the square root of the size of the sample. In the example above, our estimate for the probability that a flipped thumbtack lands pointy side up would be .

Another approach is to use what are called * confidence
intervals*. Given the estimated value of , a confidence
interval is an range of values in which the true value of is
likely to be. By ``likely'' one often means that the probability that
the true value falls in the interval is 95%. This is called the
95% confidence interval.
You might know that for a normal distribution it is expected that the
data falls within one standard error 68% of the time, and within two
standard errors about 95% of the time. One says that one has 95%
confidence that the true value is between the estimate minus two
standard errors and the estimate plus two standard errors. Now, the
distribution of the measured estimate is not normal, it is binomial,
but a normal distribution can be approximated by a normal
distribution if the value of is not too close to or . Or
one can use a binomial table. Figure shows a graph of the 95%
confidence intervals.

In the example about, we would be 95% confident that the true value for the probability of a flipped thumbtack landing pointy side up is between and . If a more accurate estimate is desired, a larger number of experiments is required. The size of the interval will decrease with the square root of the number of experiments.