A confidence interval is a notion from

probability. For a given
confidence level, it represents the probability that a random sample
will fall within that interval.

For instance, let's suppose the average flight speed of an unladened
swallow is 15 m/s with a 95% confidence interval ±1.5 m/s. This
means that for any randomly selected swallow, there is a 95% probability
that his flight speed will be between 13.5 m/s and 16.5 m/s (unless he
carries a coconut).

In experimental data, it is quite often impossible to measure the
average of the entire population (the population mean). For example,
it is impractical to measure the flight speed of *every*
swallow that is alive. In this case the sample mean is used as an
estimate of the population mean.

There is another problem with experimental data, which is that
usually we don't know the variance of a population. The variance
describes how much a random variable deviates from the population
mean. Since the sample mean is only an estimate of the population
mean, the variance is unknown.

In many experimental measurements, the data of an entire population
will follow a normal distribution. For instance, if we measure the
time it takes a ball to drop from 10 m, and remeasure it a great many
times, the measured values will fall around 1.43 s.; most of the values
will be close to the mean, fewer will be far off. If we plot the
*number of observations* of any specific time that we measured as
a function of this time, it will resemble the typical bell curve that is
described by the normal distribution.

However... As I mentioned before, we cannot measure the variance if
the entire population mean is not available. This is where William S.
Gosset, better known as *student* comes to the rescue. The so
called *student t-distribution* makes a correction to the
normal distribution, based on the number of samples that were taken
from the population. This data can be looked up in a book with
mathematical tables. You will need the *double sided* t-
distribution values. The values will be listed for a certain P-value,
corresponding to the required confidence limit (P = 0.05; confidence
interval = 1 - 0.05 = 0.95 = 95%). For each confidence interval, the
values are listed with an increasing number of degrees of freedom. Any
number of measurements, `n` will correspond to `n`-1
degrees of freedom; one degree of freedom is used to calculate the
mean. Frequently used confidence limits are the 99%, 95%, and 50%
confidence intervals:

Confidence limit
DF 99% 95% 50%
1 63.656 12.706 1.0000
2 9.9250 4.3027 0.8165
3 5.8408 3.1824 0.7649
4 4.6041 2.7765 0.7407
5 4.0321 2.5706 0.7267
6 3.7074 2.4469 0.7176
7 3.4995 2.3646 0.7111
8 3.3554 2.3060 0.7064
9 3.2498 2.2622 0.7027
10 3.1693 2.2281 0.6998
15 2.9467 2.1315 0.6912
20 2.8453 2.0860 0.6870
40 2.7045 2.0211 0.6807
100 2.6259 1.9840 0.6770
∞ 2.5758 1.9600 0.6745
values calculated using the TINV function in Excel

At infinite sample size the t-distribution values approach the normal
distribution. Assuming that the entire population represented by the
normal distribution, the confidence interval δ`q` for a
mean `q` can now be calculated with:

δ`q`= `ts` / √`n`

Where δ`q` is the confidence interval, `t` is
the value of the t-distribution for n-1 degrees of freedom,
`s` is the standard deviation, and `n` is the number
of measurements in the sample.

As an example, we're going to measure the average number of users
online as reported by the Everything Snapshot. The number of users
online for one week is:

17 December 2000: 42
18 December 2000: 58
19 December 2000: 48
20 December 2000: 45
21 December 2000: 57
22 December 2000: 43
23 December 2000: 41

The sample mean is 47.7 users. The standard deviation, s
is 7.06433. The 95% confidence limit for 6 degrees of freedom (7
measurements minus one) is 2.4469. Thus, the 95% confidence interval is:

δ`q`= 2.4469 x 7.06433 / √7 =
6.5

Therefore, the average number of users online during the
Everything Snapshot is 47.7 ± 6.5 users for a 95% confidence interval.