What is the relationship between the size of a sample and the width of the confidence interval for the mean?

Let's say you have a sample mean, you may wish to know what confidence intervals you can place on that mean. Colloquially: "I want an interval that I can be P% sure contains the true mean". (On a technical point, note that the interval either contains the true mean or it does not: the meaning of the confidence level is subtly different from this colloquialism. More background information can be found on the NIST site).

The formula for the interval can be expressed as:

Where, Ys is the sample mean, s is the sample standard deviation, N is the sample size, /α/ is the desired significance level and t(α/2,N-1) is the upper critical value of the Students-t distribution with N-1 degrees of freedom.

Note

The quantity α   is the maximum acceptable risk of falsely rejecting the null-hypothesis. The smaller the value of α the greater the strength of the test. The confidence level of the test is defined as 1 - α, and often expressed as a percentage. So for example a significance level of 0.05, is equivalent to a 95% confidence level. Refer to "What are confidence intervals?" in NIST/SEMATECH e-Handbook of Statistical Methods. for more information.

From the formula, it should be clear that:

• The width of the confidence interval decreases as the sample size increases.
• The width increases as the standard deviation increases.
• The width increases as the confidence level increases (0.5 towards 0.99999 - stronger).
• The width increases as the significance level decreases (0.5 towards 0.00000...01 - stronger).

The following example code is taken from the example program students_t_single_sample.cpp.

We'll begin by defining a procedure to calculate intervals for various confidence levels; the procedure will print these out as a table:

We'll define a table of significance/risk levels for which we'll compute intervals:

Note that these are the complements of the confidence/probability levels: 0.5, 0.75, 0.9 .. 0.99999).

Next we'll declare the distribution object we'll need, note that the degrees of freedom parameter is the sample size less one:

Most of what follows in the program is pretty printing, so let's focus on the calculation of the interval. First we need the t-statistic, computed using the quantile function and our significance level. Note that since the significance levels are the complement of the probability, we have to wrap the arguments in a call to complement(...):

Note that alpha was divided by two, since we'll be calculating both the upper and lower bounds: had we been interested in a single sided interval then we would have omitted this step.

Now to complete the picture, we'll get the (one-sided) width of the interval from the t-statistic by multiplying by the standard deviation, and dividing by the square root of the sample size:

The two-sided interval is then the sample mean plus and minus this width.

And apart from some more pretty-printing that completes the procedure.

Let's take a look at some sample output, first using the Heat flow data from the NIST site. The data set was collected by Bob Zarr of NIST in January, 1990 from a heat flow meter calibration and stability analysis. The corresponding dataplot output for this test can be found in section 3.5.2 of the NIST/SEMATECH e-Handbook of Statistical Methods..

As you can see the large sample size (195) and small standard deviation (0.023) have combined to give very small intervals, indeed we can be very confident that the true mean is 9.2.

For comparison the next example data output is taken from P.K.Hou, O. W. Lau & M.C. Wong, Analyst (1983) vol. 108, p 64. and from Statistics for Analytical Chemistry, 3rd ed. (1994), pp 54-55 J. C. Miller and J. N. Miller, Ellis Horwood ISBN 0 13 0309907. The values result from the determination of mercury by cold-vapour atomic absorption.

This time the fact that there are only three measurements leads to much wider intervals, indeed such large intervals that it's hard to be very confident in the location of the mean.