1. 4.0 Confidence Limits cdf for the z-statistic
probability and confidence intervals
confidence interval of the average
some practical comments about the number of replicates
visualizing confidence intervals
confidence intervals with s
the t-parameter and its pdf
t-based confidence intervals
replicates revisited
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2. Normal CDF
The normal pdf is given by one particular form of Gauss's equation. This equation does not have an indefinite integral, thus the cdf cannot be written in a closed form.
The value of F(m) has to be determined by numeric integration, with the results tabulated. To avoid needing a table for all values of m and s, a normalized variable, called the z-statistic, is created.
F(z=-3) = 0.001
F(z=0) = 0.500
F(z=+3) = 0.999
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3. Probability Intervals
The cdf can tell us the range about m that contains a specified fraction of all measured values. For a normal density, this range is ordinarily chosen to be symmetric about m. Using the tabulated values, probability can be determined for the listed intervals.
-k < z < +k
-1.00 < z < %
-1.96 < z < %
-2.00 < z < %
-3.00 < z < %
The above inequalities can be rewritten in terms of x by using the definition of z.
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4. Confidence Intervals
Probability intervals tell us the range in which we expect to find a certain percentage of measured random values. Of more interest would be the interval about a single measurement that would contain the mean, m. This range is called the confidence interval or confidence limits.
x - ks < m < x + ks
The confidence level used in analytical chemistry is 95%, thus we would use k = 1.96, and say that the interval from x s to
x s contains the true value 95% of the time. To simplify the calculations, 1.96 is almost always rounded up to 2. With this change, the confidence range is written m = x  2s.
It has to be recognized that 5% of the time we write such a range it will not contain m. If it were desired to increase confidence to 99.9% the range would increase to x  3.3s. This range is considered too large by the analytical community, in view of the fact that confidence has only increased ~5%.
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5. Confidence Interval of the Average
The confidence interval of an average can be written in a similar manner by remembering that if x has the pdf n(m,s), then the average, , has the pdf n(m,s/N1/2).
For 95% confidence we have the following interval.
Both the equation and the graph imply that, for a given confidence, the range can be arbitrarily decreased by averaging more data.
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6. Comments on Increasing N
the number of replicate measurements can only be increased under very strict conditions
obtaining replicates requires more time, allowing instrumental sensitivity to drift
reagent concentrations can change over time, e.g. NaOH
for some experiments replicates cost money, e.g. titrations
some samples are not amenable to replication, e.g. taking blood from a newborn child
since the interval only drops as the root of the replicates, it is often more effective to decrease s
When performing an analysis for the first time we usually take three replicates. This is the minimum number of replicates allowing a "vote," that is, recognize outliers.
when s is unknown, it is often cost effective to use 10 to 12 replicates to decrease the uncertainty in s
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7. Visualization of Confidence Intervals
The data below are the confidence intervals for the averages of three values. The random variable comes from a normal density, n(5,0.1). The averages then have a normal density, n(5,0.058). The 95% confidence interval is given by m = avg  The replicates with ranges not including m are identified. This method of graphing ranges is used most often in curve fitting.   
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8. Control Charts
In a control chart the average is plotted along with the expected upper and lower ends of the confidence range. With 60 replicates you would expect on average three of the data points to be outside the range. If more or less points are outside the range, the analysis is out of control. This plot is used extensively in quality control when analyzing calibration standards.
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9. 1.96 Confidence Interval with s
When s is substituted for s to establish the confidence limits,
m is not included 95% of the time. This is shown in the following graph, where 9 miss the mean instead of ~0.0560 = ~3.         
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10. Why Doesn't the Interval with s Work?
The numeric limits we have been using are actually values of the zstatistic. This can be seen by rearranging the limit expression.
The average has the normal pdf, n(m,s/N1/2); the numerator has the pdf, n(0,s/N1/2); and, z has the pdf, n(0,1).
What happens when s is replaced by s? To determine this, use the symbol, t, for the numeric limit and look at its functional form.
The numerator of t has the normal pdf, n(0,s/N1/2). However, the denominator has the pdf of s, which was shown in Section 3.3 to depend upon N and have the functional form of a normal moment.
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11. The PDF of t
The pdf of t is derived in a manner similar to that for the standard deviation.
The graph of f(t) is shown at the right for N = 3 (red), N = 6 (blue) and N = 101 (magenta). The pdf is symmetric about 0. As N  , the pdf approaches n(0,1).
Numeric integration of f(t) is used to find the value of t that has a probability interval of F(t) - F(-t) = 0.95.
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12. Confidence Intervals Using t
The t-limits are given in the following table for a few values of the degrees of freedom, f = N-1. f 2 3 4 9 19 29 39 
t(0.95)
4.30
3.18
2.78
2.26
2.09
2.05
2.02
1.96
The 95% confidence limits about the average of 3 replicates is given by The data set previously shown with 1.96 is now shown with 4.30.  
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13. Value of N when Using t-Intervals
When s is used to determine confidence intervals it is often more advantageous to make replicate measurements than when using s. This is because the value of t is decreasing at the same time that 1/N1/2 is decreasing. The graph below shows both z/N1/2 and t/N1/2.
With t-intervals it is often cost effective to run ~10 replicates. Much beyond this value, the reduction in interval size is dictated by the slowly changing 1/N1/2 term. In particular note the factor of ~2 reduction in going from
N = 3 to N = 5.
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