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Chapter 7

# Chapter 7 - Chapter 7 Statistical Data Treatment and...

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Chapter 7 Statistical Data Treatment and Evaluation Experimentalist use statistical calculations to sharpen their judgments concerning the quality of experimental measurements. These applications include: Defining a numerical interval around the mean of a set of replicate analytical results within which the population mean can be expected to lie with a certain probability. This interval is called the confidence interval (CI).

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Determining the number of replicate measurements required to ensure at a given probability that an experimental mean falls within a certain confidence interval. Estimating the probability that (a) an experimental mean and a true value or (b) two experimental means are different. Deciding whether what appears to be an outlier in a set of replicate measurements is the result of a gross error or it is a legitimate result. Using the least-squares method for constructing calibration curves.
CONFINENCE LIMITS Confidence limits define a numerical interval around x that contains μ with a certain probability. A confidence interval is the numerical magnitude of the confidence limit. The size of the confidence interval, which is computed from the sample standard deviation, depends on how accurately we know s, how close standard deviation is to the population standard deviation σ . Finding the Confidence Interval when s Is a Good Estimate of σ A general expression for the confidence limits (CL) of a single measurement CL = x ± z σ For the mean of N measurements, the standard error of the mean, σ / N is used in place of σ CL for μ = x ± z σ / N _ _ _

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Finding the Confidence Interval when σ Is Unknown We are faced with limitations in time or the amount of available sample that prevent us from accurately estimating σ . In such cases, a single set of replicate measurements must provide not only a mean but also an estimate of precision. s calculated from a small set of data may be quite uncertain. Thus, confidence limits are necessarily broader when a good estimate of σ is not available.

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…continued… To account for the variability of s, we use the important statistical parameter t, which is defined in the same way as z except that s is substituted for σ . t = (x - μ ) / s t depends on the desired confidence level, but t also depends on the number of degrees of freedom in the calculation of s. t approaches z as the number of degrees of freedom approaches infinity.
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Chapter 7 - Chapter 7 Statistical Data Treatment and...

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