# 4.S: Evaluating Analytical Data (Summary)

The data we collect are characterized by their central tendency (where the values cluster), and their spread (the variation of individual values around the central value). We report our data’s central tendency by stating the mean or median, and our data’s spread using the range, standard deviation or variance. Our collection of data is subject to errors, including determinate errors that affect the data’s accuracy, and indeterminate errors affecting its precision. A propagation of uncertainty allows us to estimate how these determinate and indeterminate errors will affect our results.

When we analyze a sample several times the distribution of the results is described by a probability distribution, two examples of which are the binomial distribution and the normal distribution. Knowing the type of distribution allows us to determine the probability of obtaining a particular range of results. For a normal distribution we express this range as a confidence interval.

A statistical analysis allows us to determine whether our results are significantly different from known values, or from values obtained by other analysts, by other methods of analysis, or for other samples. We can use a t-test to compare mean values and an F-test to compare precisions. To compare two sets of data you must first determine whether the data is paired or unpaired. For unpaired data you must also decide if the standard deviations can be pooled. A decision about whether to retain an outlying value can be made using Dixon’s Q-test, Grubb’s test, or Chauvenet’s criterion. You should be sure to exercise caution when deciding to reject an outlier.

Finally, the detection limit is a statistical statement about the smallest amount of analyte that we can detect with confidence. A detection limit is not exact since its value depends on how willing we are to falsely report the analyte’s presence or absence in a sample. When reporting a detection limit you should clearly indicate how you arrived at its value.

## 4.9.1 Key Terms

 alternative hypothesis bias binomial distribution box plot central limit theorem Chauvenet’s criterion confidence interval constant determinate error degrees of freedom detection limit determinate error Dixon’s Q-test dot chart error F-test Grubb’s test histogram indeterminate error kernel density plot limit of identification limit of quantitation mean median measurement error method error normal distribution null hypothesis one-tailed significance test outlier paired data paired t-test personal error population probability distribution propagation of uncertainty proportional determinate error range repeatability reproducibility sample sampling error significance test standard deviation standard error of the mean Standard Reference Material tolerance t-test two-tailed significance test type 1 error type 2 error uncertainty unpaired data variance

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