# 4.11: Chapter Summary and Key Terms

- Page ID
- 220686

## 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 that affect its precision. A propagation of uncertainty allows us to estimate how these determinate and indeterminate errors 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 variances. To compare two sets of data you first must determine whether the data is paired or unpaired. For unpaired data you also must decide if you can pool the standard deviations. 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 if you decide to reject an outlier. Finally, the detection limit is a statistical statement about the smallest amount of analyte 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.

## Key Terms

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