Read through the summarized points below carefully. As a last exercise, continue with the mixed exercises (link below), which summarize everything that you (should) have learned today.

## Error propagation

- All experiments contain experimental error
- For many analytical instruments (glassware, balance, ...), the error is shown on the equipment
- The error in a final answer can be calculated by using error propagation rules
- Errors for multiplication and division have to be calculated relatively, because these operations often involve different quantities

## The normal distribution

- Many experimental errors are normally distributed
- They can be characterised by a mean value and a standard deviation
- The mean describes the location
- The standard deviation describes the spread around the mean
- Random errors are related to the standard deviation
- Systematic errors are related to the difference between the mean and the (unknown)
*true*value (the*bias*).

## Confidence intervals

- The larger the spread around a central value, the wider a confidence interval will be
- A 95% confidence interval for an individual value will contain the next experimental value with a probability of 95%
- A crude estimate of a 95% confidence interval is the mean plus or minus twice the standard deviation
- A crude estimate of a 99% confidence interval is the mean plus or minus three times the standard deviation
- Confidence intervals for the mean are narrower than confidence intervals for individual values since errors cancel out
- The standard deviation of the mean is the standard deviation of the individual values divided by the square root of the number of measurements

## Least squares regression

- The linear relationship between two variables
*x*and*y*can be described by the abcissa and the slope of the optimal straight line - Standard errors for abcissa and slope can be calculated, and therefore confidence intervals too
- It is assumed that the error in
*x*is much smaller than the error in*y* - Residuals are assumed to be independent and normally distributed with constant variance
- Regression lines may be used for calibration: the calibration line is set up using a set of calibration samples and the concentration in an unknown sample can be predicted
- Regression lines can be used to compare methods

As a final test, prepare yourself for the mixed exercises.