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*).

## Prediction & Confidence intervals

- The larger the standard deviation (the spread around a central value), the wider a prediction interval will be.
- A 95% prediction interval will contain the next experimental value with a probability of 95%.
- A crude estimate of a 95% prediction interval is the mean plus or minus twice the standard deviation.
- A crude estimate of a 99% prediction interval is the mean plus or minus three times the standard deviation.
- Confidence intervals are narrower than their corresponding prediction intervals, 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 and can be described by the intercept and the slope of the optimal straight line.
- Standard errors for the intercept with the axis and slope can be calculated, and therefore confidence intervals too.
- It is assumed that the error in is much smaller than the error in .
- 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.