Uncertainties and errors

Random errors If a measurement is repeated many times, it can be expected that the measurement will be too large as often as it will be too small. So, if an average of these measurements is taken, the error will tend to cancel. The experimental result of the measurement of a given quantity x will thus be the average of the individual measurements: We then define the deviation of each individual measurement from the average by If the absolute magnitudes of all these deviations are smaller than the reading error, then we can quote the experimental result as

Random errors However, if the deviations from the mean are larger in magnitude than the reading error, the experimental error in the quantity x will have to include random errors as well. To estimate the random error we calculate the quantity which is called the unbiased estimate of the standard deviation. The result of the experiment is then expressed as

Systematic errors The most common source of systematic error is an incorrectly calibrated instrument. A systematic error will also arise if we use an instrument that has a zero error. Systematic errors are not always easy to estimate but sometimes the direction of the error is.

Systematic errors A systematic error will also arise if the experimenter makes the same error for all the measurement he takes.

Propagation of errors Suppose that in an experiment quantities a, b, c, etc., are measured, each with an error Δa, Δb, Δc, etc. That is a =a0 ± Δa and so on, where the subscript zero indicates the mean value of the quantity. The quantity Δa is called the absolute uncertainty or error in the measurement of the quantity a and the ratio Δa/a0 is the fractional or relative uncertainty or error. If multiplied by 100% we get the percentage error in the quantity a. If we wish to calculate a quantity Q in terms of a, b, c, etc., an error in Q will arise as a result of the individual errors in a, b and c. That is, the errors in a, b and c propagate to Q.

Multiplication, division, powers and roots Propagation of errors Addition and subtraction Multiplication, division, powers and roots

Best fit line If we have a reason to suspect that the data points we have plotted fall on a straight line, we must draw the best straight line through the points. This means using a ruler and choosing that line which goes through as many data points as possible in such a way that the distances between the line and the points on one side of it are, on average, the same as the distances between the line and points on the other side of it.

Best fit line In many experiments it will be necessary to obtain the slope (gradient) of the graph. To find the slope one must use the line of best fit and not data points. We must take two points on the line of best fit, which must be chosen to be as far apart as possible and then apply the formula

Uncertainties in the slope and intercept Having decided the line of best fit for a given set of data that are expected to fall on a straight line, it is usually necessary to calculate the slope and intercept of that straight line. However, since the data points are the result of measurements in an experiment, they are subject to experimental uncertainties. A simple way to estimate these uncertainties is by drawing two extreme straight lines and finding the slope of each. Both straight lines are made to pass through a point that is halfway in the range of the x values. The first line is then drawn so as to have the largest slope and still fit the data (this means it will pass at the extremes of the vertical error bars. The second line is made to have the least slope and still fit the data. The two slopes are then measured.