Evaluating data

In a nutshell

Evaluation is the process of explaining results of an experiment to what was expected to happen.. Conclusions are only drawn when the results have been evaluated. The evaluation of results includes identifying and dealing with anomalies, as well as calculating uncertainty in readings. 

Equations

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Definitions

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Anomalies

Anomalies are measured values that seem not to follow the expected correlation of values. They arise from experimental errors and will affect the results of an experiment in a way that decreases its accuracy.

Part of evaluating results involves identifying anomalies before drawing a conclusion. These anomalies can be identified by looking for values that don't fit the trend of the experiment as a whole.

1. Anomaly

When an anomaly is identified, it is discounted and that particular set-up is repeated so that a drawn conclusion is more accurate. They are also ignored when calculating the mean of values.

Uncertainties

The uncertainty is the amount of error a particular value may have and how far the measurement may deviate from its true value. All equipment have a limited precision, which leads to some level of uncertainty.

There are two types of uncertainty - absolute uncertainty and percentage uncertainty. A reading with absolute uncertainty is written in the following way.

​$$reading = x\, \pm \Delta x$$​​

​$$\Delta x$$​ is the absolute uncertainty on the measurement and is usually defined as half of the smallest interval a piece of equipment can measure.

Percentage uncertainty is calculated using the following equation.

$$percentage\ uncertainty = \dfrac{absolute\ uncertainty}{reading} \times 100$$

Absolute error is always fixed, as it relates to the piece of equipment used, whereas percentage uncertainty will reduce when the reading gets larger. For this reason, it's more precise to measure a larger reading then divide the reading into intervals. 

To give an example, measuring ten fringes in a diffraction pattern then dividing by the ten intervals is more precise than just measuring one fringe, as the percentage uncertainty from measuring ten fringes is lower.

Example

A weight scales can measure weight to the closest $$0.1\,kg$$, giving it an error of $$\pm0.05\,kg$$. Calculate the percentage uncertainty if an object is weighed to be $$10.4\,kg$$.

First, write down the equation:

​$$percentage\ uncertainty = \dfrac{error}{reading}\times 100$$​​

Substitute in the values:

​$$percentage\ uncertainty = \dfrac{0.05}{10.4}\times 100$$​​

​$$percentage\ uncertainty = 0.48\%$$​​

The uncertainty on the reading is $$\underline{0.48\%}$$.​

Combining uncertainties

When measurements are involved in calculations, it's important to also combine the uncertainties related to those measurements. 

When two measurements are added together or subtracted from each other, then the absolute uncertainties are added together. When two measurements are multiplied or divided, then the percentage uncertainties are added together. When a measurement is raised to a power, then the percentage uncertainty of the measurement is multiplied by that power.

Example

A rectangle is measured to have a short side of $$24.5\,\pm0.1\,cm$$ and a long side of $$33.4\,\pm0.1\,cm$$. Calculate the area and its uncertainty.

First, calculate the area.

​$$area = 24.5 \times 33.4$$

$$area = 818.3\,cm^2$$​​​

Next, calculate the uncertainty. The two measurements are being multiplied together, so the percentage uncertainties should be added. First, calculate the percentage uncertainties of the two values.

​$$percentage\ uncertainty = \dfrac{0.1}{24.5} \times 100$$​​

​$$percentage\ uncertainty = 0.4\%$$​​

​$$percentage\ uncertainty = \dfrac{0.1}{33.4}\times100$$​​

​$$percentage\ uncertainty = 0.3\%$$​​

Now, add the percentage uncertainties together.

​$$0.4\% + 0.3\% = 0.7\%$$​​

The area and uncertainty in the area of the rectangle is $$\underline{818.3\,cm\pm0.7\%}$$.​

Evaluation

The results of the experiment need to be evaluated so that they can be explained and used to back up the conclusion that was drawn. This evaluation assesses the validity of the experiment and data, and whether the experiment can be reproduced easily.

Good practices of evaluation is identifying limitations and future improvements to remove those limitations, identifying sources of error and explaining how to improve on those and comparing readings to existing data sets and explaining how they differ and how they are similar.