Processing and presenting data

In a nutshell

It is important to analyse your data so you can interpret what your results mean and how reliable they are. This can be done using a variety of mathematical techniques.

Processing data

Percentage change

Calculating percentage change is one way to process the data. It helps you quantify how much the data has changed.

$Percentage\ change = {{final\ value - original\ value}\over original\ value} \times 100$

A positive percentage change indicates a percentage increase and a negative percentage chance indicates a percentage decrease.

Averages, range and standard deviation

As you always perform at least three repeats of an experiment, the mean can be calculated. This is easily performed by adding up all the values and dividing by the total number of values.

You may also calculate the median, this is the middle number you get when your data is put into numerical order. Or the mode, this is the most common number that appears in your data.

The range is also used to process data as it tells you how spread out the data is. It is calculated by subtracting the smallest value from the largest value.

The standard deviation tells you how spread about the mean the values are. This means a small standard deviation indicates the data points are close to mean and therefore precise.

Statistical tests

Statistical tests such as the Chi-squared test and the Student's t-test can be used to analyse data.

The Student's t-test can be used to assess whether there is a significant difference between the means of two data sets. Using a critical value, you can determine whether the differences in means is due to chance.

The Chi-squared test can be used on categorical data to compare whether your observed results are statistically different from your expected results, it is used to test the null hypothesis. The null hypothesis examines the results of scientific experiments. It is based on the assumption that there is no statistically significant difference between the set of observations and any differences are due solely to chance.

$chi\ squared= sum\ of {(observed\ numbers - expected\ numbers)^2\over expected\ numbers}$

$\chi^2 = \sum{(O-E)^2\over E}$

Procedure

1.	Work out the chi-squared value using the above formula.
2.	Compare your value to the critical value. This is the value of chi-squared that corresponds to a $0.05\ (5\%)$ level of probability that the difference between the observed and expected values is due to chance. This probability level is known as the p-value.
3.	If $\chi^2$ is smaller than the critical value then there is no significant difference between observed and expected results. The null hypothesis cannot be rejected.
4.	If $\chi^2$ is larger than or equal to the critical value, then there is a significant difference between the observed and expected results. The null hypothesis can be rejected and something other than chance is causing the difference.

Tip: Remember to check whether you need to present your data to a certain number of significant figures or in standard form.

Graphs

When presenting your data, it is important that you choose an appropriate graph.

For discrete data or qualitative data you can use bar charts or pie charts, but when you have continuous data you should used histograms or line graphs. When plotting one variable against the other you should use a scatter graph.

Logarithms

When your data points are very small and very large, they can be difficult to plot on the same axis. This is why we use a logarithmic scale.

Example

On a $log_{10}$ scale, each value is ten times larger than the value before. This means your graph scale may show one, but represent ten.

Linear value	Type into calculator	$log_{10}$ value
$10$	$log_{10}(10)$	$1$
$100$	$log_{10}(100)$	$2$
$1000$	$log_{10}(1000)$	$3$
$10000$	$log_{10}(1000)$	$4$

Gradient

You can use the gradient to calculate the rate. The rate is a measure of how much something is changing over time.

For linear graphs, the rate can be calculated used the following formula:

$Gradient = {change\ in\ Y\over change\ in\ X}$

For curved graphs, you can fine the rate by drawing a tangent at the point where you want to calculate the rate from. You can then use the above formula again to calculate the gradient.