Statistical Analysis Error Bars When we collect a set of data for a given variable, it shows variation. When displaying this data in a graph, the variation can be shown using error bars. Error bars can show the range of data.

The mean gives a measure of the central value, or average, of a set of figures. The standard deviation gives a measure of the spread of data around the mean. 68% of the data falls within one standard deviation of the mean. 95% of the data falls within two standard deviations of the mean.

Standard deviation data from the two sites Standard deviation data from the two sites. Standard deviation can be calculated via Excel 2010.

One effective way to represent data is to draw a graph that shows error bars of the standard deviation. Here, each sample has the mean +/- one standard deviation. Note that this graph, unlike the original graph of the means, shows that there may be significant differences in the data from both sites.

The t test. A common form of data analysis is to compare two sets of data to see if they are the same or different. Eg/ are the mollusc shells from the two locations significantly different? Often the means are quite close and it is difficult to judge whether the two sets are the same or are significantly different. To compare two sets of data we use the t test , which tells you the probability (P) that the two sets are basically the same. This is called the null hypothesis . P varies from 0 (not likely) to 1 (certain). The higher the probability, the more likely it is that the two sets are the same, and that any differences are just due to random chance. The lower the probability, the more likely it is that that the two sets are significantly different, and that any differences are real. In biology the critical probability is usually taken as 0.05 (or 5%). This may seem very low, but the fact is that biology experiments are expected to produce quite varied results.

Once a value for t has been calculated, the degrees of freedom must be worked out. In general, the degrees of freedom of an estimate is equal to the number of independent scores that go into the estimate minus the number of parameters used as intermediate steps in the estimation of the parameter itself. Eg/ For the data on the mollusc shell length, the number of independent scores that go in to the calculation = 20 (10 from each site). The number of parameters = 2 (two sites were analysed). So the value for the degrees of freedom is 20 – 2 = 18. The value of t for the mollusc shell length has been calculated to be 1.98. The next step is to consult a table of critical values for t, and refer to the value for 18 degrees of freedom under the p = .05 probability.

We can see from the table that the critical value for t using 18 degrees of freedom is 2.10. since the calculated value for t, 1.98, is less than this figure, we are able to accept the null hypothesis (differences between the two sets of data are due to chance). In conclusion, we can say that there is no significant difference in the shell lengths from the two sites.

Correlation and Causation In biological investigations, it is often necessary to determine whether or not two variables are correlated. For example, is water temperature correlated with limpet movement on inter tidal rock platforms? Once the data is collected and plotted, a line of best fit can be applied. A correlation coefficient can then be calculated which indicates how well the data fit the line of best fit. Correlation coefficients are expressed as numbers between -1 and +1. The closer the number is to either -1 or +1, the stronger the correlation. This graph represents a strong positive correlation.

This graph shows a relatively strong negative correlation This graph shows a relatively strong negative correlation. A correlation coefficient close to zero means that there is little correlation. This graph shows the correlation between repetitive sequences of DNA and reproductive rates in animals.

Correlation vs Causation Correlation vs Causation. Two variables may show a high degree of correlation, but it does not necessarily mean that one causes the other. eg/ Since 1952, CO2 levels in the atmosphere have risen. During the same period, white collar crime has increased. Often, data will reveal that there may be a causal effect between two variables, but only time and further analysis can establish that link. For example, the link between smoking and heart disease took many years to establish.