### Maths for Biology

Love it or hate it, you’ll need to do some maths in biology. Here’s a collection of the most common mathematical techniques needed for A-Level.

Methods of displaying frequency distribution:

1. A data table usually demonstrates 2 variables. More variables can be included, but this decreases clarity.
2. Line graphs can be used to show the relationship between a fixed, independent variable (x-axis) and the measured, dependent variable (y-axis). Continuous variation produces a Gaussian or normal distribution curve.
3. Histograms can be used for measuring grouped data. Area of bar shows frequency.
4. Bar graphs are used for measuring discontinuous data. Height of bar indicates frequency.
5. Kite graphs are bar charts with frequency values- I’ve never been asked to or needed to draw one of these, so I don’t know how useful they are.
6. Pie charts are used to display percentages of a whole for discontinuous data.

Mode: the single value in a group which occurs most often. See genetics section for info on modal and bimodal distributions.

Median: the central or middle value in a set. To find the median, do median value = (n+1)/2 where n is the number of values in the set.

Mean ($\bar{x}$):the average of a group of values. The mean can be calculated either by $\bar{x}=\frac{\Sigma x}{n}$ (for a set of n values where x represents each value) or mean = $\bar{x}=\frac{\Sigma fx}{\Sigma f}$ (for a set of grouped values where x is the mid-point value of the group and f is the frequency of values in a group).

Standard deviation (σ): a measure of how spread out a set of values is. To calculate standand deviation, either use

where x represents the values and I is the total number of values (use n-1 where n is less than 30), or

where Σx2 is the sum of the square of each value, rather than the sum of the values squared.

Chi-squared test: a statistical test used to determine whether the difference between a set of recorded results and the respective expected results is due to chance, or because of a definite factor which is affecting results. The best way to explain this is through a worked example.

1. If you tossed a coin 100 times, you would expect it to lands on heads 50 times and tails 50 times. These are your expected results (E).
2. Imagine you took a coin, and tossed it 100 times. You got 46 heads and 54 tails. These are your observed results (O).
3. First, we formulate a null hypothesis. Our null hypothesis is that there is any difference between observed and expected results is only due to chance (this is always the null hypothesis for a chi-squared test). To determine whether this is true, we must now calculate the chi-squared (χ2) value, using the formula below.
4. We must work out (O-E)2/E for both sets of data- heads and tails. This gives the equation

which gives a chi-squared value of 0.64. Of course, this number isn’t much good to us right now. It’s time to refer to a chi-squared table.

The 0.05 column is in red because this is the one usually referred to. Degrees of freedom are always one less than the number of sets, i.e. the number of values you’d need to know before you could work out all values (given the total of the values). In this case, there is only degree of freedom, so the “magic number” is 3.84.

5. As chi-squared is less than 3.84, we can accept the null hypothesis- any deviation between observed and expected results is due to chance. If chi-squared was greater than 3.84, we would reject the null hypothesis- we could be 95% sure that the difference between observed and expected results was because some other factor was affecting results (e.g. using a weighted coin).

Note: specific statistical tests such as the Simpson Diversity Index and the Lincoln Index are included in the ecology section.