Mathematics and statistics: TestAS practice testlets

Biological measurements vary, so results are reported with a measure of spread or uncertainty. Two measures are often shown as error bars in figures, and they must not be confused. The standard deviation (SD) describes how much individual values vary around the mean. The standard error of the mean (SEM) describes how precisely the mean itself has been estimated; it equals the SD divided by the square root of the sample size n. As more data are collected, the SD stays roughly the same, but the SEM shrinks. A figure should always state which measure its error bars show.

SEM = SD ÷ √n approximate 95% confidence interval: mean ± 2 × SEM

A 95% confidence interval gives a range of plausible values for the true mean. For reasonably large samples it extends about two standard errors on either side of the sample mean. If the same study were repeated many times, about 95% of such intervals would contain the true mean.

To decide whether a difference between groups is larger than chance variation would explain, researchers use significance tests, such as the t-test for comparing two means. The test starts from a null hypothesis, usually that there is no real difference. The p-value is the probability of obtaining a difference at least as large as the observed one if the null hypothesis were true. If p is below a threshold chosen in advance, commonly 0.05, the result is called statistically significant. The p-value is not the probability that the null hypothesis is true.

Statistical significance is not the same as importance. With very large samples, tiny differences become significant; with small samples, important effects can be missed. Researchers therefore also report effect sizes and confidence intervals. When many tests are carried out, some will be significant by chance alone: with 20 independent tests of true null hypotheses at the 0.05 level, one false positive is expected on average.

Good practice is to decide on the sample size, the main outcome and the statistical test before collecting the data, and to report all tests that were carried out, not only the significant ones.

A t-test comparing the two groups gives p < 0.001.

Many processes in biology change by a constant factor in equal time intervals rather than by a constant amount. A bacterial population that doubles every 30 minutes grows exponentially: after 30 minutes it is twice as large, after one hour four times, after two hours sixteen times. Radioactive substances, drugs in the body and light passing through an absorbing solution decrease exponentially: in each equal interval, the same fraction is lost.

If the doubling time Td is known, the size of a growing population after time t is the initial size N0 multiplied by 2 to the power of t divided by Td. For decay, the half-life T½ takes the place of the doubling time, and the base is one half. Equivalently, exponential change can be written with the natural base e and a rate constant k; the doubling time and k are linked by Td = ln 2 ÷ k, where ln 2 is about 0.693.

growth: N = N0 × 2^(t ÷ Td) decay: N = N0 × (1/2)^(t ÷ T½) Td = ln 2 ÷ k ≈ 0.693 ÷ k

Exponential growth is easy to underestimate. A pond lily that doubles its area every day and covers the whole pond on day 30 covered only half the pond on day 29 and about one thousandth of it on day 20, since 2 to the power of 10 is 1,024. For the same reason, an epidemic that doubles every three days can go from a few cases to overwhelming numbers within weeks.

No real population can grow exponentially for long. As resources run short, growth slows. A bacterial culture in a closed flask passes through a lag phase, in which the cells adapt, an exponential (log) phase, a stationary phase, in which division and death balance, and finally a death phase.

Exponential data are often plotted with a logarithmic vertical axis. On such a semi-logarithmic plot, exponential growth or decay appears as a straight line, which makes it easy to read off doubling times and to spot deviations.

The same mathematics describes money growing at compound interest and the spread of an infection in its early phase. In each case, a useful rule of thumb is that a quantity growing by r percent per time unit doubles in about 70 ÷ r time units.