Mathematics practice questions.

Community X has a Simpson's reciprocal diversity index of 8.5 and community Y has 2.1. What does this show?

• Community X is more diverse than Y
• Community Y is more diverse than X
• The two communities are equally diverse
• Only community Y contains rare species

Answer: Community X is more diverse than Y

For the Simpson's reciprocal index a higher value means greater diversity, so X (8.5) is more diverse than Y (2.1). The index reflects both the number of species and how evenly individuals are shared among them.

A spreadsheet fits a line to a scatter graph and reports R squared as 0.86. What does that value tell you?

• Most of the variation in one variable tracks the other
• The measurements were accurate to within 86 per cent
• There is an 86 per cent chance the result is not chance
• The line passes through 86 per cent of the plotted points

Answer: Most of the variation in one variable tracks the other

R squared reports how much of the variation in the vertical variable is accounted for by the relationship with the horizontal one, so the remaining fraction is due to everything else. It says nothing about accuracy and nothing about statistical significance, which is a separate test.

Bacterial numbers rising from a few hundred to several million are plotted with a logarithmic scale on the vertical axis. What are the two advantages?

• The whole range fits, and exponential growth plots straight
• The early points are removed, and the later ones spread out
• The scale becomes linear, and the gradient becomes a rate
• The uncertainty is reduced, and the trend becomes smoother

Answer: The whole range fits, and exponential growth plots straight

On a linear axis the early hundreds are indistinguishable from zero once millions are on the same scale, and a curve that steepens without limit is hard to compare between treatments. A constant proportional rate of increase appears as a straight line, so the gradient becomes something that can be read off.

An oxygen uptake rate is reported in the units cm³ g⁻¹ h⁻¹. How should that be read?

• Cubic centimetres per gram per hour
• Cubic centimetres per gram in one hour only
• Grams of oxygen per cubic centimetre each hour
• Cubic centimetres multiplied by grams and by hours

Answer: Cubic centimetres per gram per hour

A negative index means division, so each negative index introduces a per. Dividing by the mass is what makes the rate comparable between a mouse and an elephant, and reporting an uptake without it makes the comparison meaningless.

Five lengths, each read to the nearest 0.1 mm, are averaged and the mean is reported as 7.4386 mm. What is wrong with that report?

• A mean cannot be more precise than the readings behind it
• A mean of five values should always be given as a whole number
• The mean should have been rounded up rather than truncated
• Five readings are too few for a mean to be calculated at all

Answer: A mean cannot be more precise than the readings behind it

The extra digits are produced by the calculator, not by the ruler, and quoting them claims a precision the instrument never had. The mean should be given to the same precision as the measurements, or at most one further figure.

A t-test comparing the mean leaf width of plants from two sites returns a probability (p) of 0.02. Using the 5 per cent significance level, what does this show?

• The difference between the two means is large in biological terms.
• There is a 2 per cent chance that the two sites have identical means.
• The measurements at one site are more variable than at the other site.
• The difference between the two means is significant at that level.

Answer: The difference between the two means is significant at that level.

The probability is that of obtaining a difference at least this large if the two samples came from populations with the same mean, so a value below 0.05 leads to that explanation being rejected. Significance says how confidently a difference can be claimed, not how big or how important it is.