Mathematics practice questions.

The pH scale is logarithmic. If the water of a lake changes from pH 6.0 to pH 5.0, the concentration of hydrogen ions has

• fallen to one tenth of its earlier value
• risen by one unit of concentration
• risen to ten times its earlier value
• risen to twice its earlier value because the scale doubles

Answer: risen to ten times its earlier value

Each whole pH unit is a factor of ten in hydrogen ion concentration, so a fall of one unit is a tenfold rise in acidity. Reading a logarithmic scale as if it were linear badly understates how large the chemical change is.

Two sets of ten quadrat counts have the same mean of 24 plants, but set A has a standard deviation of 2 and set B a standard deviation of 11. This shows that

• set B was sampled from a larger area than set A
• the mean of set A must be closer to the true population mean
• set A contains more species than set B
• the counts in set B vary far more widely about the mean

Answer: the counts in set B vary far more widely about the mean

Standard deviation measures the spread of the values about the mean, not the area sampled or the number of species. A large spread warns that the species is patchy and that a difference between two site means may be smaller than the variation within each site.

Nine nitrate readings from a stream lie between 2 and 6 mg dm-3, while a tenth, taken just after a slurry spill, is 61 mg dm-3. To describe the nitrate concentration typical of the stream it is best to quote

• the median, because one extreme value distorts the mean
• the mean, because it uses every value in the data set
• the range, because it shows how variable the stream water is
• the mode, because it is the value recorded most often

Answer: the median, because one extreme value distorts the mean

The single reading of 61 mg dm-3 pulls the mean above every other value, so the mean would misrepresent normal conditions; the median is resistant to such outliers. The spill should be reported separately rather than hidden inside an average.

Country X emits 480 Mt of carbon dioxide a year and has 60 million inhabitants; country Y emits 900 Mt and has 300 million inhabitants. Expressed per capita,

• Y emits more per person because its total is larger
• the two countries emit about the same per person
• X emits about one third as much per person as Y
• X emits more than twice as much per person as Y

Answer: X emits more than twice as much per person as Y

Dividing by population gives 8 t per person for X and 3 t for Y, so the ranking by total emissions is the opposite of the ranking per person. Per capita figures are what make countries of very different size comparable.

A kick sample from a polluted stretch of river contains 40 individuals, all belonging to one species of tubificid worm. Using D = N(N - 1) / sum of n(n - 1), the value of Simpson's reciprocal diversity index for this sample is

• 1.00, which is the lowest value the index can take
• 0.00, because there is no variety between the individuals counted
• 40.00, which is the total number of individuals in the sample
• impossible to obtain, because the index needs at least two species present

Answer: 1.00, which is the lowest value the index can take

With one species, sum of n(n - 1) equals N(N - 1), so D = 1. The index has a floor of 1, not 0, and a higher value means a more diverse community.

In a study of pond snails, 50 snails are marked and released and 40 snails are caught in the second sample. The Lincoln index N = (n1 x n2) / n3 gives an estimated population of 250 snails. The number of marked snails in the second sample was

• 5 snails
• 8 snails
• 10 snails
• 20 snails

Answer: 8 snails

Rearranging N = (n1 x n2) / n3 gives n3 = (50 x 40) / 250 = 8. Working the formula backwards is the quickest check that a quoted population estimate is arithmetically possible.