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Statistics for biologists: spread, error bars and the three tests questions
Mean, median and mode; why a standard deviation beats a range; error bars and what overlapping bars do and do not show; writing a null hypothesis; choosing between chi-squared, Student's t-test and Spearman's rank; degrees of freedom and critical values; and what p = 0.05 licenses you to write in a conclusion.
6 original questions · 22 marks · the statistics for biologists: spread, error bars and the three tests notes · Scientific method and quantitative biology
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Two heterozygous pea plants are crossed and the offspring are expected in a ratio of 3 tall to 1 dwarf. Of 400 offspring, 280 are tall and 120 are dwarf. Calculate chi-squared and use it to reach a conclusion. The critical value at p = 0.05 with 1 degree of freedom is 3.84.
Mark scheme
- M1 expected values are 400 × 3/4 = 300 tall and 400 × 1/4 = 100 dwarf, both comfortably above 5
- M1 (280 − 300)² ÷ 300 = 1.33 and (120 − 100)² ÷ 100 = 4.00
- A1 chi-squared = 1.33 + 4.00 = 5.33
- B1 degrees of freedom = 2 categories − 1 = 1, and the calculated value of 5.33 is greater than the critical value of 3.84 at p = 0.05
- B1 the null hypothesis is rejected: there is a significant difference between the observed numbers and a 3:1 ratio, so the difference is unlikely to have arisen by chance alone
Five leaves taken from one shrub have widths of 12, 14, 15, 17 and 22 mm. Calculate the mean width and the standard deviation of these widths, giving the standard deviation to three significant figures.
Mark scheme
- M1 mean = (12 + 14 + 15 + 17 + 22) ÷ 5 = 80 ÷ 5 = 16 mm
- M1 the deviations from the mean are −4, −2, −1, +1 and +6, and the squares of these total 16 + 4 + 1 + 1 + 36 = 58
- M1 divide by n − 1 rather than by n: 58 ÷ 4 = 14.5, then take the square root
- A1 standard deviation = 3.81 mm
A student writes that the error bars on two mean values do not overlap, so the difference between the means is significant. Explain why this reasoning is unsound.
Mark scheme
- B1 an error bar may show one standard deviation, one standard error or a 95 per cent confidence interval, and the three have different lengths drawn on the same data, so the same means can appear to overlap or not depending on the choice
- B1 standard-deviation bars describe variation between individuals and barely narrow however many organisms are measured, so two genuinely different means can easily have overlapping bars
- B1 standard-error bars describe how precisely each mean has been estimated and shrink roughly as one over the square root of the sample size, so with a large enough sample almost any two means separate
- B1 significance is decided only by comparing a calculated test value with a critical value at a stated probability, so bars can suggest that a test is worth doing and no more
Compare Student's t-test with Spearman's rank correlation, referring to the data each one needs and the question each one answers.
A researcher measures 2000 fish from each of two lakes and reports that the mean masses differ by 0.4 g, a difference a t-test finds significant at p = 0.05. Suggest why this significant result may still be of little biological interest.
An investigation compares the mean number of stomata per square millimetre on leaves grown in sun and on leaves grown in shade. State a suitable null hypothesis, and state the significance level conventionally used in biology.
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