Cambridge International AS and A Level Marine Science Practical Skills 6: Descriptive statistics and statistical tests
Cambridge International AS and A Level Marine Science Practical Skills 6: Descriptive statistics and statistical tests
Study guide//Updated 16 Oct 2026
Cambridge Marine Science 9693 practical notes on averages, spread, Lincoln and Simpson indices, Spearman correlation, error bars and chi-squared tests.
Descriptive Statistics and Statistical Tests reviews summaries, ecological indices, correlation, uncertainty intervals and frequency tests for Marine Science practical study. Check the current official syllabus and question instructions for the assessed methods and level-specific scope. Statistics summarise evidence or test a defined hypothesis; they do not rescue a biased design.
Statistical study scope
This note practises mean, median, mode, range, the Lincoln population estimate, Simpson diversity, Spearman correlation, standard deviation, standard error, confidence intervals and chi-squared goodness of fit.
Read the formulae, symbols, tables and instructions provided in each question. Check the current syllabus for what must be recalled and what is supplied.
Distinguish a descriptive population quantity from a sample estimate and use the specified formula. Correct substitution does not replace interpretation or the method's assumptions.
Begin with question and data type
Ask what the investigation needs:
a typical value
spread or precision
a population estimate
diversity comparison
evidence of monotonic correlation
evidence that observed nominal frequencies differ from expectation
Then check that sampling units are independent and representative. A correct calculation on pseudoreplicated or biased data produces a precise-looking but weak conclusion.
State a null hypothesis before a significance test and return to it after using the critical value or probability information.
Mean
The arithmetic mean is total of the values divided by number of values. It uses every observation and works well as a centre for roughly balanced numerical data.
An extreme value can shift the mean strongly. Inspect raw data and justify treatment of anomalies before calculation.
Keep unrounded values until the final result and include the measured quantity's unit.
A mean without sample size or spread does not show reliability.
Median
The median is the middle value after ordering the data. For an even number of values, it is the mean of the two central values.
It is less affected by extremes and can be useful for skewed abundance or size data. It does not use the magnitude of every observation in the same way as the mean.
Do not select the middle entry before sorting.
The median's unit is the same as the original measurement.
Check this topic from memory
Attempt the matching topic bank before reopening the notes. Use each missed idea to decide what to review next.
The mode is the most frequent value or category. It can describe numerical or categorical data.
A dataset can have more than one mode or no unique mode. Grouped modes depend on class boundaries.
Mode is useful for common shell-size class or dominant category but may ignore most of the distribution.
Do not calculate it by adding or dividing values.
Range
Range is maximum minus minimum. It gives a simple measure of spread.
It depends only on two observations and tends to grow as sample size increases, so compare it cautiously across unequal samples.
An anomaly can inflate range greatly. Retain the raw value and justify any exclusion.
Range has the same unit as the data.
Choosing a centre and spread
Use mean with a spread measure when data and question justify it. Use median when skew or valid extremes make the mean unrepresentative. Use mode for most common value or category.
Explain the choice from the distribution, not from which answer looks more favourable.
Mean or median with an appropriate spread measure can describe a sample. Standard deviation uses deviations from the mean, whereas range uses only the extremes; choose according to the purpose and distribution.
No summary replaces the raw pattern or graph.
Lincoln index for mark-release-recapture
The Lincoln estimate is:
N=m2n1n2
Here, N is estimated population size, n1 is the number initially marked and released, n2 is the total second sample and m2 is the number of recognised marked individuals recaptured.
The estimate assumes marks remain and are recognised, marking does not alter survival or catchability, marked organisms mix fully, the population is effectively closed to births, deaths, immigration and emigration during the study, and every individual has similar capture probability.
A very small recapture count makes the estimate unstable. Zero recaptures makes the formula unusable.
Interpreting Lincoln bias
For given first- and second-sample sizes, fewer recognised marked recaptures make the estimate larger. Trap-shy behaviour can therefore lead to overestimation under the model assumptions. If marked organisms are attracted to traps, recaptures rise and the estimate may be too low.
Immigration of unmarked organisms can reduce marked proportion and inflate the estimate. Mark loss has a similar effect.
Mortality or emigration of marked individuals changes the population and recapture pool.
State the direction through the formula rather than labelling an assumption merely "unreliable".
Simpson's index of diversity
One commonly used diversity form is:
D=1−∑(Nn)2
Here, n is the number of individuals of each species and N is the total number of individuals across species.
For this formula, total abundance must be positive. It uses sampled species proportions and corresponds to the probability that two independent draws with replacement from those proportions have different species. A version based on n(n-1) and N(N-1) instead describes draws without replacement, requires N greater than one and is numerically different for finite samples; use the stated definition. The index reflects richness and evenness. In this form, a larger value indicates greater diversity because dominance by one species makes the squared-proportion sum larger and D smaller.
Compare sites only when sampling effort, identification and detectability are sufficiently comparable.
Richness, evenness and sampling
Species richness is number of species. Evenness is how similarly individuals are distributed among species.
Two sites can have equal richness but different Simpson values if one is dominated by a single species. Rare species contribute little to abundance-weighted indices but may still matter ecologically.
Larger sampling effort often detects more species. Standardise area, time, mesh, tide and identification method.
An index is a summary of the sampled community, not proof that every habitat function is healthy.
Spearman's rank correlation
Spearman's rank coefficient measures association between paired ranked variables. In general, calculate the Pearson correlation of the two sets of ranks, using average ranks for ties. The following simplified formula applies when neither variable has tied ranks:
rs=1−n3−n6∑D2
Here, D is the difference between paired ranks and n is number of pairs.
For inference, check independence between pairs, the sampling design and a test procedure suitable for the sample size and ties. There is no universal sample-size range that guarantees validity. A monotonic pattern is the relationship this coefficient is designed to summarise; zero rank correlation does not exclude every form of dependence.
Ranking and tied values
Rank each variable consistently in the same direction. For tied values, assign the mean of the ranks they would occupy, then use a tie-aware coefficient and inference method. Do not assume average ranking alone makes the no-ties shortcut exact.
Keep original pairs together after ranking. Calculate rank difference and its square for every pair.
In the no-ties formula, a small squared-rank-difference sum gives rs near positive one; opposing ranks give a negative value. If either variable is constant, rank correlation is undefined.
Check that rank totals match the expected sum to find ranking errors.
Spearman interpretation
Values range from negative one for perfect negative correlation through zero for no monotonic correlation to positive one for perfect positive correlation.
Magnitude alone does not establish statistical significance. Compare the calculated value with the supplied critical value for sample size and chosen probability level according to the table rule.
If the chosen test is significant, reject its null hypothesis, such as zero population rank correlation, under that test's assumptions. The null is a population/model statement, not a statement of "no significant correlation". Then describe direction and context.
Correlation does not prove that one variable causes the other; a third variable or reverse relationship may exist.
Standard deviation
Standard deviation describes spread of observations around the mean. Use the stated sample or population formula; the usual sample variance denominator is n minus one, not n.
A smaller standard deviation means values are more tightly clustered around their mean when units and scale are comparable. It does not show whether the mean is accurate.
One extreme value can increase standard deviation. Sample design and distribution should be inspected.
Standard deviation has the same unit as the original variable.
Standard error
Standard error describes sampling variability of an estimator. For the mean of independent, identically distributed observations, estimated using the sample standard deviation s and sample size n greater than one:
SE=ns
It decreases as sample size increases when standard deviation is comparable. For n greater than one it is no larger than s, and is strictly smaller when s is positive. Repeated readings or clustered subsamples cannot automatically be counted as independent observations in this formula.
Standard error does not describe the spread of individual organisms and does not correct sampling bias.
Report it with the same unit as the mean.
95 per cent confidence interval
A large-sample approximation sometimes uses a multiplier of two:
95%,CI≈xˉ±2(SE)
This is not a universal 95 per cent interval. For independent normal observations with unknown population variance, the usual mean interval uses a t critical value with n minus one degrees of freedom; for small samples it can be much larger than two. Follow the specified valid method and its assumptions. The confidence level describes long-run coverage of the interval procedure, not a 95 per cent probability assigned to the fixed population mean after seeing one interval.
Narrower intervals indicate a more precise mean estimate. They do not mean 95 per cent of individual observations lie inside.
Compare intervals together with independence, sample size and design. Do not use non-overlap as an infallible universal significance test.
Error bars
Plot mean at the point or bar height and extend bars by the stated statistic. A standard-error bar runs one SE above and below unless the prompt says otherwise; a confidence-interval bar reaches the interval limits.
Label or state what bars represent. Do not mix SD and SE among treatments.
Long bars show greater spread or lower mean precision according to the statistic used.
Graph scale must allow bar endpoints to be read accurately.
Chi-squared goodness-of-fit test
Pearson's chi-squared goodness-of-fit test compares observed category counts with counts expected under a specified probability model. Its statistic is:
χ2=∑E(O−E)2
For each category, subtract expected from observed, square, divide by expected and sum. This note uses a one-way goodness-of-fit setting, not a general contingency-table test.
The data are counts in mutually exclusive categories, observations are independent and expected frequencies must be positive and sufficiently large for the chi-squared approximation. When expected counts are too small, use an appropriate alternative or justified grouping rather than applying the approximation automatically.
Expected frequencies and null hypothesis
Expected values come from the stated theoretical ratio, equal distribution or relevant model, not from copying the observations.
A null hypothesis might be: population habitat-allocation probabilities follow the specified distribution. Whether the observed discrepancy is statistically significant is the test decision, not part of the null statement.
Expected frequencies should sum to the same total as observed frequencies. If they do not, the setup is wrong.
Chi-squared uses counts, not percentages or means substituted into the formula without converting to valid expected counts.
Degrees of freedom and critical values
For a one-way goodness-of-fit test with c categories, a fixed total and fully specified category probabilities, with no parameters estimated from these counts:
v=c−1
where c is number of classes. This degrees-of-freedom rule is conditional. Estimating model parameters can reduce degrees of freedom, and contingency tables have a different rule; check the actual model and method.
Use degrees of freedom and the stated probability level to find the supplied critical value. If calculated chi-squared exceeds the critical value under the table convention, reject the null hypothesis; otherwise do not reject it.
"Do not reject" is not proof that observed and expected distributions are identical.
Chance, significance and effect size
A p-value describes how extreme the observed statistic is under the specified null model and assumptions, including outcomes at least as extreme as the observed one. It is not the probability that the null hypothesis is true. It does not measure ecological importance directly.
Large samples can make a small difference statistically detectable. Small samples may lack power to detect an important effect.
Report the actual pattern, statistic, decision and biological context. Avoid claiming certainty from one threshold.
Statistical validity still depends on sampling independence, suitable categories and honest data handling.
Choosing among the methods
Use:
mean, median, mode and range to summarise centre or spread
Lincoln index to estimate a marked population
Simpson's index to compare sampled diversity
Spearman's rank to test monotonic association in paired rankable data
SD to describe observation spread
SE or an appropriate confidence interval to describe mean precision
chi-squared goodness of fit to compare observed counts with a specified categorical probability model
Do not choose a test merely because its formula is provided.
Worked application: selecting and interpreting statistics
In this illustrative study, limpet counts and shore height are recorded at twelve positions sampled under a design supporting independence. Separately, 80 snails are classified into two habitat categories under a specified null model assigning probability one half to each category. Equal expected counts require that model and its sampling assumptions, not merely the existence of two habitats. Use mean and standard deviation to summarise limpet abundance, but use Spearman's rank for the monotonic height-abundance relationship after checking the scatter plot and using a tie-aware coefficient and test if needed. State a null hypothesis of zero population rank correlation and compare calculated rs with the supplied critical value. For snail habitat, expected counts are 40 and 40. Calculate each (O−E)2/E contribution, sum chi-squared and use one degree of freedom. Compare with the supplied critical value before rejecting or not rejecting the frequency null hypothesis. Neither test proves shore height or habitat alone causes the pattern; sampling, tide and substrate remain relevant.
Common misconceptions and corrections
Calling the mean the most frequent value. That is the mode.
Finding the median without sorting. Order the data first.
Calling range maximum plus minimum. It is maximum minus minimum.
Using mean alone to claim reliability. Spread and sample design matter.
Choosing the statistic that supports the prediction. Choose from data and question.
Using Lincoln index without marked recaptures. Zero makes the estimate unusable.
Assuming marked organisms mix instantly. Mixing is an assumption to evaluate.
Saying trap-shy behaviour underestimates population. Fewer recaptures inflate the estimate.
Ignoring mark loss. It reduces recognised recaptures.
Calling Simpson's index species richness only. It includes evenness.
Assuming identical richness means identical diversity. Dominance can differ.
Comparing diversity under unequal effort automatically. Standardise sampling.
Using Spearman for unpaired data. Each observation needs a pair.
Ranking the two variables in opposite directions accidentally. Use one convention.
Giving tied observations arbitrary consecutive ranks. Use mean rank.
Calling rs=0.8 significant without a critical value. Sample size and threshold matter.
Calling correlation causation. Alternative mechanisms remain.
Calling standard deviation uncertainty in the mean. Standard error serves that role.
Calling standard error spread among organisms. It is mean precision.
Saying larger sample always lowers SD. It does not guarantee less biological spread.
Saying a confidence interval contains 95 per cent of observations. It concerns the mean estimate.
Calling any overlapping intervals no difference. Interpretation is not an absolute visual rule.
Plotting error bars without identifying the statistic. Meaning is ambiguous.
Using chi-squared on continuous measurements directly. It tests nominal counts.
Using percentages in place of expected frequencies. Convert to counts.
Letting expected totals differ from observed totals. The model setup is inconsistent.
Forgetting to square observed-minus-expected. Contributions cannot remain signed.
Dividing by observed instead of expected. The supplied formula uses expected.
Using category count as degrees of freedom. The fixed-probability one-way case here uses categories minus one; other models may differ.
Rejecting the null when calculated value is below critical. Follow the supplied table rule.
Accepting the null as proven. Use "do not reject".
Calling statistical significance ecological importance. Effect size and context differ.
Applying a correct test to pseudoreplicates. Design invalidity remains.
Memorising formulae but not assumptions. Test selection and interpretation earn meaning.
Assessment guidance
State the question, data type and null hypothesis before selecting a method. Use the current syllabus and question instructions to check which methods belong to the assessed level. Show substitution and intermediate steps with guard digits, then report a justified final precision. For Lincoln and Simpson, explain sampling assumptions and bias direction. For Spearman, preserve pairs, use average ranks and a tie-aware coefficient when needed, and apply a suitable critical-value or p-value procedure. For chi-squared goodness of fit, use category counts, a justified expected model, appropriate totals and model-specific degrees of freedom before comparing with a suitable critical value. Finish with a contextual conclusion that separates statistical significance, causation and ecological importance.
Retrieval practice
Given six marine datasets, select and justify a centre, spread, population estimate, diversity index, correlation test or frequency test. Complete one Lincoln and one Simpson calculation, rank tied Spearman pairs, interpret a critical value, calculate SD, SE and a confidence interval from supplied formulae, then complete a one-row chi-squared test with degrees of freedom. For every result, audit independence, sampling bias and biological meaning.
