1.1 Biostatistics

Biostatistics is the application of statistical reasoning and methods to biological observations. It helps biologists describe variation, design investigations, estimate population characteristics and evaluate whether observed patterns are compatible with chance variation.

1.2 Population and Sample

A population is the complete set of biological units about which a conclusion is intended. A sample is the subset actually observed. Valid inference depends on how well the sample represents the target population.

1.3 Variables

A variable is a characteristic that can take different values among observations. Examples include body mass, leaf length, genotype, species, blood group and number of offspring.

1.4 Continuous Data

Continuous variables can, in principle, take any value within an interval and arise commonly from measurement. Length, mass, temperature and concentration are typical examples.

1.5 Discontinuous Data

Discontinuous or discrete data occur as separated values, commonly counts or categories. Number of seeds and number of colonies are counts; sex category or phenotype class is categorical.

1.6 Sources of Variation

Biological variation may arise from genetics, environment, development, measurement error and sampling. Statistics describes this variation rather than assuming that organisms are identical.

2.1 Why Sampling Is Necessary

Complete enumeration is often impossible, expensive or destructive. Sampling permits information about a larger population to be obtained from a manageable number of observations.

2.2 Simple Random Sampling

In simple random sampling, each eligible unit has a known and equal opportunity of selection. Random selection reduces deliberate or unconscious choice bias.

2.3 Systematic Sampling

Systematic sampling selects units at regular intervals after a suitable starting point. It is efficient but can be biased if the interval coincides with a hidden spatial pattern.

2.4 Stratified Sampling

A heterogeneous population may be divided into meaningful strata, followed by sampling within each stratum. This can improve representation of important subgroups.

2.5 Quadrats and Transects

Ecological studies often use quadrats for organisms associated with an area and transects for change along an environmental gradient. Placement rules should match the research question.

2.6 Sampling Error and Bias

Sampling error is random difference between a sample estimate and the population value. Bias is systematic distortion caused by selection, measurement or procedure and is not corrected simply by increasing sample size.

3.1 Observation

An observation is a recorded value or category obtained from a biological unit. Clear operational definitions are essential so that different observers measure the same property consistently.

3.2 Parameter and Statistic

A parameter describes a population, while a statistic is calculated from sample data. Sample statistics are used to estimate unknown population parameters.

3.3 Point Estimation

A point estimate gives one numerical estimate of a parameter, such as the sample mean used to estimate a population mean.

3.4 Uncertainty

An estimate should be interpreted with its sampling uncertainty. Large, representative samples usually provide more stable estimates than small samples, although poor design can remain biased.

3.5 Measurement Error

Random measurement error increases scatter; systematic error shifts measurements consistently. Calibration, standardisation and repeated measurements help control these problems.

4.1 Data Tables

A good table has an informative title, clear row and column headings, units and a logical arrangement. Raw data should remain traceable to the observations from which summaries are calculated.

4.2 Graphs

Bar charts suit categorical comparisons, histograms display continuous frequency distributions, scatter plots show paired quantitative variables and line graphs are useful for ordered or time-based measurements.

4.3 Frequency Distribution

A frequency distribution records how many observations fall in each value or class interval. Grouping simplifies large datasets but sacrifices some individual detail.

4.4 Relative Frequency

Relative frequency is the class frequency divided by total frequency. Multiplication by 100 expresses it as a percentage.

4.5 Cumulative Frequency

Cumulative frequency is the running total of observations up to successive values or class boundaries and is useful for locating medians and percentiles.

5.1 Arithmetic Mean

The arithmetic mean is the sum of observations divided by their number. It uses every value but can be strongly affected by extreme observations.

5.2 Median

The median is the middle ordered value, or the average of the two middle values when the sample size is even. It is resistant to extreme values.

5.3 Mode

The mode is the most frequently occurring value or category. A dataset may have one mode, more than one mode or no unique mode.

5.4 Choosing a Measure

The mean is useful for many quantitative datasets; the median is valuable for skewed data; and the mode can describe the most common category. Choice should reflect data type and distribution.

5.5 Weighted Mean

When observations or subgroup means contribute unequally, a weighted mean gives each value a specified weight rather than treating all contributions as equal.

6.1 Range

The range is maximum minus minimum. It is simple but depends only on two observations and is sensitive to extremes.

6.2 Variance

Variance measures average squared departure from the mean, with sample variance commonly using n−1 in the denominator.

6.3 Standard Deviation

Standard deviation is the square root of variance and is expressed in the original units of measurement.

6.4 Coefficient of Variation

The coefficient of variation expresses standard deviation relative to the mean, commonly as a percentage. It is useful only when the scale and mean make such a ratio meaningful.

6.5 Interquartile Range

The interquartile range spans the middle half of ordered observations and is less sensitive to extremes than the range.

7.1 Probability

Probability quantifies uncertainty on a scale from 0 to 1. A probability of 0 represents an impossible event under the model; 1 represents certainty.

7.2 Complement

If A is an event, P(not A)=1−P(A). This rule is often simpler than adding probabilities of many alternative outcomes.

7.3 Addition Rule

For mutually exclusive events A and B, P(A or B)=P(A)+P(B). For events that can overlap, their intersection must be subtracted.

7.4 Multiplication and Independence

If A and B are independent, P(A and B)=P(A)P(B). Independence is an assumption about events, not merely a convenient calculation.

7.5 Conditional Probability

Conditional probability P(A|B) is the probability of A given that B has occurred. Biological sampling and diagnostic reasoning often involve conditional probabilities.

8.1 Conditions

A binomial model applies to a fixed number of independent trials, each with two outcomes and constant success probability p.

8.2 Probability Function

For X successes in n trials, P(X=x)=C(n,x)p^x(1−p)^(n−x).

8.3 Mean and Variance

For a binomial variable, mean=np and variance=np(1−p).

8.4 Biological Use

The model can describe counts such as germinated versus non-germinated seeds when its assumptions are reasonable.

8.5 Limitations

Changing probabilities, dependence among trials or more complex outcomes violate the simple binomial model.

9.1 Meaning

The Poisson distribution models counts of events occurring within a defined interval of time, area, volume or other exposure under specified assumptions.

9.2 Parameter

The distribution is governed by λ, the expected number of events in the interval. Its mean and variance are both λ.

9.3 Probability Function

P(X=x)=e^(−λ)λ^x/x! for x=0,1,2,...

9.4 Biological Applications

It may approximate counts of relatively uncommon independent events such as colonies or occurrences per sampling unit when the assumptions are suitable.

9.5 Model Checking

Strong clustering, inhibition or varying event rates can produce departures from a Poisson model.

10.1 Shape

The normal distribution is continuous, symmetric and bell-shaped. Its location is determined by the mean and its spread by the standard deviation.

10.2 Standardisation

A value x can be standardised as z=(x−μ)/σ when population mean μ and standard deviation σ are known.

10.3 Empirical Pattern

For an exact normal distribution, approximately 68% of observations lie within one standard deviation of the mean and about 95% within two standard deviations.

10.4 Biological Relevance

Many measurements are approximately normal under suitable conditions, and normal-based methods are important in statistical inference.

10.5 Caution

Normality should not be assumed merely because a variable is continuous. Skewness, outliers and bounded measurements may make other descriptions more appropriate.

11.1 Null Hypothesis

The null hypothesis states the reference claim tested by the statistical procedure, commonly no difference or no association of a specified kind.

11.2 Alternative Hypothesis

The alternative describes the departure of interest and may be two-sided or directional when justified in advance.

11.3 Test Statistic

A test statistic summarises how far observed data depart from what is expected under the null model, relative to sampling variability.

11.4 P-value

A p-value is the probability, assuming the null hypothesis and model are true, of obtaining a result at least as incompatible with the null as the observed result. It is not the probability that the null hypothesis is true.

11.5 Statistical and Biological Importance

A statistically detectable difference can be biologically trivial, while an important biological effect may be estimated imprecisely in a small study. Effect size, uncertainty and context should accompany significance testing.

12.1 One-Sample t-test

A one-sample t-test compares a sample mean with a specified reference mean when the method's assumptions are sufficiently satisfied.

12.2 Independent-Samples t-test

An independent-samples t-test compares means from two independent groups. The appropriate variance treatment depends on assumptions about group variances.

12.3 Paired t-test

A paired t-test analyses within-pair differences, as in measurements before and after treatment on the same biological units.

12.4 Test Statistic

For a one-sample test, t=(x̄−μ0)/(s/√n). The denominator is the estimated standard error of the mean.

12.5 Interpretation

The calculated t is interpreted with appropriate degrees of freedom and the preselected significance framework, but conclusions should also report the observed difference.

13.1 Variance Ratio

An F statistic is a ratio of variance estimates. Under the relevant null model, its sampling distribution follows an F distribution.

13.2 Two-Variance Context

A classical F-test can compare two population variances under assumptions including independent samples and normality.

13.3 ANOVA Connection

Analysis of variance also uses an F ratio, comparing variation explained by group differences with residual variation within groups.

13.4 Degrees of Freedom

The F distribution depends on numerator and denominator degrees of freedom, so an F value cannot be interpreted without them.

13.5 Caution

Variance tests can be sensitive to departures from normality. Biological interpretation should not reduce a study to a variance ratio alone.

14.1 Goodness of Fit

A chi-square goodness-of-fit test compares observed categorical counts with counts expected under a specified model.

14.2 Independence

A chi-square test of independence evaluates whether two categorical variables show evidence of association in a contingency table.

14.3 Statistic

χ²=Σ(O−E)²/E, where O is observed frequency and E is expected frequency.

14.4 Expected Counts

Expected counts in a contingency table are calculated from row totals, column totals and the grand total under the independence model.

14.5 Interpretation

A large chi-square statistic indicates stronger discrepancy between observed and expected counts, but the reference distribution and degrees of freedom determine statistical significance.

15.1 Purpose

ANOVA compares variation among group means with variation within groups and is commonly used when more than two means are studied.

15.2 Partitioning Variation

Total variability is partitioned into components attributable to specified sources, such as treatment and residual error.

15.3 F Ratio

In one-way ANOVA, F=MS_between/MS_within. Mean squares are sums of squares divided by their corresponding degrees of freedom.

15.4 Null Hypothesis

The usual one-way ANOVA null hypothesis is that all population means are equal under the model.

15.5 After ANOVA

A significant overall F test indicates evidence that not all means are equal; it does not by itself identify every pair that differs. Planned contrasts or suitable multiple-comparison procedures may be needed.

16.1 Meaning

Analysis of covariance combines comparison of groups with adjustment for a quantitative covariate related to the response.

16.2 Covariate

A covariate is a measured variable included to account for systematic variation that is not the primary treatment factor.

16.3 Purpose

Adjustment can improve precision and separate treatment comparisons from predictable variation associated with the covariate when assumptions are appropriate.

16.4 Relationship to Regression

ANCOVA can be viewed within a linear-model framework containing both categorical predictors and quantitative covariates.

16.5 Caution

A covariate measured after treatment or affected by treatment can complicate interpretation. Adjustment is not automatically equivalent to experimental control.

17.1 Replication

Replication applies treatments to independent experimental units, allowing estimation of experimental variability.

17.2 Randomisation

Random assignment protects against systematic allocation bias and supports probability-based inference.

17.3 Control

A control provides an appropriate reference against which treatment effects can be interpreted.

17.4 Blocking

Blocking groups similar experimental units so that known nuisance variation can be accounted for in the design.

17.5 Pseudoreplication

Repeated measurements or subsamples from one experimental unit are not automatically independent replicates. The experimental unit must be identified correctly.

18.1 Meaning

Correlation measures the strength and direction of association between two quantitative variables.

18.2 Pearson Correlation

Pearson's r describes linear association and ranges from −1 to +1. Values near ±1 indicate strong linear association, while values near zero indicate weak linear association.

18.3 Sign

A positive sign means larger values of one variable tend to accompany larger values of the other; a negative sign indicates an inverse tendency.

18.4 Correlation Is Not Causation

Association may result from direct influence, reverse influence, common causes or chance. Correlation alone does not establish mechanism.

18.5 Outliers and Nonlinearity

A single extreme point can strongly influence r, and a strong curved relationship may have a modest linear correlation.

19.1 Regression Model

Simple linear regression models a response Y as Y=a+bX+error, where a is the intercept and b is the slope.

19.2 Slope

The slope estimates the expected change in the response for a one-unit increase in the predictor within the range supported by the data.

19.3 Intercept

The intercept is the predicted response at X=0. It may lack biological meaning if zero lies outside the relevant range.

19.4 Residuals

Residuals are observed minus fitted values. Their patterns help assess whether a linear model and constant variability are reasonable.

19.5 Prediction

Regression can be used for prediction, but extrapolation far beyond observed predictor values is risky because the relationship may change.

20.1 Nonlinear Pattern

Biological relationships are often curved because responses can saturate, accelerate, decline or approach limits.

20.2 Polynomial Terms

Curvature can sometimes be represented by adding terms such as X² to a regression model, but the equation should be chosen for scientific and statistical reasons.

20.3 Biological Models

Mechanistic models may be preferable when biology suggests a specific functional relationship.

20.4 Model Fit

A visually closer curve is not automatically a better scientific model. Complexity should be justified and residual behaviour examined.

20.5 Interpretation

Coefficients in curved models must be interpreted jointly because the effect of X can vary across its range.

21.1 Purpose

Transformation changes the numerical scale of a variable and can sometimes improve linearity, stabilise variance or reduce severe skewness.

21.2 Log Transformation

Logarithmic transformation compresses large values more strongly than small values and can be useful for positive right-skewed data.

21.3 Square-Root Transformation

Square-root transformation is sometimes considered for count data, depending on the analytical setting.

21.4 Interpretation

Analysis occurs on the transformed scale, so biological interpretation and back-transformation require care.

21.5 Not a Cure-All

Transformation cannot repair biased sampling, dependence, poor experimental design or incorrect biological assumptions.

22.1 Start with the Question

The biological question determines the population, variables, sampling design and statistical method. Calculations should follow the design rather than dictate it.

22.2 Identify Data Type

Determine whether variables are categorical, discrete counts or continuous measurements and whether observations are independent or paired.

22.3 Describe Before Testing

Inspect tables, graphs, centre, spread and unusual observations before performing inferential tests.

22.4 Choose a Method

Match the method to the question and assumptions: t procedures for appropriate mean comparisons, chi-square for categorical counts, ANOVA for appropriate multi-group comparisons, and correlation/regression for quantitative relationships.

22.5 Report Completely

State the method, sample information, estimates, test statistic where required, degrees of freedom where relevant, p-value or decision criterion, and biological interpretation without overstating causation.

1. National Universities Commission. Core Curriculum and Minimum Academic Standards for the Nigerian University System: Sciences. BIO 208: Biostatistics.
2. Zar, J. H. Biostatistical Analysis.
3. Fowler, J., Cohen, L. and Jarvis, P. Practical Statistics for Field Biology.
4. Whitlock, M. C. and Schluter, D. The Analysis of Biological Data.