Complete Theory & Notes for B.Sc. Microbiology Students
Applied Statistics deals with the collection, organization, presentation, analysis and interpretation of numerical data. It provides statistical methods that are useful for understanding biological, medical and microbiological data.
This section covers the major theoretical concepts required for B.Sc. level study, including data summarization, correlation, regression, probability, probability distributions, estimation, hypothesis testing and non-parametric tests.
Data are the observations or measurements collected for a particular purpose. In biological sciences, data may include bacterial counts, body temperature, blood glucose level, colony counts, height, weight or laboratory results.
A frequency distribution is a systematic arrangement of data showing how many observations fall into each value or class interval.
Arithmetic mean is the most commonly used measure of central tendency. It is obtained by dividing the sum of all observations by the total number of observations.
Where:
Median is the middle value of an ordered dataset. It divides the data into two equal parts.
Arrange the observations in ascending or descending order.
When the number of observations is even, the median is the average of the two middle observations.
Mode is the value that occurs most frequently in a dataset. A distribution may have one mode, more than one mode or no mode.
Variance measures the average squared deviation of observations from their mean.
A larger variance indicates greater dispersion of the observations around the mean.
Standard deviation is the positive square root of variance. It is one of the most important measures of dispersion.
A small standard deviation indicates that observations are relatively close to the mean, while a large standard deviation indicates greater variability.
Coefficient of variation (CV) is a relative measure of dispersion expressed as a percentage.
CV is useful for comparing variability between datasets having different means.
Quartiles divide an ordered dataset into four equal parts.
Percentiles divide data into one hundred equal parts.
Mean deviation is the arithmetic mean of the absolute deviations of observations from a central value such as mean or median.
Here, A may represent the mean or median depending on the method used.
Skewness describes the degree and direction of asymmetry in a distribution.
Kurtosis describes the degree of peakedness or flatness of a distribution in comparison with a normal distribution.
Correlation is a statistical method used to measure the degree and direction of relationship between two variables.
For example, in microbiology, bacterial concentration and optical density may show a relationship.
Pearson's correlation coefficient measures the strength and direction of a linear relationship between two quantitative variables.
The value of r ranges from −1 to +1.
| Value of r | Interpretation |
|---|---|
| +1 | Perfect positive correlation |
| 0 | No linear correlation |
| −1 | Perfect negative correlation |
Spearman's rank correlation is used when observations can be ranked or when the assumptions required for Pearson's correlation are not appropriate.
Here, d is the difference between paired ranks.
When two or more observations have the same value, they receive the average of the ranks they would occupy.
Kendall's tau is a rank-based measure of association based on concordant and discordant pairs.
C represents concordant pairs and D represents discordant pairs.
The sign of the coefficient indicates the direction of association, while the magnitude indicates the strength of the relationship.
Regression analysis is a statistical method used to describe the relationship between a dependent variable and one or more independent variables and to make predictions.
A simple linear regression model describes a straight-line relationship between two variables.
The least squares method determines the regression line by minimizing the sum of squared differences between observed and predicted values.
Regression coefficients indicate the expected change in the dependent variable associated with a one-unit change in the independent variable.
Once a regression equation is obtained, the equation can be used to estimate the value of the dependent variable for a given value of the independent variable.
A residual is the difference between an observed value and its corresponding predicted value.
R² represents the proportion of variation in the dependent variable that is explained by the regression model.
For example, an R² of 0.64 indicates that 64% of the variation is explained by the fitted model in a simple linear regression setting.
Multiple correlation measures the relationship between one dependent variable and two or more independent variables.
Partial correlation measures the relationship between two variables while controlling the effect of another variable or variables.
Categorical data represent observations divided into distinct categories. Examples include sex, blood group, disease status and test result.
A contingency table displays the frequency distribution of two or more categorical variables.
Expected frequency is the frequency expected in a cell when the null hypothesis is true.
The chi-square test is commonly used to determine whether there is a significant difference between observed and expected frequencies.
This test determines whether observed categorical frequencies fit a specified theoretical distribution.
This test determines whether two categorical variables are statistically associated or independent.
Yates' correction, also called the continuity correction, may be applied to a 2 × 2 contingency table when the sample or expected frequencies are small.
Yule's coefficient is used to measure the degree of association between two attributes in a 2 × 2 table.
It provides a measure of association between categorical variables.
Probability is a numerical measure of the likelihood that an event will occur.
A probability of 0 indicates an impossible event, while a probability of 1 indicates a certain event.
For two events A and B:
If A and B are mutually exclusive:
For independent events:
Conditional probability is the probability of an event occurring when another event is already known to have occurred.
Joint probability describes the probability of two events occurring together.
Marginal probability is the probability of a single event without considering the outcome of another variable.
Bayes' theorem is used to calculate the probability of a cause or condition based on available evidence.
It has important applications in diagnostic testing and medical statistics.
A probability distribution describes the probabilities associated with possible values of a random variable.
Binomial distribution is used when an experiment consists of a fixed number of independent trials and each trial has two possible outcomes, commonly called success and failure.
Where q = 1 − p.
Poisson distribution is commonly used for counting the number of events occurring in a fixed interval of time, area or space when events occur independently.
The normal distribution is a continuous probability distribution with a symmetrical bell-shaped curve.
Probability distributions are useful for studying biological counts, disease occurrence, laboratory measurements, population characteristics and experimental outcomes.
Estimation is the statistical process of using sample information to estimate an unknown population parameter.
Point estimation provides a single numerical value as an estimate of a population parameter.
For example, the sample mean can be used as an estimate of the population mean.
A confidence interval provides a range of values within which a population parameter is estimated to lie at a specified confidence level.
For a large sample with known population standard deviation:
When the population standard deviation is unknown and the sample is small, the t-distribution is commonly used.
Here q = 1 − p.
Sample size calculation determines the number of observations required to estimate a parameter with a desired level of precision.
Sample size depends on factors such as confidence level, expected variability and allowable error.
Hypothesis testing is a statistical procedure used to determine whether sample evidence provides sufficient evidence against a stated population assumption.
The null hypothesis generally states that there is no difference, association or effect.
The alternative hypothesis represents a difference, association or effect.
The level of significance, commonly represented by α, is the probability of rejecting a true null hypothesis.
Common levels are 5% and 1%.
A one-sample Z test is used to compare a sample mean with a known or hypothesized population mean when the relevant large-sample conditions are satisfied.
The one-sample t test is used to compare a sample mean with a hypothesized population mean when the population standard deviation is unknown, particularly for small samples under appropriate assumptions.
The paired t test is used when two measurements are taken from the same subjects or when observations naturally form matched pairs.
Examples include measurements before and after treatment.
The two-sample t test is used to compare the means of two independent groups under appropriate assumptions.
The F test is commonly used to compare variances and also forms the basis of several analysis of variance procedures.
One-way analysis of variance (ANOVA) is used to compare the means of three or more groups based on one factor.
ANOVA separates total variability into variability between groups and variability within groups.
Two-way ANOVA examines the effects of two factors on a dependent variable and may also assess their interaction when the design permits.
A sample correlation coefficient can be tested to determine whether the observed relationship is statistically significant in the population.
Regression significance testing determines whether the regression relationship provides evidence of an association between the variables.
After calculating the test statistic, it is compared with the appropriate critical value or evaluated using the p-value.
Non-parametric tests are statistical methods that do not require the same distributional assumptions as many parametric tests. Several non-parametric methods are based on ranks rather than the original numerical values.
They are particularly useful for ordinal data, small samples or data that do not satisfy the assumptions required for a parametric test.
The sign test is used to test differences based on the direction of paired observations rather than their numerical magnitude.
It can be used as a non-parametric alternative to a paired t test under appropriate conditions.
The Wilcoxon signed-rank test is used for paired or matched observations and considers both the magnitude and direction of differences after ranking their absolute values.
The Mann–Whitney U test is used to compare two independent groups when the assumptions of the independent-samples t test are not appropriate.
It is based on the ranks of observations from both groups.
The Kruskal–Wallis test is a rank-based non-parametric method for comparing three or more independent groups.
It is commonly considered a non-parametric alternative to one-way ANOVA.
| Situation | Common Test |
|---|---|
| Paired observations | Sign test / Wilcoxon signed-rank test |
| Two independent groups | Mann–Whitney U test |
| Three or more independent groups | Kruskal–Wallis test |
After studying the theory, practice chapter-wise numerical problems with complete step-by-step calculations.
The Solved Numericals section will contain basic, moderate and exam-level questions from all major topics.