METHODS IN BIOLOGY • L12

Probability Distribution, Confidence Interval, Hypothesis Testing, Correlation & Regression

CSIR-NET Life Sciences • GATE Biotechnology • DBT-BET • ICAR-JRF • Research Methodology

Exam Focus: This chapter explains the statistical concepts required to understand biological data, experimental variation, probability distributions, confidence intervals, hypothesis testing, correlation and regression. Special attention is given to formulas, interpretation, assumptions and common CSIR-NET/GATE traps.

📚 Index / Table of Contents

Probability Distribution
Discrete and Continuous Distributions
Binomial Distribution
Poisson Distribution
Normal Distribution
Sampling and Sampling Distribution
Confidence Interval
Hypothesis Testing
Type I and Type II Errors
P-value and Significance Level
Common Statistical Tests
Correlation
Regression
Correlation vs Regression
CSIR-NET/GATE Quick Revision
10 MCQs with Interactive Quiz

1. Probability Distribution

Probability describes the likelihood of occurrence of an event. In biological research, probability is used to deal with uncertainty associated with measurements, sampling, inheritance, disease occurrence, experimental outcomes and random variation.

Important basic terms

Experiment: A procedure that produces an observable outcome.
Sample space: The complete set of possible outcomes.
Outcome: A particular result of an experiment.
Event: A collection of one or more outcomes.
Probability: A numerical measure between 0 and 1 indicating the likelihood of an event.

Key point: Probability can never be less than 0 or greater than 1.
0 = impossible event; 1 = certain event.

Probability distribution

A probability distribution describes how probabilities are assigned to possible values of a random variable.
A random variable is a numerical variable whose value depends on the outcome of a random process.
Random variables can be discrete or continuous.
The total probability across all possible outcomes must equal 1.

2. Discrete and Continuous Distributions

Discrete random variable

Has countable possible values.
Values are usually integers.
Examples in biology:
- Number of bacterial colonies on a plate.
- Number of mutations detected in a gene.
- Number of offspring produced.
- Number of infected individuals in a sample.
A discrete random variable is described by a probability mass function (PMF).

Continuous random variable

Can take any value within a continuous interval.
Examples include height, body weight, blood glucose concentration, temperature and enzyme activity.
Continuous variables are described using a probability density function (PDF).
The probability of obtaining one exact value in a truly continuous distribution is theoretically zero; probabilities are assigned to intervals.

Expected value

The expected value represents the long-term average value of a random variable.

E(X) = Σ xP(x)

For a discrete variable, multiply every possible value by its probability and add the results.
Expected value does not necessarily have to be an observed value.

Variance

Variance measures the spread of observations around the mean.

Var(X) = E[(X − μ)²]

Standard deviation is the square root of variance:

SD = √Variance

3. Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent trials when each trial has only two possible outcomes.

Conditions for binomial distribution

There is a fixed number of trials, n.
Each trial has two outcomes, commonly called success and failure.
The probability of success, p, remains constant.
The trials are independent.

Binomial probability formula

P(X = x) = C(n,x) p^x q^n−x

where:

n = number of trials
x = number of successes
p = probability of success
q = probability of failure = 1 − p
C(n,x) = n!/[x!(n−x)!]

Mean and variance

Mean = np
Variance = npq
SD = √npq

Exam trap: Binomial distribution requires independence and constant probability of success.

4. Poisson Distribution

The Poisson distribution is commonly used to describe the number of events occurring in a fixed interval of time, space or area when events occur independently and at a relatively constant average rate.

Biological examples

Number of mutations occurring in a particular DNA region.
Number of bacterial colonies appearing in a defined area under suitable assumptions.
Number of rare events observed in a fixed time interval.
Number of defects in a defined biological sample.

Formula

P(X = x) = e^−λ λ^x / x!

λ is the average number of events.
x is the observed number of events.
Mean = λ
Variance = λ
Therefore, for a Poisson distribution, mean = variance.

Remember: In Poisson distribution, mean and variance are equal.

5. Normal Distribution

The normal distribution is one of the most important continuous probability distributions in biology and statistics. Many biological measurements approximately follow a normal distribution under appropriate conditions.

Characteristics

It is continuous.
It is bell-shaped.
It is symmetrical around the mean.
Mean = median = mode in a perfectly normal distribution.
The total area under the curve is 1.
The two tails theoretically extend indefinitely.
The curve is determined by its mean and standard deviation.

Empirical rule

Approximately 68% of observations lie within ±1 SD of the mean.
Approximately 95% lie within ±2 SD.
Approximately 99.7% lie within ±3 SD.

Z-score

Z = (X − μ) / σ

Z-score tells how many standard deviations an observation lies above or below the mean.
Positive Z indicates a value above the mean.
Negative Z indicates a value below the mean.

6. Sampling and Sampling Distribution

In biological research, it is often impossible to study an entire population. Therefore, researchers collect a sample and use it to make conclusions about the population.

Population

The complete group about which we want information.
Example: all individuals of a particular species in a defined study population.

Sample

A subset selected from the population.
A well-designed sample should represent the population as closely as possible.

Common sampling methods

Simple random sampling: Every member has an equal probability of selection.
Systematic sampling: Individuals are selected at a regular interval.
Stratified sampling: Population is divided into meaningful subgroups and samples are taken from each.
Cluster sampling: Groups or clusters are selected rather than individual units.
Convenience sampling: Easily available individuals are selected; this can introduce substantial bias.

Sampling distribution

A sampling distribution is the probability distribution of a statistic calculated from repeated samples of the same size from a population.
The statistic could be a sample mean, proportion, variance or correlation coefficient.
The standard deviation of the sampling distribution of the mean is called the standard error of the mean (SEM).

SEM = s / √n

As sample size increases, the standard error generally decreases.

Important: Standard deviation describes variation among individual observations, whereas standard error describes uncertainty/variability of a sample statistic such as the mean.

7. Confidence Interval

A confidence interval provides a range of plausible values for a population parameter based on sample data.

General interpretation

A 95% confidence interval is constructed using a method that, under repeated sampling and correct assumptions, would capture the true parameter approximately 95% of the time.
It does not strictly mean that there is a 95% probability that the already-fixed population parameter lies inside a particular calculated interval.
A wider confidence interval indicates greater uncertainty.
A larger sample size generally produces a narrower confidence interval.
Greater variability generally produces a wider confidence interval.

Confidence interval for a mean

When population standard deviation is known or a normal approximation is appropriate:

CI = x̄ ± Z × (σ / √n)

When σ is unknown and sample size/assumptions require use of the t distribution:

CI = x̄ ± t × (s / √n)

Factors affecting CI width

Higher confidence level → wider interval.
Larger sample size → narrower interval.
Higher variability → wider interval.
Lower variability → narrower interval.

CSIR-NET shortcut: If sample size increases by a factor of 4, the standard error of the mean decreases by a factor of 2, assuming the sample SD is otherwise comparable.

8. Hypothesis Testing

Hypothesis testing is a formal statistical procedure used to evaluate whether observed data provide sufficient evidence against a specified null hypothesis.

Null hypothesis (H₀)

Represents the default statistical assumption.
Often states that there is no difference, no association or no effect.
Example: A treatment has no effect on enzyme activity.

Alternative hypothesis (H₁ or Hₐ)

Represents a departure from the null hypothesis.
May be two-sided or one-sided depending on the research question.
Example: The treatment changes enzyme activity.

Basic steps

Define the research question.
State H₀ and H₁.
Select an appropriate statistical test.
Choose a significance level, commonly α = 0.05.
Calculate the test statistic.
Determine the P-value or compare with a critical value.
Make a statistical decision regarding H₀.
Interpret the result in the biological context.

Reject vs fail to reject

If P < α, the result is statistically significant under the chosen testing framework, and H₀ is rejected.
If P ≥ α, there is insufficient evidence to reject H₀.
It is better to say “fail to reject H₀” rather than “accept H₀” in standard statistical interpretation.

Important: Statistical significance does not automatically mean biological significance. A tiny effect can become statistically significant with a sufficiently large sample.

9. Type I and Type II Errors

Type I error

Occurs when a true null hypothesis is rejected.
It is commonly represented by α.
Also called a false positive.
Example: Concluding that a treatment works when in reality it has no effect.

Type II error

Occurs when a false null hypothesis is not rejected.
It is commonly represented by β.
Also called a false negative.
Example: Concluding that there is insufficient evidence for a treatment effect when a real effect exists.

Statistical power

Power = 1 − β

Power is the probability of detecting an effect when a real effect exists, under the specified model and assumptions.
Power generally increases with larger sample size.
Power increases when the true effect size is larger.
Power can also increase with lower variability and, depending on the design, a higher significance level.

Reality	Decision	Result
H₀ true	Reject H₀	Type I error
H₀ true	Fail to reject H₀	Correct decision
H₀ false	Reject H₀	Correct detection
H₀ false	Fail to reject H₀	Type II error

10. P-value and Significance Level

P-value

The P-value measures how incompatible the observed data, or something more extreme according to the test, are with the null hypothesis under the specified statistical model.
A small P-value provides stronger evidence against H₀.
P-value is not the probability that H₀ is true.
P-value is not the size of the biological effect.
P-value should be interpreted together with effect size, confidence intervals, study design and biological relevance.

Significance level (α)

α is the preselected threshold for statistical decision-making.
A common value is 0.05.
If α = 0.05, the researcher accepts a 5% Type I error rate under the testing framework when H₀ is true.

One-tailed vs two-tailed test

One-tailed: Tests for an effect in a specified direction.
Two-tailed: Tests for a difference in either direction.
The choice should be made based on the research question and study design, not after examining the results.

Example: If P = 0.03 and α = 0.05, then P < α, so the result is statistically significant at the 5% level.

11. Common Statistical Tests

t-test

Used for comparing means under appropriate assumptions.
One-sample t-test: compares a sample mean with a reference value.
Independent-samples t-test: compares means of two independent groups.
Paired t-test: compares two related measurements, such as before-and-after measurements from the same subjects.

ANOVA

Analysis of variance is used to compare means across more than two groups.
The null hypothesis in one-way ANOVA generally states that all group means are equal.
A significant ANOVA result indicates that not all means are equal, but does not by itself identify which specific groups differ.
Post-hoc tests may be used to determine pairwise differences.

Chi-square test

Frequently used for categorical data.
Can test association between categorical variables.
Can also test whether observed frequencies differ from expected frequencies in suitable goodness-of-fit settings.

Non-parametric tests

Used when assumptions of certain parametric tests are not appropriate or when data structure calls for rank-based methods.
Mann–Whitney U test is commonly used for two independent groups.
Wilcoxon signed-rank test is commonly used for paired data.
Kruskal–Wallis test is commonly used for comparing more than two independent groups.

Situation	Common test
One mean vs reference	One-sample t-test
Two independent means	Independent t-test
Before vs after same subjects	Paired t-test
More than two group means	ANOVA
Categorical variables	Chi-square
Two independent non-parametric groups	Mann–Whitney U

12. Correlation

Correlation measures the strength and direction of association between two variables. The most commonly discussed measure is Pearson's correlation coefficient.

Pearson correlation coefficient

r = Cov(X,Y) / (SD_X × SD_Y)

Range of r

−1 ≤ r ≤ +1
r = +1 → perfect positive linear correlation.
r = −1 → perfect negative linear correlation.
r = 0 → no linear correlation, although nonlinear association may still exist.
Positive r means that larger values of one variable tend to be associated with larger values of the other.
Negative r means that larger values of one variable tend to be associated with smaller values of the other.

Examples

Height and body mass may show positive association in an appropriate population.
Temperature and enzyme activity may show a relationship that is not necessarily linear across the entire temperature range.
Drug concentration and response may show a positive or negative relationship depending on the biological system and response definition.

Major exam concept: Correlation does NOT prove causation.

Coefficient of determination

R² = r²

For a simple linear regression, R² describes the proportion of variation in the response variable explained by the linear model with the predictor variable, within the context and assumptions of the model.

13. Regression

Regression is used to describe and quantify the relationship between a dependent variable and one or more predictor variables and, under appropriate assumptions, to make predictions.

Simple linear regression

Y = a + bX

Y = predicted/dependent variable.
X = predictor/independent variable.
a = intercept.
b = slope.

Interpretation of slope

The slope represents the expected change in Y associated with a one-unit increase in X, according to the fitted linear model.
Positive slope → predicted Y increases as X increases.
Negative slope → predicted Y decreases as X increases.
Slope equal to zero → no linear change in predicted Y with X in the fitted model.

Residuals

A residual is the difference between an observed value and its fitted/predicted value.
Residual = Observed Y − Predicted Y
Residual analysis is important for checking model adequacy.
Patterns in residuals can indicate that a simple linear model may be inappropriate.

Multiple regression

Uses two or more predictor variables.
Example: predicting plant biomass from light intensity, nutrient concentration and temperature.
It allows researchers to examine the association of one predictor while statistically accounting for other predictors in the model.

14. Correlation vs Regression

Feature	Correlation	Regression
Main purpose	Measure association	Model relationship/predict
Typical output	r	Equation and coefficients
Range of coefficient	−1 to +1	Slope is not restricted to −1 to +1
Direction	Positive/negative	Determined by slope
Causation	Does not establish causation	Does not automatically establish causation

15. CSIR-NET / GATE Quick Revision

Probability: always lies between 0 and 1.
Expected value: weighted average of possible values.
Variance: measure of squared deviation from the mean.
SD: square root of variance.
Binomial: fixed n, two outcomes, independent trials, constant p.
Binomial mean: np.
Binomial variance: npq.
Poisson mean: λ.
Poisson variance: λ.
Normal distribution: mean = median = mode for a perfect normal curve.
Normal distribution: approximately 68%, 95%, 99.7% within 1, 2 and 3 SD respectively.
Z-score: (X − μ)/σ.
SEM: s/√n.
Increasing sample size: generally decreases standard error.
Confidence interval: gives a range of plausible parameter values based on a specified procedure.
Higher confidence: generally produces a wider interval.
Type I error: reject true H₀.
Type II error: fail to reject false H₀.
Power: 1 − β.
α: significance level / Type I error rate under the testing framework.
P < α: reject H₀ under the selected decision rule.
Correlation r: ranges from −1 to +1.
r = +1: perfect positive linear association.
r = −1: perfect negative linear association.
r = 0: no linear correlation; nonlinear association may still exist.
Correlation ≠ causation.
Regression: models a response in relation to predictor variables.
Simple linear regression: Y = a + bX.
Residual: observed − predicted.
R²: proportion of variation explained by the fitted model in the relevant context.

🧮 Important Formula Sheet

Mean: x̄ = Σx/n
Variance: σ² = E[(X−μ)²]
SD: σ = √σ²
Binomial: P(X=x) = C(n,x)pˣqⁿ⁻ˣ
Binomial mean: np
Binomial variance: npq
Poisson: P(X=x)=e⁻λ λˣ/x!
Poisson mean: λ
Poisson variance: λ
Z-score: Z=(X−μ)/σ
SEM: s/√n
Confidence interval: estimate ± critical value × standard error
Power: 1−β
Correlation: −1 ≤ r ≤ +1
Regression: Y=a+bX
Residual: observed Y − predicted Y
Simple regression: R²=r²

🎯 Final Take-Home Points

Probability quantifies uncertainty.
Binomial distribution is appropriate for fixed independent trials with two outcomes.
Poisson distribution is especially useful for counting events occurring at an average rate under appropriate assumptions.
Normal distribution is symmetrical and characterized by mean and standard deviation.
Sampling allows researchers to make population-level inferences from samples.
Standard error generally decreases as sample size increases.
Confidence intervals communicate the uncertainty associated with parameter estimation.
Hypothesis testing evaluates evidence against a null hypothesis.
Type I error = false positive; Type II error = false negative.
Power = 1 − β.
P-value is not the probability that the null hypothesis is true.
Correlation describes association, not causation.
Pearson r ranges from −1 to +1.
Regression models the relationship between predictors and a response.
For simple linear regression, Y = a + bX.
Always interpret statistical results together with biological meaning and experimental design.

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Saturday, 8 August 2026

PROBABILITY DISTRIBUTION, CI, HYPOTHESIS, CORRELATION & REGRESSION