Statistics for Biology

Introduction to Statistics

Statistics helps us make sense of data by summarizing it, estimating unknowns, and testing ideas.

• Descriptive statistics: Summarize data
• Inferential statistics: Make predictions or generalizations
• Applied in biology, medicine, engineering, psychology, economics, etc.

Describing Distributions

Measures of Location

• Mean – average value
• Median – middle value
• Mode – most frequent value

These describe the “center” of the data.

Measures of Spread

• Range – difference between max and min
• Quartiles – divide data into four parts
• Percentiles – value below which X% of data falls
• Deviation – difference from the mean
• Variance – average of squared deviations
• Standard deviation – square root of variance

These show how scattered the data is.

Probability Foundations

What is Probability?

• The likelihood of an event occurring
• Expressed between 0 (impossible) and 1 (certain)

Law of Total Probability

If A can happen through exclusive events B and C:

P(A) = P(A|B) × P(B) + P(A|C) × P(C)

Bayesian Thinking

Bayes’ Theorem

Helps update belief in a hypothesis given new data.

P(H|D) = [P(D|H) × P(H)] / P(D)

Example: COVID-19 Testing

Even with a 99% accurate test, if prevalence is low (e.g. 1%), many positives may be false due to the base rate.

Probability Distributions

PMF (Probability Mass Function)

• For discrete variables
• Assigns probabilities to individual outcomes
• Example: Coin toss

PDF (Probability Density Function)

• For continuous variables
• Probability = Area under the curve
• Example: Human height

Normal Distribution

• Symmetrical bell curve
• Defined by mean (μ) and standard deviation (σ)
• Most biological measurements follow this

Sampling: Population vs Sample

• Population – entire group (e.g., all adults in India)
• Sample – subset used to infer about the population

Sampling must be random and unbiased to ensure valid conclusions.

Estimating Parameters

Point Estimate

• A single value (e.g., sample mean) used to estimate a population parameter

Central Limit Theorem

• The distribution of the sample mean becomes normal as sample size increases, even if the original data isn’t normal

Confidence Intervals

Used to show a range where the population value likely falls

• Confidence level – how sure we are (e.g., 95%)
• Margin of error – how much the sample estimate may vary

CI = sample mean ± z × (σ / √n)

Hypothesis Testing

Goal:

Test if a difference/relationship is statistically significant

• Null hypothesis (H₀) – no effect/difference
• Alternative hypothesis (H₁) – there is an effect

Errors:

• Type I (α): False positive (rejecting true H₀)
• Type II (β): False negative (failing to reject false H₀)

P-value

• Probability that observed data is due to chance
• If p ≤ α (e.g. 0.05) → reject H₀

Parametric Tests

ANOVA (Analysis of Variance)

Used to test differences across 3+ groups

Types:

• One-Way ANOVA – one factor
• Two-Way ANOVA – two factors, also tests interaction

Assumptions:

• Independence
• Normality
• Equal variances

Key Terms:

• SSB: Variation between groups
• SSW: Variation within groups
• F-ratio:

  F = MSB / MSW
  

If F is large, likely that group means differ significantly.

Non-Parametric Tests

Used when assumptions of normality or equal variance don’t hold

Real-Life Examples

• Body Fat Study: Use 2-sample t-test to compare men vs women
• Teaching Effectiveness: F-test to compare variance in student scores
• Diet Study: One-way ANOVA to compare 3 diets’ weight loss
• HIV Drug Trial: Mann-Whitney test for viral load comparison
• Learning Preference vs Gender: Chi-square test for association

Practice Problems

1. Normal range: What is the central 95% range for μ = 80, σ² = 144?
2. Cholesterol study: Do Asian immigrants differ significantly in cholesterol?
3. Normality check: Use a sample of weights to test for normal distribution.
4. CLT demonstration: Use different sample sizes to show CLT in action.

✅ Summary Table: Test Selection

“Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write.” – H.G. Wells