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শিক্ষামূলক নোট: এই পৃষ্ঠা একাডেমিক জীববিজ্ঞান শেখা ও পরীক্ষার প্রস্তুতির সহায়ক।

Chi-square Test: কাই-বর্গ পরীক্ষা

Concept Overview

Chi-square test বা কাই-বর্গ পরীক্ষা হলো categorical frequency data বিশ্লেষণের একটি inferential statistical test। এটি observed frequency এবং expected frequency-এর মধ্যে পার্থক্য random variation দিয়ে ব্যাখ্যা করা যায় কি না তা যাচাই করে।

Biology-তে chi-square test বিশেষভাবে useful যখন data count বা category আকারে থাকে: যেমন tall vs dwarf plants, male vs female, diseased vs healthy, present vs absent, survived vs died.

Why This Matters

Biological data সবসময় mean-based নয়। অনেক সময় researcher frequency বা category count নিয়ে কাজ করেন। Mendelian genetics-এ 3:1 ratio, ecology-তে species presence/absence, public-health data-তে disease frequency, behaviour study-তে response category—এসব ক্ষেত্রে chi-square thinking দরকার।

LBFL Educational Framework

Use the central framework pages below for the full method. This page keeps only the topic-specific learning path so learners do not meet the same boilerplate repeatedly.

LOLO / LALA Objectives, outcomes, activities, and applications. Bloom Taxonomy Remember, understand, apply, analyze, evaluate, and create. CQ Studio Board-style creative question structure and answer logic. Assessment Rubric MCQ, short answer, CQ, diagram, practical, and reflection quality. Practical Framework Observation, evidence, life application, and reflective responsibility.

Chi-square Learning Focus

এই lecture central LBFL framework-কে categorical-data hypothesis testing-এ প্রয়োগ করে। Learner-এর focus হবে observed frequency, expected frequency, goodness of fit, contingency table, degrees of freedom, assumptions, chi-square formula, and cautious biological interpretation.

Main Formula

χ² = Σ[(O − E)² / E]

Where:

• O = observed frequency
• E = expected frequency
• Σ = sum over all categories

Type 1: Goodness of Fit

Goodness of fit test checks whether observed data fit a theoretical ratio or expected distribution.

Example question:

Do observed plant counts fit Mendelian 3:1 ratio?

Worked Example: Mendelian 3:1 Ratio

Problem: 400 pea plants show 300 tall and 100 dwarf plants. Does this fit the expected 3:1 ratio?

Expected values:

Total plants = 400
Expected tall  = 3/4 × 400 = 300
Expected dwarf = 1/4 × 400 = 100

Calculation table:

Category	O	E	O − E	(O − E)² / E
Tall	300	300	0	0
Dwarf	100	100	0	0
Total	400	400		χ² = 0

Interpretation:

χ²calculated = 0

In this exact dataset, observed frequency matches expected frequency. Therefore, there is no statistical evidence against the 3:1 model from these data.

Degrees of Freedom for Goodness of Fit

For a simple goodness-of-fit test:

df = number of categories − 1

For tall/dwarf categories:

df = 2 − 1 = 1

Type 2: Test of Independence

A chi-square test of independence checks whether two categorical variables are associated.

Example questions:

Is habitat type associated with species presence?
Is smoking status associated with disease category?
Is sex associated with response category?

Contingency Table Logic

In a contingency table, expected frequency is calculated from row total, column total and grand total.

Expected frequency = (Row total × Column total) / Grand total

Degrees of freedom:

df = (r − 1)(c − 1)

Where:

• r = number of rows
• c = number of columns

Decision Logic

State H₀ and H₁
  ↓
Calculate expected frequencies
  ↓
Calculate χ²
  ↓
Find df
  ↓
Compare with critical value or p-value
  ↓
Reject or fail to reject H₀
  ↓
Interpret in biological context

Assumptions and Conditions

Categorical data

Data should be frequency counts, not continuous measurements.

Independent observations

One observation should not be counted repeatedly or improperly linked.

Expected frequency

Expected counts should not be too small; very small expected counts can weaken the test.

Clear categories

Categories should be mutually exclusive and biologically meaningful.

Chi-square vs t-test

Feature	Chi-square test	t-test
Data type	categorical frequency	continuous measurement
Main comparison	observed vs expected counts	means
Example	3:1 genetic ratio	mean plant height
Output	χ² value	t value
Key concern	expected frequency and categories	mean, SD, sample size

Common Mistakes to Avoid

Mistake 1

Using chi-square test for continuous data such as weight or height without categorization.

Mistake 2

Forgetting to calculate expected frequency before using the formula.

Mistake 3

Interpreting failure to reject H₀ as proof that H₀ is absolutely true.

Mistake 4

Ignoring biological design, sample source, and category validity.

Synaptic Bridge

Chi-square test teaches that categories also carry evidence. A biological pattern may look acceptable or suspicious, but scientific thinking asks: how far are observed counts from expected counts, and is that difference larger than random variation would usually allow?

Critical Thinking Questions

1. Why is chi-square test suitable for Mendelian ratio data?
2. What is the difference between goodness of fit and test of independence?
3. Why must expected frequencies be calculated before χ²?
4. Why does failure to reject H₀ not prove the expected model absolutely true?
5. When would t-test be more appropriate than chi-square test?

Related Learning Paths

• Biostatistics Hub
• Basic Concepts of Biostatistics
• T-test: Significant Difference Between Means
• Measures of Dispersion
• MCQ Arena

References

• Standard HSC Zoology Biostatistics notes.
• General biostatistics references on chi-square goodness of fit, contingency table and categorical data analysis.