The best examples of chi-square test examples in genetics

When instructors talk about examples of chi-square test examples in genetics, they almost always start with Mendel’s peas. There’s a good reason: the predictions are clean, the math is simple, and you can see exactly how expected and observed numbers line up.

Imagine a monohybrid cross: Yy × Yy for seed color, where Y = yellow (dominant) and y = green (recessive). Mendelian genetics predicts a 3:1 phenotypic ratio of yellow to green seeds.

Say you grow 160 seeds and observe:

• 118 yellow
• 42 green

Your hypothesis (H₀): the data follow a 3:1 ratio.

Expected counts under 3:1 for 160 seeds:

• Yellow: 160 × 3/4 = 120
• Green: 160 × 1/4 = 40

Now apply the chi-square formula:

χ² = Σ ( (Observed − Expected)² / Expected )

For yellow: (118 − 120)² / 120 = 4 / 120 ≈ 0.0333
For green: (42 − 40)² / 40 = 4 / 40 = 0.10

Total χ² ≈ 0.1333 with 1 degree of freedom (2 phenotypic classes − 1). The p-value is large (well above 0.05), so you fail to reject the Mendelian 3:1 model. This is one of the cleanest, best examples of chi-square test examples in genetics: it shows how small deviations from the expected ratio are usually just sampling variation.

For background on Mendelian inheritance itself, see resources from the National Human Genome Research Institute (NHGRI).

Dihybrid cross: testing a 9:3:3:1 ratio with chi-square

If you want richer examples of chi-square test examples in genetics, move from one gene to two genes. Consider a classic dihybrid cross: YyRr × YyRr, where

• Y = yellow, y = green
• R = round, r = wrinkled

Mendel’s law of independent assortment predicts a 9:3:3:1 phenotypic ratio:

• 9 yellow round
• 3 yellow wrinkled
• 3 green round
• 1 green wrinkled

Suppose you collect 320 seeds and observe:

• Yellow round: 180
• Yellow wrinkled: 82
• Green round: 46
• Green wrinkled: 12

Expected counts for 320 seeds:

• 9/16 × 320 = 180
• 3/16 × 320 = 60
• 3/16 × 320 = 60
• 1/16 × 320 = 20

Now calculate χ²:

• Yellow round: (180 − 180)² / 180 = 0
• Yellow wrinkled: (82 − 60)² / 60 = 484 / 60 ≈ 8.07
• Green round: (46 − 60)² / 60 = 196 / 60 ≈ 3.27
• Green wrinkled: (12 − 20)² / 20 = 64 / 20 = 3.2

Total χ² ≈ 14.54 with 3 degrees of freedom (4 classes − 1).

A χ² of 14.54 with 3 df gives a p-value below 0.01. That’s strong evidence that your data do not fit the 9:3:3:1 ratio. This real example of a chi-square test in genetics might indicate:

• Linkage between the two genes
• Selection against certain genotypes
• Misclassification of phenotypes

This is where chi-square goes from classroom exercise to a genuine scientific tool: it flags that the simple Mendelian model may not describe what’s happening biologically.

Testing a 1:2:1 genotypic ratio in F2 generations

Another standard example of chi-square test examples in genetics involves genotypes rather than phenotypes. For a monohybrid cross Yy × Yy, the expected genotypic ratio is 1:2:1 (YY:Yy:yy).

Say you genotype 200 F2 offspring in a lab using PCR or SNP arrays and observe:

• 48 YY
• 110 Yy
• 42 yy

Expected counts under 1:2:1:

• YY: 200 × 1/4 = 50
• Yy: 200 × 1/2 = 100
• yy: 200 × 1/4 = 50

Compute χ²:

• YY: (48 − 50)² / 50 = 4 / 50 = 0.08
• Yy: (110 − 100)² / 100 = 100 / 100 = 1
• yy: (42 − 50)² / 50 = 64 / 50 = 1.28

Total χ² ≈ 2.36 with 2 degrees of freedom.

The p-value is comfortably above 0.1, so you keep the 1:2:1 model. In modern genetics labs, this kind of test is a routine quality check: are the genotyping results consistent with simple Mendelian segregation, or did something go wrong with the cross, the DNA extraction, or the assay?

For a more formal treatment of this kind of segregation analysis, you can compare with population genetics material from places like Harvard’s online genetics resources.

Sex-linked inheritance: chi-square with X-linked traits

Not all genes behave the same way in males and females. That makes sex-linked traits a great source of real examples of chi-square test examples in genetics.

Consider an X-linked recessive trait in humans, like classic color blindness. If a carrier woman (XᴺXᶜ) has children with a normal man (XᴺY), the expected offspring proportions are:

• Daughters: 1/2 normal (XᴺXᴺ), 1/2 carrier (XᴺXᶜ)
• Sons: 1/2 normal (XᴺY), 1/2 affected (XᶜY)

If you’re only tracking sons, the expected ratio is 1:1 normal:affected.

Say you look at 80 sons in an extended pedigree and find:

• 54 normal
• 26 affected

Expected under 1:1 for 80 sons:

• 40 normal
• 40 affected

χ²:

• Normal: (54 − 40)² / 40 = 196 / 40 = 4.9
• Affected: (26 − 40)² / 40 = 196 / 40 = 4.9

Total χ² = 9.8 with 1 degree of freedom. The p-value is below 0.01, so the data strongly disagree with the simple 1:1 model. In practice, this could suggest:

• Misclassification (mildly affected sons labeled as normal)
• Reduced fertility or survival of affected sons
• The trait is not purely X-linked recessive

This kind of chi-square test is widely used in medical genetics to sanity-check inheritance models before more expensive molecular testing. For background on X-linked conditions, see the NIH Genetic and Rare Diseases Information Center.

Human blood groups: chi-square and population genetics

Blood group data offer some of the best examples of chi-square test examples in genetics at the population level, rather than within a single family or cross.

Take the ABO blood group system. Suppose a researcher samples 1,000 individuals in a city and observes:

• Type A: 420
• Type B: 90
• Type AB: 50
• Type O: 440

Using allele frequency estimates (p for Iᴬ, q for Iᴮ, r for i), you can compute expected genotype and phenotype frequencies under Hardy–Weinberg equilibrium. Then you use a chi-square test to see whether observed blood type counts match those expectations.

If the chi-square test shows a large χ² and a very small p-value, it suggests:

• Non-random mating
• Selection for or against certain blood types
• Population structure (subgroups with different allele frequencies)

This is a classic population genetics example of chi-square in action, and it’s still used in modern epidemiology and forensic genetics. For an accessible explanation of Hardy–Weinberg and blood groups, the CDC’s genetics resources are a good starting point.

Chi-square in modern GWAS-style thinking

By 2024–2025, most large-scale human genetics studies use massive datasets, but the underlying logic still leans on the same chi-square idea: compare observed counts to expected counts.

In a simplified case–control setup for a single SNP:

• You have 5,000 cases with a disease and 5,000 controls.
• At a particular SNP, the minor allele appears 2,000 times in cases and 1,600 times in controls.

You can arrange this in a 2×2 contingency table (allele vs. case/control), then use a chi-square test of independence:

Expected minor alleles in cases under no association:

• 10,000 × (3,600 / 20,000) = 1,800

You repeat for each cell, compute χ², and get a p-value. In real GWAS, this is done for millions of SNPs with corrections for multiple testing, but conceptually it’s the same chi-square test you use in a pea-plant lab.

This is one of the best examples of chi-square test examples in genetics that bridges classroom problems and current research. The statistic is the same; the scale is just much larger. For an overview of how association tests are used in modern genomics, the NHGRI GWAS catalog resources are helpful.

Segregation distortion and transmission ratio distortion

Another modern application, often discussed in 2024–2025 literature, is segregation distortion or transmission ratio distortion (TRD). These are situations where alleles are passed on to offspring at frequencies that deviate from the expected 50:50.

Imagine a heterozygous mouse (A/a) where you expect half the gametes (and thus half the offspring alleles) to be A and half to be a. You genotype 400 offspring and get:

• 260 A alleles
• 140 a alleles

Under equal transmission, expected counts:

• A: 200
• a: 200

χ²:

• A: (260 − 200)² / 200 = 3,600 / 200 = 18
• a: (140 − 200)² / 200 = 3,600 / 200 = 18

Total χ² = 36 with 1 df, which gives a tiny p-value. This is a strong signal of TRD. Researchers then dig into biological explanations, like meiotic drive, gamete competition, or embryo lethality.

This is another real example of chi-square test in genetics where the statistic is not just a homework answer; it points you toward interesting biology.

Chi-square in linkage mapping and recombination analysis

Genetic linkage mapping is packed with examples of chi-square test examples in genetics. When you look at offspring from a testcross, you can classify them as parental or recombinant types.

Suppose you cross an individual with genotype AB/ab to ab/ab and score 1,000 offspring:

• Parental (AB, ab): 460 AB, 460 ab
• Recombinant (Ab, aB): 40 Ab, 40 aB

Under no linkage (independent assortment), you’d expect 500 parental and 500 recombinant (a 1:1 parental:recombinant ratio). Instead, you see 920 parental and 80 recombinant.

You can set up a chi-square test comparing observed counts to expected under independence:

• Expected parental: 500
• Expected recombinant: 500

χ²:

• Parental: (920 − 500)² / 500 = 176,400 / 500 = 352.8
• Recombinant: (80 − 500)² / 500 = 176,400 / 500 = 352.8

Total χ² = 705.6 with 1 df, which is astronomically significant. This confirms strong linkage and allows you to estimate recombination frequency (80/1,000 = 8%).

Again, the math is the same chi-square framework; the interpretation is genetic distance and map construction.

How to think about chi-square results in genetics

Across all these examples of chi-square test examples in genetics, the interpretation pattern is consistent:

• Small χ², large p-value (typically > 0.05):

  • Your observed data are consistent with the genetic model you proposed (3:1, 1:2:1, 1:1, Hardy–Weinberg, no association, etc.).
  • It does not prove the model is true; it just means you don’t have strong evidence against it.

• Large χ², small p-value (typically < 0.05):

  • Your data disagree with the model more than you’d expect from random sampling.
  • This is a prompt to reconsider your assumptions: maybe the trait is linked, sex-influenced, under selection, or misclassified.

A few practical tips that show up repeatedly in the best examples of chi-square test examples in genetics:

• Check sample size: Very small counts (especially expected counts < 5) can make chi-square unreliable; exact tests (like Fisher’s) may be better.
• Define hypotheses carefully: You must be clear about the expected ratio or model before you calculate χ².
• Look beyond the p-value: In genetics, the pattern of which classes deviate (e.g., too few homozygous mutants) often tells the biological story.

For statistical background, many university biostatistics notes (for example, from major U.S. universities like Harvard or state schools) walk through these tests in a genetics context.

FAQ: Common questions about chi-square test examples in genetics

Q1. What are some standard examples of chi-square test examples in genetics used in teaching?
Common classroom examples include Mendelian monohybrid and dihybrid crosses (3:1 and 9:3:3:1 ratios), testing a 1:2:1 genotypic ratio, and simple sex-linked traits with 1:1 expectations in sons. These are the best examples for building intuition before moving on to blood groups, Hardy–Weinberg equilibrium, and linkage mapping.

Q2. Can you give an example of using chi-square for human disease genetics?
Yes. A typical example of a chi-square test in human genetics is a case–control study of a candidate gene variant. You compare allele or genotype counts between patients and healthy controls using a chi-square test of independence. A significant result suggests the variant is associated with disease risk, which can then be followed up with larger studies or functional experiments.

Q3. How are chi-square tests used in modern genomics, not just Mendel-style problems?
Modern genomics uses chi-square logic in genome-wide association studies (GWAS), transmission ratio distortion analyses, and large-scale quality control checks (for example, testing whether genotype frequencies match Hardy–Weinberg expectations in control samples). The datasets are huge, but the core idea—comparing observed and expected counts—remains the same.

Q4. When should I avoid using a chi-square test in genetics?
If expected counts in any category are very small (often below 5), the chi-square approximation can be unreliable. In that case, exact tests like Fisher’s exact test are preferred. Also, chi-square does not handle continuous traits directly; for quantitative traits (like height), you’d typically use regression or ANOVA instead.

Q5. Where can I learn more about genetic examples that use chi-square tests?
Authoritative sources include the National Human Genome Research Institute, the NIH Genetic and Rare Diseases Information Center, and genetics and biostatistics courses from major universities. Many of these resources walk through real examples of chi-square test applications in both classical and modern genetics.