Biology IA · D · Continuity and change · topic 24 of 25

Corn ear: does counting more kernels change the 3 : 1? · IB Biology IA idea: guide, sample graph and step-by-step plan

easy kit, tricky stats. examiners notice.

Mendel got 3 : 1 on a cob, but only if you count enough kernels. How many is enough, and what does the chi-squared test actually measure?

Difficulty●●○ Medium
KitBasic: a genetics corn ear (or good photographs), a spreadsheet
TimeAbout 10 hours
FitsSL and HL (the dihybrid twist is HL content)
  • ✓ you like statistics that actually need thinking
  • ✓ you want to link a big idea (Mendel) to a number
  • ✓ you can use random numbers in a spreadsheet

The big idea

← the test depends on n!

A cross between two heterozygotes is expected to give about three purple kernels to one yellow, but a small sample can stray from that by chance. A chi-squared test asks whether the counts fit the expected ratio. Here is the subtle part: chi-squared grows with the number counted if the ear really differs from 3 : 1, so it is not a pure measure of deviation. Your investigation takes random samples of different sizes from one ear and works out what the test is really telling you, and whether the 3 : 1 is itself a fair hypothesis.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

Swipe for more twists →

Use proportion, not chi-squared

Compare the observed proportion of purple kernels, with a confidence interval, at each sample size.

→ Your dependent variable no longer scales with n; you compare spread with the expected proportion, and discuss why this is easier to interpret.
Dihybrid ear

Use an ear with two traits (colour and smooth or shrunken), expected 9 : 3 : 3 : 1.

→ Four categories and df = 3; counts for the rarest class need a larger sample, so your sample sizes must go higher.
Many random samples

For each sample size, draw many random samples from the same ear using a spreadsheet and record the chi-squared of each.

→ You get a distribution at each n, not a single value; you can plot the spread and the proportion that exceed the critical value.
Compare several ears

Repeat the whole-ear count on several ears and compare the proportions with 3 : 1.

→ Ear becomes the unit of replication; you can ask whether different ears agree, and what a different ratio might mean about the parents.
Test a different expected ratio

Test the observed counts against 1 : 1 as well as 3 : 1.

→ Your analysis becomes a comparison of two hypotheses; you must explain why one fits better and how you judge it.
Change the context

Link it to crop breeding, genetic counselling or Mendel's pea experiments.

→ Gives the introduction a real purpose; the method and the graph stay the same.

Your Biology IA research question: run the check

Write your own question first. Then make sure it ticks every box:

  • Names the independent variable as the number of kernels counted, with at least five sample sizes and how you choose them.
  • Names the dependent variable (the test statistic) and the formula, and says what the expected counts are and why.
  • Says how kernels are selected at random, and what happens to the ear between samples (replaced or not).
  • Says how many random samples you take at each size, so each value has a spread.
  • States the degrees of freedom and the critical value, and what a result above or below it would mean.
  • Says what you will do if an expected count is small (below 5), since the test is unreliable there.

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The set-up

One way to do it. Draw your own ear and your own sampling plan for the Method section.

OEpurpleyellowtotalpick kernels withrandom numbersE comes from thecross you assumethe ear is a finite number ofkernels: n cannot exceed ittally O, work out E,then χ² with df = 1

How your Biology IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 3,000 words it takes. The words are all yours.

Research question and context~250
Your question; Mendel's 3 : 1 and why a sample can stray from it; what you predict about chi-squared as the sample grows.
Background biology~450
Monohybrid cross and the 3 : 1 ratio; why the parents are assumed heterozygous; the chi-squared test, its formula and conditions; why the test statistic depends on n.
Variablestable
Independent, dependent and controlled variables; how randomness is achieved.
Method~450
How the whole ear is counted first; how the numbers of kernels are chosen; how random samples are drawn; how many per size; how chi-squared is calculated; df and critical value.
Safety, ethics, environment~40
Short: no real hazards, but say how the ear is handled and what happens to it.
Raw datatable
Whole-ear counts; observed counts for each random sample at each size, with the chi-squared of each.
Processing~300
Chi-squared for each sample; mean and spread at each size; one worked example, including the expected counts and df.
Graphs and analysis~400
Chi-squared against n with error bars and the critical value line; what the shape tells you; a second graph with proportion if you use one.
Conclusion~400
Answer with values and spread; say whether chi-squared grew with n and what that means; say whether the 3 : 1 hypothesis is supported, and limit it to this ear.
Evaluation~600, the biggest part!
Weaknesses ranked by their effect on chi-squared: sampling method, small expected counts, counting errors on colour, one ear only, finite ear size.
Referencesnot counted
Genetics background, chi-squared method and anything on the ear's source.

What your Biology IA graph could look like

bars = spread of repeated samples

Made-up data, to show you the form of a strong graph, not the answer. This sketch assumes an ear that differs slightly from 3 : 1. example only

050100150200250300350400012345Number of kernels counted, n (± 0)χ² (mean of 20 random samples ± SE), df = 1expected for an ear slightly off 3 : 1critical value, df = 1, p = 0.05 (3.84)mean χ² of random samples, with SEif the ear is exactly 3 : 1,χ² stays near 1 at every nILLUSTRATIVE · MADE-UP DATA
  • The bars are the spread over repeated random samples, not an instrument uncertainty. Say that.
  • If the ear truly fits 3 : 1, chi-squared stays near 1 however many kernels you count. If it does not, it climbs roughly in proportion to n.
  • The critical value line shows when the test would reject 3 : 1 at the 5% level.
  • With very small n, an expected count under 5 means the test is unreliable: say so and treat those points with care.
  • Another graph to consider: the observed proportion of purple kernels against n with its spread, which shows the same idea without the scaling.

How big is the sampling error?

The spread in the observed proportion shrinks as you count more. This chart shows the standard error for a true proportion of 75% purple. It is the reason your small samples jump around.

10 kernels13.7 pp25 kernels8.7 pp50 kernels6.1 pp100 kernels4.3 pp400 kernels2.2 ppsmall samples: huge spread!standard error of the % purple, in percentage points (pp)ILLUSTRATIVE · MADE-UP NUMBERS

Your game plan

count the whole ear once, first

Tick steps off as you go. We'll remember on this device.

0 of 10 done

  1. Read and predictLearn the monohybrid cross, expected counts and chi-squared. Work out what happens to chi-squared as n grows, both if the ear fits 3 : 1 and if it doesn't. Write down your prediction.
  2. Trial run (1 lesson)Count a section of the ear. Is it easy to tell colours apart? How long does a full count take? How will you pick kernels randomly, with or without replacement?
  3. Count the whole earNumber the kernels or photograph the ear with a grid, and count every kernel once. Record the totals: this is your reference for later samples.
  4. Decide sample sizes and repeatsChoose at least five sample sizes from small to large, and how many random samples at each. Check the expected count for yellow is at least 5 at the smallest size, or explain what you do instead.
  5. Draw random samplesUse a random number generator to pick kernels. Record counts for each sample. Check you are not drawing the same kernel twice by accident.
  6. ProcessExpected counts, chi-squared and df for each sample, with one worked example. Then the mean and spread at each size.
  7. Graph and interpretPlot chi-squared against n with error bars and the critical value. Say what the shape tells you about the ear, and about the test.
  8. Compare with theoryDoes the whole-ear count fit 3 : 1? Is the expected ratio reasonable for the parents? What would a different ratio mean?
  9. EvaluateRank weaknesses by their effect on chi-squared. Tie each improvement to a problem you saw.
  10. Check before your teacher's draftRun the checklist below, then get your RQ or draft checked. Your teacher reads one draft only, so make it count.

Biology you'll need

Traps to avoid in this Biology IA

  • Counting one section of the ear and calling it a random sample. A section is not random.
  • Wrong degrees of freedom: two categories gives df = 1, not 2.
  • Using chi-squared when an expected count is below 5, without saying the test is unreliable.
  • Treating a high chi-squared at large n as a 'bigger deviation' when the test statistic simply grows with n.
  • Forgetting that 3 : 1 is a hypothesis about the parents, not a fact about the ear.
  • Using the same few kernels again and again so the 'random samples' are not independent.
  • Using percentages in the test instead of counts.

More Biology IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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