IB Biology How to Solve: IB Biology – Chi-Squared & Statistical Testing for Lab Investigations

How to Solve: IB Biology – Chi-Squared & Statistical Testing for Lab Investigations

Introduction

Mastering chi-squared unlocks 6–8 marks on your IB Biology IA—and could be the difference between a 6 and a 7. Examiners love testing this because it proves you can analyze real lab data, not just memorize facts. Whether you’re comparing genetic crosses, enzyme reactions, or economic models, this skill applies across IB Physics, Chemistry, Biology, and Economics.

WHAT YOU NEED TO KNOW FIRST

Before diving in, you must understand:
1. Null Hypothesis (H₀): The default assumption that there’s no difference or no effect in your data.
2. Degrees of Freedom (df): How many categories in your data can vary independently (e.g., for a 2×2 table, df = 1).
3. Critical Value: The cutoff from the chi-squared table that determines if your result is significant.

(If any of these are fuzzy, pause and review them now—this guide won’t make sense without them.)

KEY TERMS & FORMULAS

1. Chi-Squared Test (χ²)

Purpose: Tests if observed data matches expected data (e.g., genetic ratios, enzyme activity, survey responses).

Formula: MEMORISE THIS [ \chi^2 = \sum \frac{(O - E)^2}{E} ] - O = Observed frequency (your actual data) - E = Expected frequency (what you’d predict under H₀) - ∑ = Sum of all categories

2. Degrees of Freedom (df)

Formula: MEMORISE THIS [ df = (\text{number of rows} - 1) \times (\text{number of columns} - 1) ] - For a 1×2 table (e.g., coin flips): df = 1 - For a 2×2 table (e.g., dihybrid cross): df = 1

3. Critical Value Table

• Given on exam sheet (IB provides a chi-squared table).
• How to use it:
• Find your df on the left.
• Move right to the column for your significance level (usually p = 0.05).
• If your χ² value ≥ critical value, reject H₀.

STEP-BY-STEP METHOD

(Follow these steps exactly for every problem.)

Step 1: State the Null Hypothesis (H₀)

• Write: "There is no significant difference between observed and expected results."
• Example: "The ratio of pea plant phenotypes is 3:1 (dominant:recessive)."

Step 2: Calculate Expected Frequencies (E)

• Use the ratio from H₀ to split your total observations.
• Example: If H₀ predicts a 3:1 ratio and you have 100 plants:
• E(dominant) = 75
• E(recessive) = 25

Step 3: Plug into the χ² Formula

• For each category, calculate ((O - E)^2 / E).
• Sum all values to get χ².

Step 4: Determine Degrees of Freedom (df)

• Use the formula: (df = (\text{rows} - 1) \times (\text{columns} - 1)).

Step 5: Find the Critical Value

• Use the IB chi-squared table (p = 0.05).
• Compare your χ² to the critical value.

Step 6: Make a Decision

• If χ² ≥ critical value → Reject H₀ (significant difference).
• If χ² < critical value → Fail to reject H₀ (no significant difference).

Step 7: Write a Conclusion

• Example: "Since χ² (4.2) > critical value (3.84), we reject H₀. The observed data does not fit the 3:1 ratio."

WORKED EXAMPLES

Example 1 – Basic (Monohybrid Cross)

Problem: A student crosses pea plants and gets: - 78 yellow seeds (dominant) - 22 green seeds (recessive) Test if this fits a 3:1 ratio.

Step 1: H₀ "The ratio of yellow to green seeds is 3:1."

Step 2: Expected (E) Total = 78 + 22 = 100 - E(yellow) = 100 × (3/4) = 75 - E(green) = 100 × (1/4) = 25

Step 3: χ² Calculation [ \chi^2 = \frac{(78 - 75)^2}{75} + \frac{(22 - 25)^2}{25} = \frac{9}{75} + \frac{9}{25} = 0.12 + 0.36 = 0.48 ]

Step 4: df 1×2 table → df = 1

Step 5: Critical Value (p = 0.05, df = 1) = 3.84

Step 6: Decision
0.48 < 3.84 → Fail to reject H₀

Step 7: Conclusion "The observed data fits the 3:1 ratio (χ² = 0.48 < 3.84)."

What we did and why: We tested if the student’s data matched Mendel’s predicted ratio. Since χ² was small, the results were consistent with H₀.

Example 2 – Medium (Dihybrid Cross)

Problem: A student crosses pea plants and gets: - 315 round yellow - 108 round green - 101 wrinkled yellow - 32 wrinkled green Test if this fits a 9:3:3:1 ratio.

Step 1: H₀ "The ratio of phenotypes is 9:3:3:1."

Step 2: Expected (E) Total = 315 + 108 + 101 + 32 = 556 - E(round yellow) = 556 × (9/16) = 312.75 - E(round green) = 556 × (3/16) = 104.25 - E(wrinkled yellow) = 556 × (3/16) = 104.25 - E(wrinkled green) = 556 × (1/16) = 34.75

Step 3: χ² Calculation [ \chi^2 = \frac{(315 - 312.75)^2}{312.75} + \frac{(108 - 104.25)^2}{104.25} + \frac{(101 - 104.25)^2}{104.25} + \frac{(32 - 34.75)^2}{34.75} ] [ = 0.016 + 0.135 + 0.101 + 0.218 = 0.47 ]

Step 4: df 2×2 table → df = (2-1)(2-1) = 1

Step 5: Critical Value (p = 0.05, df = 1) = 3.84

Step 6: Decision
0.47 < 3.84 → Fail to reject H₀

Step 7: Conclusion "The observed data fits the 9:3:3:1 ratio (χ² = 0.47 < 3.84)."

What we did and why: We tested a more complex genetic cross. The small χ² value means the data matches the expected ratio.

Example 3 – Exam-Style (Disguised Problem)

Problem: A student investigates if a coin is fair. They flip it 50 times and get: - 30 heads - 20 tails Test at p = 0.05.

Step 1: H₀ "The coin is fair (50:50 ratio)."

Step 2: Expected (E) - E(heads) = 50 × 0.5 = 25 - E(tails) = 50 × 0.5 = 25

Step 3: χ² Calculation [ \chi^2 = \frac{(30 - 25)^2}{25} + \frac{(20 - 25)^2}{25} = \frac{25}{25} + \frac{25}{25} = 1 + 1 = 2 ]

Step 4: df 1×2 table → df = 1

Step 5: Critical Value (p = 0.05, df = 1) = 3.84

Step 6: Decision 2 < 3.84 → Fail to reject H₀

Step 7: Conclusion "The coin is likely fair (χ² = 2 < 3.84)."

What we did and why: This looks like a coin flip, but it’s the same math as genetics! The χ² test works for any categorical data.

COMMON MISTAKES

EXAM TRAPS

1-MINUTE RECAP

(Spoken naturally, like a coach the night before the exam.)

"Okay, listen up—this is your 60-second chi-squared survival guide. First, write your null hypothesis: ‘No difference between observed and expected.’ Then, calculate expected values using the ratio from H₀. Plug into the formula: sum of (O-E)²/E. Degrees of freedom? For a 2×2 table, it’s 1. Grab the critical value from the table—p = 0.05, df = 1, it’s 3.84. If your χ² is bigger, reject H₀. Smaller? Fail to reject. That’s it. No shortcuts, no guessing. Write every step, label your table, and you’ll get full marks. Now go practice!