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The chi-squared test

Edexcel IAL Biology · WBI16 Practical Skills in Biology II

Chi-squared tests whether observed counts differ significantly from the counts expected by a hypothesis, such as an even spread or a genetic ratio.

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The formula

χ² = Σ (O - E)² / E

O = observed count in each category; E = expected count.
Degrees of freedom = number of categories - 1.
Use actual counts, not percentages or means.

If χ² ≥ the critical value at p = 0.05, reject H₀: the observed numbers differ significantly from those expected.

Degrees of freedom	1	2	4
Critical value (p = 0.05)	3.84	5.99	9.49

Critical values of chi-squared at p = 0.05

Worked layout

40 woodlice are placed in a choice chamber with a damp side and a dry side. If they have no preference, 20 are expected on each side.

Side	O	E	O - E	(O - E)²	(O - E)²/E
Damp	28	20	8	64	3.2
Dry	12	20	-8	64	3.2
				χ² =	6.4

Interpreting

Degrees of freedom = 2 - 1 = 1; critical value 3.84. χ² = 6.4 > 3.84, so reject H₀: there is a significant difference between observed and expected numbers; woodlice prefer damp conditions.

For a genetic cross, expected numbers come from the ratio: with 160 offspring and an expected 3 : 1 ratio, E = 120 and 40.

In ecology (CP11), chi-squared can test whether a species is spread evenly between zones.

Worked example

160 offspring of a cross are 112 tall and 48 short. Test whether they fit a 3 : 1 ratio.

E: 3/4 x 160 = 120 tall, 1/4 x 160 = 40 short
χ² = (112 - 120)²/120 + (48 - 40)²/40 = 0.533 + 1.6
= 2.13; df = 1; critical value 3.84
2.13 < 3.84

Answer: χ² = 2.13: accept H₀; there is no significant difference from a 3 : 1 ratio, so the difference may be due to chance.

Key idea: χ² = Σ(O - E)²/E using counts; df = categories - 1; p = 0.05 critical values 3.84 (1 df), 5.99 (2 df), 9.49 (4 df). χ² ≥ critical: reject H₀.

Check yourself

A plant is counted in three zones: 30, 18 and 12 plants. Calculate χ² for an even spread (expected 20 in each).
Show answer
8.4. (100 + 4 + 64) / 20 = 8.4.
For the three-zone data, χ² = 8.4. With 2 degrees of freedom (critical value 5.99), what is the conclusion?
- Accept the null hypothesis: the spread is even
- Reject the null hypothesis: the plant is not evenly spread between the zones
- There is a significant correlation
- The test needs means
Show answer
Reject the null hypothesis: the plant is not evenly spread between the zones. 8.4 > 5.99.
A chi-squared test uses 5 categories. How many degrees of freedom?
Show answer
4. 5 - 1 = 4.
In a 1 : 2 : 1 cross with 200 offspring, how many are expected in the middle class?
Show answer
100. 2/4 x 200 = 100.
Observed = 46, expected = 40. Calculate (O - E)²/E for this category.
Show answer
0.9. 36 / 40 = 0.9.
Which data are suitable for chi-squared?
- Numbers of seeds that germinated and did not germinate in two conditions
- Mean root lengths of two groups
- Paired light and temperature readings
- Percentage cover in quadrats
Show answer
Numbers of seeds that germinated and did not germinate in two conditions. Chi-squared uses counts in categories.
Chi-squared for a choice chamber is 1.2 (1 df, critical value 3.84). What does this show?
- A significant preference
- A significant correlation
- No significant preference: the difference from expected may be due to chance
- The animals all died
Show answer
No significant preference: the difference from expected may be due to chance. 1.2 < 3.84.

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Sources

Pearson Edexcel International Advanced Level in Biology specification (XBI11/YBI11, first teaching 2018): Unit 3, Unit 6 and Appendix 6 mathematical skills

Last updated 8 November 2026. Spotted a mistake? Tell us.