4.1 Chi-squared test
IB Biology Skill:
• Skill: Testing for association between two species using the chi-squared test with data obtained by quadrat
sampling.
• Skill: Recognizing and interpreting statistical significance.
Background Information:
There are numerous uses for this test. Our example shows its use in the simplest way possible.
All chi-squared tests are concerned with counts of things (frequencies) that you can put into
categories. For example, you might be investigating flower colour and have frequencies of red flowers
and white flowers. Or you might be investigating human health and have frequencies of smokers and
non-smokers.
The test looks at the frequencies you obtained and compares them with the frequencies you might
expect given your null hypothesis. The null hypothesis is this:
There is no significant difference between the observed and expected frequencies
The only difficult part of a chi-squared test is working out what the expected values are. In our example
this is very simple. It can get more complicated in different circumstances. If you use this test, each
measurement has to be independent of the others and all your expected frequencies should be bigger
than 5.
Problem :
Let us surmise that you have been wandering about on a seashore and you have noticed that a small
snail (the flat periwinkle) seems to live only on seaweeds of various kinds. You decide to investigate
whether the animals prefer certain kinds of seaweed by counting numbers of animals on different
species. You end up with the following data:
1. Null Hypothesis:
2. alternative hypothesis (HA):
3. Calculated EXPECTED Frequencies if our Null Hypothesis:
4. Calculate the value of our test statistic that is specific to our data. The formula for Chi-squared
is this:
An easy way to do this is to tabulate the calculations:
Seaweed
serrated
wrack
bladder
wrack
egg wrack
spiral wrack
other algae
TOTALS
Observed
frequencies
45
38
10
5
2
100
Expected
frequencies
O–E
O - E squared
O-E squared/E
5. Identify the Chi Square Calculated Value in the above table (HINT:The total of our final column
represents the value of the test statistic that is specific to our data) =
.
6. Calculate the degrees of Freedom: Degrees of freedom is one less than the number of
categories/ types of seaweed:
7. Identify the Significance Level: There is no rule about this but with fieldwork data most people
reckon that 5% significance is an adequate level of acceptance or rejectance of the null
hypothesis. Remember this means that you would expect to be correct in accepting or rejecting
your null hypothesis 95% of the time, 5% of the time you might get a different result due to
chance.
As with all tests of this kind, we next compare our calculated value with the critical value (obtained
from a table of critical values of chi-squared).
.

Our calculated Value:

Critical value of Chi-squared at 5% significance and 4 degrees of freedom is

Do you accept of Reject the NULL HYPOTHESIS?
.
o If our calculate value is bigger that the critical value then we must reject the null
hypothesis.
.
o If our calculate value is less than the critical value then we accept the null hypothesis
8. Discuss what it means to accept or reject the null hypothesis