4.2 Applications and skills

4.2.2 Chi-square statistics and quadrat samples

  • A quadrat sample is a method used to collect random data in order to estimate population size or distribution of species in an ecological community.
  • Quadrats are square or rectangular frames, like picture frames, which are placed on the ground so that scientists can count organisms within the frame.
  • Quadrats are useful for counting plants, fungi or very slow-moving animals. The dimensions of a quadrat can range from 10 x 10cm when studying lichens or moss, to 10 x 10m when studying trees.

How to set up a quadrat sample

  1. Choose or make suitably sized quadrats for your sample. 1 x 1m squares are appropriate for most samples.
  2. Measure the test site and divide it into a grid of zones equal to the size of your quadrat. For example, if your quadrat is 1 x 1m, and your test site is 8 x 10m, you should have 80 zones. Number the zones.
  3. Use a random method to generate numbers, for example roll a die, or use the MS Excel RAND function.
  4. Place your quadrats in the areas represented by the numbers generated and start counting species. Make a note of which species are present or absent in each quadrat.
  5. Your sample is complete when you have counted in about 5% of the zones, for example 4 zones out of 80, or when you are consistently finding the same species in all quadrats (i.e. no new species). Record the species as either present or absent for each quadrat as shown in the table below.
Species Quadrat 1 Quadrat 2 Quadrat 3 ... and so on
A + + +  
… and so on        

Figure 4.2.2a – Results table for quadrat sample

Testing for association using a chi-square (x2) test

A chi-squared test is a statistical tool used to determine whether there is an association between two sets of frequency data. The test involves calculating the probability (p) of an association by comparing observed values to values expected if there was no association. It is important that the data used is categorical, not continuous.

Categorical data
(can be used in a chi-square test)
Continuous data
(cannot be used in a chi-square test)
  • Species name
  • Sex
  • Presence of a certain gene or trait
  • Yes/No category
  • Height
  • Abundance
  • Density

For example, let’s say you wanted to know if the presence of one species is associated with the presence of another. This could mean either that the two species tend to appear together, or that the presence of one species is associated with the presence or absence of another. Either way, you have two possible hypotheses:

  • There is no association between the two species. This is the null hypothesis. If there is no association, at the end of the test we will see that p >0.05.
  • There is an association between the two species. If there is an association, at the end of the test we will see that p <0.05. 

Let’s assume we have collected data about many species, as shown in Figure 4.2.2a. Highlighting the relevant data about the two species we are interested in, leaves us with a table like Figure 4.2.2b.

  Species 2 present Speciesnot present Total
Species 1 present 12 5 17
Species 1 not present 8 13 21
Total 20 18 38

Figure 4.2.2b – Observed frequencies of species in a quadrat sample
This is called a 2x2 contingency table. It shows the different combinations of outcomes and the frequency observed of each.

Now we can calculate the value we expect if the null hypothesis were true, using this formula:

Expected space frequency space equals fraction numerator Column space total space straight x space Row space total over denominator Total space number space of space quadrats end fraction

  Species 2 present Species 2 not present Total
Species 1 present 8.95 8.05 17
Species 1 not present 11.05 9.95 21
Total 20 18 38

Figure 4.2.2c – Expected frequencies if no association (null hypothesis)

At this point, you may choose to use an online chi-square calculation tool to determine the p value. (You can also use the ‘CHITEST’ function on Excel, to compare the expected to observed values.)

Alternatively, you can calculate the chi-square ( x2 )value by using the following formula:

begin mathsize 20px style X squared equals sum subscript blank superscript blank fraction numerator left parenthesis straight O subscript straight n minus straight E subscript straight n right parenthesis squared over denominator straight E subscript straight n end fraction

end style

Where O is the observed frequency (from Figure 4.2.2b) and E is the expected frequency (from Figure 4.2.2c) for each outcome (n).

For the data set above, the calculated chi-squared value is 3.98. We can now compare the chi-square value to a table of critical values to see if our result is significant. A 2x2 contingency table has one degree of freedom, so we read the first line of the table.

table of critical valuesFigure 4.2.2d – Table of critical values for the chi-square test
The calculated chi-squared value falls between 95% and 97% confidence. This means our null hypothesis is void.

In our example, we can conclude, with more than 95% confidence, that there is an association between the presence of species 1 and the presence of species 2.

homemade quadratFigure 4.2.2e – Quadrats are very easy to make using wood or plastic tubing

In the lab

  • The most important thing to remember when setting up a quadrat is that you should be sampling randomly. Don’t look for areas that appear to be ‘good’ test sites.
  • Make sure you define very clearly what you are sampling. Are you counting the number of individuals of each species to estimate population size? Are you counting the presence of different species? Be clear in your aim and method.

underwater quadratFigure 4.2.2f – Underwater quadrat
This researcher has divided his quadrat into 100 zones in order to make it easier for him to calculate the percentage of the sample area where different species are found.

Concept tip

  • The aim of every statistical test is to prove that the null hypothesis is wrong. The null hypothesis predicts that the experimental values we observed are completely random.
  • An association between the two species means that the probability of these numbers being random is less than 5%. We can state our conclusion (i.e. that there is an association) with 95% confidence. 


An example of a free on-line statistical tool can be found at:


Using this tool, our data has a p value of 0.046.

plant associationsFigure 4.2.2g – Plant associations
Is the diversity in this field a coincidence or are these plants species associated with each other? Use a chi-square test to find out!

Extended essay

Think about ways you could determine if there are associations of different abiotic factors on species diversity and/or populations. A good way to do this would be to use quadrats or transects at different locations. Learn about transects in 14.2.4 Field techniques: Using transects.

Course link

There are different circumstances in which to use chi-square tests. HL students will use chi-squared values to compare observed values to expected theoretical ratios of genes in 10.1.2 Inheritance.