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The Chi-Squared test is used to compare what you have measured (observed) against what may be anticipated (expected).We establish a hypothesis for the feature under investigation and then convert it to a null hypothesis. The null hypothesis states that no relationship between the two population parameters exists. We use it because it helps us see if our hypothesis has validity.…
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The Chi-Squared test is used to compare what you have measured (observed) against what may be anticipated (expected).
We establish a hypothesis for the feature under investigation and then convert it to a null hypothesis. The null hypothesis states that no relationship between the two population parameters exists. We use it because it helps us see if our hypothesis has validity. It is impossible to prove something with absolute certainty. However, we can disprove a null hypothesis, which allows us to accept that our hypothesis is valid, and we use ‘confidence levels’ and ‘critical values’ to do this.
Null hypothesis: There is no significant difference between specified populations, any observed difference being due to sampling or experimental error.
The difference between expected and observed results in experiments can be described in two ways:
When results are significant, this suggests that something is happening that wasn’t accounted for.
The chi-squared test is a statistical test commonly used for biological hypotheses to determine if the results are statistically significant.
We can also define our hypothesis as one-tailed or two-tailed. One-tailed hypotheses are based on uni-directional hypotheses and two-tailed on bi-directional hypotheses.
In terms of our earlier hypotheses, this would be:
Chi-squared tests should only be done using categorical data and if specific criteria are met.
Categorical data: values that can be sorted into groups or categories. It can be further divided into nominal (values you can count but not order, e.g., eye colour) and ordinal (values you can count and order e.g., house numbers).
This is different from the chi-squared contingency test, which tests for the association between two categorical variables.
Additionally, the chi-squared test makes several assumptions:
The formula looks very scary - but don’t panic! We can break it down into steps.
In other words, chi-squared is the sum of the square of the difference between the observed values and expected values , divided by the expected values (E).
To help you understand how we would calculate the chi-squared, we will use flower phenotype as an example.
Table 1. Example of a table to find values for Chi-Squared calculation.
Observed number (O)
Expected number (E)
(total number x ratio/16)
The Chi-Squared test has little meaning on its own – it needs to be compared to ‘critical values’, which are found in tables or on graphs as calculated by statistical experts.
First, you must decide the confidence level you want to use. The most common is generally 95% and/or 99%, meaning for every 100 times you carried out the test, you would get chance results on five occasions or one occasion.
Table 2. Confidence, uncertainty and probability levels.
Probability level (p-value)
We then use the value we have obtained in the Chi-Squared test to see if the data is statistically significant. A distribution table is used for this. The distribution table relates the chi-squared value with probabilities. We also use degrees of freedom to determine the number of comparisons made.
For a chi-squared test, the degrees of freedom equal the number of categories minus one (n-1). You will also need to determine your p-value.
The degrees of freedom used for the Chi-Squared test is always n-1
Here is an example of a standard chi-squared table. You read the table by looking at the row corresponding to the degrees of freedom used in your experiment and the column corresponding to your p-value. You will find your critical value at the intersection of these rows and columns.
Table 3. Standard distribution table.
The probability that the difference between observed and expected is due to chance
Degrees of freedom
If your chi-squared test value is greater than the critical value (the value found from the table), then the deviation between your expected and observed results is statistically significant. If it is not greater than the critical value, the difference is not significant.
Chi-squared tests are used across biology. For instance, they can be very useful for determining whether the results of a genetic cross are significantly different from the theoretical predictions.
Genetic cross: The deliberate breeding of two different individuals that results in offspring that carry part of the genetic material of each parent.
Let’s take, for example, the actual results that Gregor Mendel obtained during his pea experiments on the inheritance of seed type. Mendel performed experiments on pea plants to determine patterns of inheritance for some of the plants’ observable traits. For more information on his experiments, check out our articles on Inheritance!
A single gene determines seed type with a dominant allele that produces smooth seeds and a recessive allele that produces wrinkled seeds. Mendel’s experiment resulted in 5474 smooth and 1850 wrinkled seeds. Allowing for some statistical error, how can we tell if this result fits our expected ratio?
Statistical error: The difference between a measured value and the actual value of the collected data. If the error value is more significant, the data will be considered less reliable.
When following these steps, it is useful to summarise your calculations in a table like so:
Table 4. Another example of how to obtain the values for the equation.
Total = 7324
When you square values, any negatives cancel out.
Suppose the observed values are significantly different to expected values. If the result is smaller than or equal to the stated p-value, there are some things we may want to consider. As discussed in the article on sex-linkage, autosomal linkage, and epistasis, there are several reasons why we might observe patterns of inheritance that do not fit Mendelian ratios.
Autosomal linkage: Autosomal linkage occurs if two or more genes are located on the same autosome (non-sex chromosome). The two genes are less likely to be separated during crossing over, resulting in the alleles of the linked genes being inherited together.
The chi-squared (χ2) tests the null hypothesis that there is no statistically significant difference between the observed and expected results of an experiment.
It can be performed on large sample sizes (>20), using raw counts of categorical data.
Chi-squared is the sum of the square of the difference between the observed and expected values, divided by the expected values.
A chi-squared distribution table is used to determine the correct critical value for the given degrees of freedom and p-value.
When chi-squared is higher than the critical value, the difference between the expected and observed results is significant.
Degrees of freedom are calculated by subtracting one from the number of categories.
A chi-squared test is used to see if the difference between the observed and expected results of an experiment is statistically significant. Chi-squared tests are often used on biological data.
The degrees of freedom are equivalent to the number of categories (n) minus one. df = n-1
The chi-squared test is calculated with the equation
To identify if results are statistically significant when comparing observed and expected results.
When we are comparing observed and expected results.
We use the value we have obtained in the Chi-Squared test to see if the data is statistically significant. A distribution table is used for this. The distribution table relates the chi-squared value with probabilities. We also use degrees of freedom to determine the number of comparisons made.
What is the null hypothesis of the chi-squared test?
There is no statistically significant difference between the observed and expected results.
What does it mean when the result of the chi-squared test is statistically significant?
The difference between the observed and expected results did not occur due to chance.
Which of the following is not a criterion for performing the chi-squared test?
- The sample size has to be large (>20)
- The data must be theoretical
- Only raw counts can be used – not ratios, rates, fractions or percentages
- The comparison is being made between theoretical (expected) and experimental (observed) results
The data must be theoretical
Which is not an assumption of the chi-squared test?
- The comparisons are made on random samples
- The expected count of each cell is greater than 5 (>5)
- No more than 20% of the cells have expected counts less than 5 (<5)
The expected count of each cell is greater than 5 (>5)
What is the formula for the chi-squared test?
chi-squared X2is the sum of the square of the difference between the observed values and expected values (O-E)2, divided by the expected values (E).
In genetic crosses, how do you calculate the expected values for each phenotype?
Find the total number of offspring and divide it according to the phenotypic ratio.
How do you calculate the degrees of freedom?
What is a null hypothesis?
The hypothesis that there is no significant difference between specified populations, any observed difference being due to sampling or experimental error.
What is the critical value?
The value that the test statistic must exceed in order to reject the null hypothesis.
What is the p-value?
The probability level which forms basis for deciding if results are statistically significant (not due to chance).
What are the degrees of freedom?
What is a one-tailed hypothesis?
It is based on uni-directional hypotheses.
What is a two-tailed hypothesis?
It is based on bi-directional hypotheses.
If the confidence level is 95%, what is the p-value?
What happens when you square negative values?
The negatives cancel out.
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