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:
 Statistically significant
 Statistically insignificant (happened by chance)
When results are significant, this suggests that something is happening that wasn’t accounted for.
The chisquared 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 onetailed or twotailed. Onetailed hypotheses are based on unidirectional hypotheses and twotailed on bidirectional hypotheses.
In terms of our earlier hypotheses, this would be:
 Onetailed: The density of trees per 5m^2 increases towards the forest’s centre.
 Twotailed: The density of trees per 5m^2 changes towards the forest’s centre.
Chisquared 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 chisquared contingency test, which tests for the association between two categorical variables.
What are the criteria for performing a chisquared test?
 The sample size has to be large (>20).
 The data must be categorical.
 Only raw counts can be used – not ratios, rates, fractions or percentages.
 The comparison between theoretical (expected) and experimental (observed) results is being made.
What are the assumptions of the chisquared test?
Additionally, the chisquared test makes several assumptions:
 The comparisons are made on random samples.
 The expected count of each cell is greater than one (>1).
 No more than 20% of the cells have expected counts less than 5 (<5).
How do you calculate chisquared?
The formula looks very scary  but don’t panic! We can break it down into steps.
What is the formula for chisquared?
${X}^{2}=\sum _{}\frac{{(OE)}^{2}}{E}\phantom{\rule{0ex}{0ex}}{X}^{2}=Theteststatistic\phantom{\rule{0ex}{0ex}}\sum _{}=Thesumof\phantom{\rule{0ex}{0ex}}O=Observedfrequency\phantom{\rule{0ex}{0ex}}E=Expectedfrequency$
In other words, chisquared ${X}^{2}$ is the sum of the square of the difference between the observed values and expected values ${(OE)}^{2}$, divided by the expected values (E).
To help you understand how we would calculate the chisquared, we will use flower phenotype as an example.
To calculate:
 Obtain the expected and observed results for the experiment (as shown in the table below)
 Calculate the difference between each set of results
 Square each difference
 Divide each squared difference by the expected value
 Use the sum of these answers to obtain the chisquared value
Table 1. Example of a table to find values for ChiSquared calculation.
Flower Phenotype  Observed number (O)  Expected Ratio  Expected number (E)(total number x ratio/16)  OE  (OE)^2/E 
Pink/Round  296  9  240  56  13.067 
Pink/Long  19  3  80  61  46.513 
Purple/Round  27  3  80  53  35.113 
Purple/long  85  1  27  58  124.593 
Total  427  X^2  219.29 
How do you calculate degrees of freedom and use a chisquared distribution table?
The ChiSquared 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.
Highly confident  Very confident  Extremely confident  
Confidence level  95%  99%  99.9% 
Uncertainty level  5%  1%  0.1% 
Probability level (pvalue)  0.05  0.01  0.001 
We then use the value we have obtained in the ChiSquared test to see if the data is statistically significant. A distribution table is used for this. The distribution table relates the chisquared value with probabilities. We also use degrees of freedom to determine the number of comparisons made.
For a chisquared test, the degrees of freedom equal the number of categories minus one (n1). You will also need to determine your pvalue.
The degrees of freedom used for the ChiSquared test is always n1
Here is an example of a standard chisquared 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 pvalue. 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  0.1  0.05  0.01  0.001 
1  2.27  3.84  6.64  10.83 
2  4.60  5.99  9.21  13.82 
3  6.25  7.82  11.34  16.27 
4  7.78  9.49  13.28  18.46 
If your chisquared 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.
How is the chisquared test used in genetics?
Chisquared 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.
Category  Observed  Expected  OE  (OE)2  (OE)2/E 
Smooth  5474  5493  19  361  0.0657 
Wrinkled  1850  1831  19  361  0.1972 
 Total = 7324 


 0.2629 
 First, let’s calculate the expected values. In this case, we would find the total number of offspring (5474+1850 = 7324) and divide it according to the 3:1 ratio. This gives us expected values of (7324 x ¾ =) 5493 smooth seeds and (7324 x ¼ =)1831 wrinkled seeds.
 We now need to know the difference between the observed and expected values. For the smooth category, the difference is (54745493 =) 19, while for the wrinkled category, the difference is (18501831 =) 19.
 We get 361 for smooth seeds and 361 for wrinkled seeds when we square these differences.
When you square values, any negatives cancel out.
What if the observed values are significantly different from the expected values?
Suppose the observed values are significantly different to expected values. If the result is smaller than or equal to the stated pvalue, there are some things we may want to consider. As discussed in the article on sexlinkage, autosomal linkage, and epistasis, there are several reasons why we might observe patterns of inheritance that do not fit Mendelian ratios.
 The trait might be sexlinked, meaning the sex of the individual affects whether they can inherit the trait.
 Two genes might be found on the same chromosome and thus exhibit linkage.
 Epistasis might also affect the phenotypes expressed by the individual.
Sex linkage: Sex linkage is the phenotypic expression of an allele that is dependent on the gender of the individual and is directly tied to the sex chromosomes.
Autosomal linkage: Autosomal linkage occurs if two or more genes are located on the same autosome (nonsex chromosome). The two genes are less likely to be separated during crossing over, resulting in the alleles of the linked genes being inherited together.
Epistasis: Epistasis is a circumstance where the expression of one gene is affected by the expression of one or more independently inherited genes.
ChiSquare Test  Key takeaways
The chisquared (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.
Chisquared is the sum of the square of the difference between the observed and expected values, divided by the expected values.
A chisquared distribution table is used to determine the correct critical value for the given degrees of freedom and pvalue.
When chisquared 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.
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Frequently Asked Questions about ChiSquare Test
What is chisquare in genetics?
A chisquared test is used to see if the difference between the observed and expected results of an experiment is statistically significant. Chisquared tests are often used on biological data.
How do you determine the degrees of freedom?
The degrees of freedom are equivalent to the number of categories (n) minus one. df = n1
How do you calculate the chi square test?
The chisquared test is calculated with the equation
What is a chisquare test used for in biology?
To identify if results are statistically significant when comparing observed and expected results.
When would you do a chisquare test?
When we are comparing observed and expected results.
How do you interpret a chisquare test?
We use the value we have obtained in the ChiSquared test to see if the data is statistically significant. A distribution table is used for this. The distribution table relates the chisquared value with probabilities. We also use degrees of freedom to determine the number of comparisons made.
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