# Chi squared test

The chi-square independence test is a procedure for testing if two categorical variables are related in some population. Example: a scientist wants to know if education level and marital status are related for all people in some country. A good first step for these data is inspecting the contingency table of marital status by education. Such a table -shown below- displays the frequency distribution of marital status for each education category separately.

So let's take a look at it. The numbers in this table are known as the observed frequencies. They tell us an awful lot about our data. For instance. Although our contingency table is a great starting point, it doesn't really show us if education level and marital status are related. This question is answered more easily from a slightly different table as shown below.

This table shows -for each education level separately- the percentages of respondents that fall into each marital status category. Before reading on, take a careful look at this table and tell me is marital status related to education level and -if so- how? In short, more highly educated respondents marry more often than less educated respondents. Our last table shows a relation between marital status and education. This becomes much clearer by visualizing this table as a stacked bar chartshown below.

If we move from top to bottom highest to lowest education in this chart, we see the dark blue bar never married increase. Marital status is clearly associated with education level. So what about the population?

The null hypothesis for a chi-square independence test is that two categorical variables are independent in some population. Now, marital status and education are related -thus not independent- in our sample. However, we can't conclude that this holds for our entire population.

The basic problem is that samples usually differ from populations. If marital status and education are perfectly independent in our population, we may still see some relation in our sample by mere chance.

However, a strong relation in a large sample is extremely unlikely and hence refutes our null hypothesis.The Chi-Square test is a statistical procedure used by researchers to examine the differences between categorical variables in the same population. For example, imagine that a research group is interested in whether or not education level and marital status are related for all people in the U. After collecting a simple random sample of U. The researchers could then perform a Chi-Square test to validate or provide additional context for these observed frequencies.

Market researchers use the Chi-Square test when they find themselves in one of the following situations:. The Chi-Square test is most useful when analyzing cross tabulations of survey response data. Because cross tabulations reveal the frequency and percentage of responses to questions by various segments or categories of respondents gender, profession, education level, etc.

## Chi-Square Test Calculator

Chi-Square testing does not provide any insight into the degree of difference between the respondent categories, meaning that researchers are not able to tell which statistic result of the Chi-Square test is greater or less than the other.

Second, Chi-Square requires researchers to use numerical values, also known as frequency counts, instead of using percentages or ratios.

This can limit the flexibility that researchers have in terms of the processes that they use. Chi-Square tests can be run in either Microsoft Excel or Google Sheets, however, there are more intuitive statistical software packages available to researchers.

Start a free 7-day trial or speak with a member of our sales team. Blog Subscribe to our blog. Subscribe to our Blog Thank you for subscribing to our Blog. Ben Foley. What is the Chi-Square Test? Market researchers use the Chi-Square test when they find themselves in one of the following situations: They need to estimate how closely an observed distribution matches an expected distribution. They need to estimate whether two random variables are independent.

Start making smarter decisions Start a free 7-day trial or speak with a member of our sales team. Request a Demo. By accessing and using this page, you agree to the Terms of Use.

Your information will never be shared.Experiments test predictions. These predictions are often numerical, meaning that, as scientists gather data, they expect the numbers to break down in a certain way. Real-world data rarely match exactly the predictions scientists make, so scientists need a test to tell them whether the difference between observed and expected numbers is because of random chance, or because of some unforeseen factor that will force the scientist to adjust the underlying theory.

A chi-square test is a statistical tool that scientists use for this purpose.

Chi Squared Test

You need categorical data to use a chi-square test. An example of categorical data is the number of people who answered a question "yes" versus the number of people who answered the question "no" two categoriesor the numbers of frogs in a population that are green, yellow or gray three categories.

You cannot use a chi-square test on continuous data, such as might be collected from a survey asking people how tall they are. From such a survey, you would get a broad range of heights.

However, if you divided the heights into categories such as "under 6 feet tall" and "6 feet tall and over," you could then use a chi-square test on the data. A goodness-of-fit test is a common, and perhaps the simplest, test performed using the chi-square statistic.

In a goodness-of-fit test, the scientist makes a specific prediction about the numbers she expects to see in each category of her data. She then collects real-world data -- called observed data -- and uses the chi-square test to see whether the observed data match her expectations.

For example, imagine a biologist is studying the inheritance patterns in a species of frog. Among offspring of a set of frog parents, the biologist's genetic model leads her to expect 25 yellow offspring, 50 green offspring and 25 gray offspring. What she actually observes is 20 yellow offspring, 52 green offspring and 28 gray offspring. Is her prediction supported or is her genetic model incorrect?

She can use a chi-square test to find out. Begin calculating the chi-square statistic by subtracting each expected value from its corresponding observed value and squaring each result. The calculation for the example of the frog offspring would look like this:.

The chi-square statistic tells you how different your observed values were from your predicted values. The higher the number, the greater the difference. You can determine whether your chi-square value is too high or low enough to support your prediction by seeing whether it is below a certain critical value on a chi-square distribution table. This table matches chi-square values with probabilities, called p-values. Specifically, the table tells you the probability that the differences between your observed and expected values are simply due to random chance or whether some other factor is present.It is the most widely used of many chi-squared tests e.

Its properties were first investigated by Karl Pearson in It tests a null hypothesis stating that the frequency distribution of certain events observed in a sample is consistent with a particular theoretical distribution.

The events considered must be mutually exclusive and have total probability 1. A common case for this is where the events each cover an outcome of a categorical variable.

A simple example is the hypothesis that an ordinary six-sided die is "fair" i. Pearson's chi-squared test is used to assess three types of comparison: goodness of fithomogeneityand independence.

### Pearson's chi-squared test

A simple application is to test the hypothesis that, in the general population, values would occur in each cell with equal frequency. The "theoretical frequency" for any cell under the null hypothesis of a discrete uniform distribution is thus calculated as.

When testing whether observations are random variables whose distribution belongs to a given family of distributions, the "theoretical frequencies" are calculated using a distribution from that family fitted in some standard way.

The degrees of freedom are not based on the number of observations as with a Student's t or F-distribution. The number of times the dice is rolled does not influence the number of degrees of freedom. The chi-squared statistic can then be used to calculate a p-value by comparing the value of the statistic to a chi-squared distribution.

The result about the numbers of degrees of freedom is valid when the original data are multinomial and hence the estimated parameters are efficient for minimizing the chi-squared statistic.

In Bayesian statisticsone would instead use a Dirichlet distribution as conjugate prior. If one took a uniform prior, then the maximum likelihood estimate for the population probability is the observed probability, and one may compute a credible region around this or another estimate.

In this case, an "observation" consists of the values of two outcomes and the null hypothesis is that the occurrence of these outcomes is statistically independent. Each observation is allocated to one cell of a two-dimensional array of cells called a contingency table according to the values of the two outcomes. If there are r rows and c columns in the table, the "theoretical frequency" for a cell, given the hypothesis of independence, is. The term " frequencies " refers to absolute numbers rather than already normalised values. For the test of independence, also known as the test of homogeneity, a chi-squared probability of less than or equal to 0. The chi-squared test, when used with the standard approximation that a chi-squared distribution is applicable, has the following assumptions: [ citation needed ]. A test that relies on different assumptions is Fisher's exact test ; if its assumption of fixed marginal distributions is met it is substantially more accurate in obtaining a significance level, especially with few observations.

In the vast majority of applications this assumption will not be met, and Fisher's exact test will be over conservative and not have correct coverage. This approximation arises as the true distribution, under the null hypothesis, if the expected value is given by a multinomial distribution. For large sample sizes, the central limit theorem says this distribution tends toward a certain multivariate normal distribution. In the special case where there are only two cells in the table, the expected values follow a binomial distribution.

In the above example the hypothesised probability of a male observation is 0. Thus we expect to observe 50 males. If n is sufficiently large, the above binomial distribution may be approximated by a Gaussian normal distribution and thus the Pearson test statistic approximates a chi-squared distribution. Let O 1 be the number of observations from the sample that are in the first cell. The Pearson test statistic can be expressed as.

By the normal approximation to a binomial this is the squared of one standard normal variate, and hence is distributed as chi-squared with 1 degree of freedom. Note that the denominator is one standard deviation of the Gaussian approximation, so can be written. So as consistent with the meaning of the chi-squared distribution, we are measuring how probable the observed number of standard deviations away from the mean is under the Gaussian approximation which is a good approximation for large n.

The chi-squared distribution is then integrated on the right of the statistic value to obtain the P-valuewhich is equal to the probability of getting a statistic equal or bigger than the observed one, assuming the null hypothesis.This tutorial provides a simple explanation of the difference between the two tests, along with when to use each one. There are actually a few different versions of the chi-square test, but the most common one is the Chi-Square test for independence.

Null hypothesis H 0 : There is no significant association between the two variables. Alternative hypothesis: Ha : There is a significant association between the two variables. Here are some examples of when we might use a chi-square test for independence:. To test this, we might survey random people and record their gender and political party preference. Then, we can conduct a chi-square test for independence to determine if there is a statistically significant association between gender and political party preference.

## Chi-squared test

To test this, we might survey random students from each grade level at a certain school and record their favorite movie genre.

Then, we can conduct a chi-square test for independence to determine if there is a statistically significant association between class level and favorite movie genre. To test this, we might survey random people and ask them what type of place they grew up in and what their favorite sport is. Before we can conduct a chi-square test for independence, we first need to make sure the following assumptions are met to ensure that our test will be valid:.

If these assumptions are met, then we can then conduct the test. There are also a few different versions of the t-test, but the most common one is the t-test for a difference in means. Null hypothesis H 0 : The two population means are equal.

Alternative hypothesis: Ha : The two population means are not equal. Here are some examples of when we might use a t-test for a difference in means:.

We can conduct a t-test for a difference in means to determine if there is a statistically significant difference in average weight loss between the two groups. We randomly assign 50 students to use one study plan and 50 students to use another study plan for one month leading up to an exam. We can conduct a t-test for a difference in means to determine if there is a statistically significant difference in average exam scores between the two study plans.

We measure the height of random students from one school and random students from another school. We can conduct a t-test for a difference in means to determine if there is a statistically significant difference in average height of students between the two schools. Before we can conduct a hypothesis test for a difference between two population means, we first need to make sure the following conditions are met to ensure that our hypothesis test will be valid:.

If these assumptions are met, then we can then conduct the hypothesis test. When you reject the null hypothesis of a t-test for a difference in means, it means the two population means are not equal. The easiest way to know whether or not to use a chi-square test vs. If you have two variables that are both categorical, i. But if one variable is categorical e. Your email address will not be published. Skip to content Menu. Posted on June 23, by admin. Chi-Square Test There are actually a few different versions of the chi-square test, but the most common one is the Chi-Square test for independence.

The hypotheses of the test are as follows: Null hypothesis H 0 : There is no significant association between the two variables. The hypotheses of the test are as follows: Null hypothesis H 0 : The two population means are equal. Published by admin. View all posts by admin.Chi Square Test in Excel is one such statistical function which is used to calculate the expected value from a dataset which has observed values. Excel is a versatile tool to analyze data visually as well as statistically.

It is one of the few spreadsheet tools around which supports advanced statistical functions. Using these functions, we can gain insights from a dataset which may not be possible by just visually analyzing them.

In this article, we will learn how to calculate the Chi Square from a database using excel. Before going into detail with the Chi Square Test, let us go through a few examples. Start Your Free Excel Course. Chi Square Test is a test of the validity of a hypothesis. The Chi Square P Value tells us if our observed results are statistically significant or not. A statistically significant result means that we reject the null hypothesis null hypothesis in statistics is a statement or hypothesis which is likely to be incorrect.

A Chi Square P Value is a number between 0 and 1. A Chi Square P Value less than 0. Chi Square test can tell us whether the proportion of a given number of items is in one place based on a random sample are statistically independent of each other or not. Suppose your company has pieces of furniture. About one by tenth of them are distributed over four halls. We can find out what proportion of the total furniture is in one hall as shown below:. Observe that we have about pieces of furniture in each hall.

If we want to get the expected number of furniture by type, we will calculate it as follows:. Which would give us the value 0. Similarly, we will find the values for each quantity and the sum of these values is the test statistic.

This statistic has an approximate Chi Squared distribution if each quantity is independent of the other. The degree of freedom for each quantity would be determined by the following formula:. We find the Chi Square P value for the first value that is the number of chairs. The null hypothesis is that the location of the furniture is independent of the type of furniture. As is clear from the above example, calculating Chi Square and testing for significance of hypothesized data in statistics is a painstaking process and demands high accuracy.

TEST Function to get the Chi Square value directly and check if our assumption that the location of the furniture is independent of the type of furniture is correct r not. In this case:. The Chi Square value is approximately 0. From our earlier discussion, we now know that this rejects the null hypothesis.

Here we discuss How to do Chi Square Test in excel along with practical examples and downloadable excel template.After reading this article you will learn about:- 1. Meaning of Chi-Square Test 2. Levels of Significance of Chi-Square Test 3.

Chi-Square Test under Null Hypothesis 4. Conditions for the Validity 5. Additive Property 6. Applications 7. Thus Chi-square is a measure of actual divergence of the observed and expected frequencies. It is very obvious that the importance of such a measure would be very great in sampling studies where we have invariably to study the divergence between theory and fact.

### Chi-Square Test: Meaning, Applications and Uses | Statistics

Chi-square as we have seen is a measure of divergence between the expected and observed frequencies and as such if there is no difference between expected and observed frequencies the value of Chi-square is 0. If there is a difference between the observed and the expected frequencies then the value of Chi-square would be more than 0. That is, the larger the Chi-square the greater the probability of a real divergence of experimentally observed from expected results.

If the calculated value of chi-square is very small as compared to its table value it indicates that the divergence between actual and expected frequencies is very little and consequently the fit is good. If, on the other hand, the calculated value of chi-square is very big as compared to its table value it indicates that the divergence between expected and observed frequencies is very great and consequently the fit is poor.

To evaluate Chi-square, we enter Table E with the computed value of chi- square and the appropriate number of degrees of freedom. Thus in 2 x 2 table degrees of freedom are 2 — 1 2 — 1 or 1. Similarly in 3 x 3 table, degrees of freedom are 3 — 1 3 — 1 or 4 and in 3 x 4 table the degrees of freedom are 3 — 1 4 — 1 or 6. The divergence of theory and fact is always tested in terms of certain probabilities.

The probabilities indicate the extent of reliance that we can place on the conclusion drawn. These levels are called levels of significance. In other words, the discrepancy between the observed and expected frequencies cannot be attributed to chance and we reject the null hypothesis.

Thus we conclude that the experiment does not support the theory. This implies that the discrepancy between observed values experiment and the expected values theory may be attributed to chance, i.

Suppose we are given a set of observed frequencies obtained under some experiment and we want to test if the experimental results support a particular hypothesis or theory. Karl Pearson indeveloped a test for testing the significance of the discrepancy between experimental values and the theoretical values obtained under some theory or hypothesis.

Under the Null Hypothesis we state that there is no significant difference between the observed experimental and the theoretical or hypothetical values, i. Thus chi-square is the sum of the values obtained by dividing the square of the difference between observed and expected frequencies by the expected frequencies in each case.

Several illustrations of the chi-square test will clarify the discussion given above. Testing the divergence of observed results from those expected on the hypothesis of equal probability null hypothesis :. Following steps may be followed for the computation of x 2 and drawing the conclusions:. Compute the expected frequencies f e corresponding to the observed frequencies in each case under some theory or hypothesis. In our example the theory is of equal probability null hypothesis.

In the second row the distribution of answers to be expected on the null hypothesis is selected equally. Compute the deviations f o — f e for each frequency.

Are these frequencies in agreement with the belief that accident conditions were the same during this week period? Null Hypothesis—Set up the null hypothesis that the given frequencies of number of accidents per week in a certain community are consistent with the belief that the accident conditions were same during the week period.

It is significant and the null hypothesis rejected at. Hence we conclude that the accident conditions are certainly not uniform same over the week period. Testing the divergence of observed results from those expected on the hypothesis of a normal distribution:.