For a person without a background in stats, it can be difficult to understand the difference between fundamental statistical tests (not to mention when to use them). Here are the differences between the most common tests, how to use null value hypotheses in these tests and the conditions under which you should use each particular test.

## Statistical Tests: The T-Test, the Chi-Square and More

1. Z-Test
2. T-Test
3. Chi-Square Test
4. ANOVA

Before we learn about the tests, let’s dive into some key terms.

## Defining Our Terms

### Null Hypothesis and Hypothesis Testing

Before we venture into the differences between common statistical tests, we need to formulate a clear understanding of a null hypothesis.

The null hypothesis proposes that no significant difference exists between a set of given observations.

In other words:

• Null: Two sample means are equal.
• Alternate: Two sample means are not equal.

To reject a null hypothesis, one needs to calculate test statistics, then compare the result with the critical value. If the test statistic is greater than the critical value, we can reject the null hypothesis.

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### Critical Value

A critical value is a point (or points) on the scale of the test statistic beyond which we reject the null hypothesis. We derive the level of significance (`α`) of the test.

Critical value can tell us the probability of two sample means belonging to the same distribution. The higher the critical value means the lower the probability of two samples belonging to the same distribution.

The general critical value for a two-tailed test is 1.96, which is based on the fact that 95 percent of the area of a normal distribution is within 1.96 standard deviations of the mean.

Critical values can be used to do hypothesis testing in the following ways:

• Calculate test statistic.
• Calculate critical values based on significance level alpha.
• Compare the test statistic with critical values.

If the test statistic is lower than the critical value, accept the null hypothesis; otherwise reject it.

Note: Some statisticians would use p-value instead of critical value for conducting null hypothesis.

### Sample vs. Population

In statistics, population refers to the total set of observations we can make. For example, if we want to calculate the average human height, the population will be the total number of people actually present on Earth.

A sample, on the other hand, is a set of data collected or selected from a predefined procedure. For our example above, a sample is a small group of people selected randomly from different regions of the globe.

To draw inferences from a sample and validate a hypothesis,  the sample must be random.

For instance, if we select people randomly from all regions on Earth, we can assume our  sample mean is close to the population mean, whereas if we make a selection just from the United States, then our average height estimate/sample mean cannot be considered close to the population mean. Instead, it will only represent the data of a particular region (the United States). That means our sample is biased and is not representative of the population.

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### Distribution

Another important statistical concept to understand is distribution. When the population is infinitely large, it’s not feasible to validate any hypothesis by calculating the mean value or test parameters on the entire population. In such cases, we assume a population is some type of a distribution.

While there are many forms of distribution, the most common are binomial, Poisson and discrete.

You must determine the distribution type to calculate the critical value and decide on the best test to validate any hypothesis.

Now that we’re clear on population, sample and distribution, let’s learn about different kinds of tests and the distribution types for which they are used.

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## Statistical Tests

### P-value, Critical Value and Test Statistic

As we know, critical value is the point beyond which we reject the null hypothesis. P-value, on the other hand, is the probability to the right of the respective statistic (z, t or chi). The benefit of using p-value is that it calculates a probability estimate, which means we can test at any desired level of significance by comparing this probability directly with the significance level.

For example, assume the z-value for a particular experiment comes out to be 1.67 which is greater than the critical value at five percent (1.64). Now, to check for a different significance level of one percent, we calculate a new critical value.

However, if we calculate p-value for 1.67 and it comes to be 0.047, we can use this p-value to reject the hypothesis at a five percent significance level since 0.047 < 0.05. However, with a more stringent significance level of one percent, we’ll fail to reject the hypothesis since 0.047 > 0.01. It’s important to note here that there’s no double calculation required.

### Z-Test

In a z-test, we assume the sample is normally distributed. A z-score is calculated with population parameters such as population mean and population standard deviation. We use this test to validate a hypothesis that states the sample belongs to the same population.

• Null: Sample mean is same as the population mean.
• Alternate: Sample mean is not same as the population mean.

The statistic used for this hypothesis testing is called z-statistic, the score for which we calculate as:

`z = (x — μ) / (σ / √n)`, where

`x`=sample mean

`μ`=population mean

`σ / √n`= population standard deviation

If the test statistic is lower than the critical value, accept the hypothesis.

### T-Test

We use a t-test to compare the mean of two given samples. Like a z-test, a t-test also assumes a normal distribution of the sample. When we don’t know the population parameters (mean and standard deviation), we use t-test.

## The Three Versions of a T-Test

1. Independent sample t-test: compares mean for two groups
2. Paired sample t-test: compares means from the same group at different times
3. One sample t-test: tests the mean of a single group against a known mean

The statistic for this hypothesis testing is called t-statistic, the score for which we calculate as:

`t=(x1 — x2) / (σ / √n1 + σ / √n2)`, where

`x1`=mean of sample 1

`x2`=mean of sample 2

`n1`=sample size 1

`n2`=sample size 2

There are multiple variations of the t-test.

Note: This article focuses on normally distributed data. You can use z-tests and t-tests for data which is non-normally distributed as well if the sample size is greater than 20, however there are other preferable methods to use in such a situation.

### Chi-Square Test

We use the chi-square test to compare categorical variables.

## The Two Types of Chi-Square Test

1. Goodness of fit test: determines if a sample matches the population
2. A chi-square fit test for two independent variables: used to compare two variables in a contingency table to check if the data fits

A small chi-square value means that data fits.

A large chi-square value means that data doesn’t fit.

The hypothesis we’re testing is:

• Null: Variable A and Variable B are independent.
• Alternate: Variable A and Variable B are not independent.

The statistic used to measure significance, in this case, is called chi-square statistic. The formula we use to calculate the statistic is:

`Χ2 = Σ [ (Or,c — Er,c)2 / Er,c ]` where

`Or,c`=observed frequency count at level r of Variable A and level c of Variable B

`Er,c`=expected frequency count at level r of Variable A and level c of Variable B

## T-Test vs. Chi-Square

We use a t-test to compare the mean of two given samples but we use the chi-square test to compare categorical variables.

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### ANOVA

We use analysis of variance (ANOVA) to compare three or more samples with a single test.

## The Two Major Types of ANOVA

1. One-way ANOVA: Used to compare the difference between three or more samples/groups of a single independent variable.
2. MANOVA: Allows us to test the effect of one or more independent variables on two or more dependent variables. In addition, MANOVA can also detect the difference in correlation between dependent variables given the groups of independent variables.

The hypothesis we’re testing with ANOVA is:

• Null: All pairs of samples are the same (i.e. all sample means are equal).
• Alternate: At least one pair of samples is significantly different.

The statistics used to measure the significance in this case are F-statistics. We calculate the F-value using the formula:

`F= ((SSE1 — SSE2)/m)/ SSE2/n-k`, where

`SSE`=residual sum of squares

`m`=number of restrictions

`k`=number of independent variables

There are multiple tools available such as SPSS, R packages, Excel etc. to carry out ANOVA on a given sample.

## The Takeaway

If you learn only one thing from this article, let it be this: In all of these tests we’re comparing a statistic with a critical value to accept or reject a hypothesis. However, the statistic and the way to calculate it differ depending on the type of variable, the number of samples you’re analyzing and whether or not we know the population parameters. We can thus choose a suitable statistical test and null hypothesis. This principle is instrumental to understanding these basic statistical concepts.

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