If you are from a non-statistical background it is essentially the most complicated aspect of statistics, are always the basic statistical tests, and how to choose statistical test?

In this article I have tried to mark out the distinction between the common statistical tests, the use of null value hypothesis in these tests by outlining the circumstances below which a particular test needs to be used.

## Null Hypothesis and Testing

Before going to the difference between different test and choose a statistical test you must understand what a null hypothesis is.

A null hypothesis tells that no significant difference exists in a set of given observations.

**Null:** Given two sample means are equal

**Alternate:** Given two sample means usually are not equal

For rejecting a null hypothesis, a test statistic is calculated. This test-statistic is then compared with a critical value and if it is greater than the critical value then the hypothesis is rejected.

“In the theoretical aspect, hypothesis tests are based on the notion of critical regions: the null hypothesis is rejected if the test statistic falls in the critical region. The critical values are the boundaries of the critical region. If the test is one-sided then there will be just one critical value, but in other cases (like a two-sided t-test) there will be two”

## Critical Value

A critical value is a point on the scale of the test statistic past which we reject the null hypothesis, and, is derived from the level of significance \alpha of the test.

Critical value tells us, what’s the likelihood of two sample means belonging to the same distribution.

Higher, the critical value means lower the likelihood of two samples belonging to the same distribution.

The common critical value for a two-tailed test is 1.96, which relies on the fact that 95% 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 mostly on significance level alpha
- Compare test statistic with critical values.

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

Before I move ahead with different statistical tests it’s crucial to understand the difference between a sample and a population.

In statistics “population” refers to the overall set of observations that may be made. For eg, if I would like to calculate the average height of people currently on the earth, “population” would be the “total number of people actually present on the earth”.

A sample, alternatively, is a set of data collected from a pre-defined process. As in the example above, it will likely be a small group of individuals chosen randomly from some parts of the earth.

To draw inferences from a sample by validating a hypothesis it’s crucial that the sample is random.

For instance, if we choose individuals randomly from all areas(Asia, America, Europe, Africa and so forth.)on earth, our estimate can be close to the actual estimate and will be assumed as a sample mean, whereas if we make selection let’s say solely from the United States, then our average height estimate is not going to be correct however would solely represent the data of a selected area (United States).

Such a sample is then known as a **biased sample** and isn’t a representative of “population”.

Another important aspect to understand in statistics is “distribution”. When “population” is infinitely massive it’s inconceivable to validate any hypothesis by calculating the mean value or test parameters on your entire population.

In such instances, a population is assumed to be of some kind of distribution.

The commonest types of distributions are Binomial, Poisson and Discrete. However, there are numerous different types which are given in detail here.

The determination of distribution type is important to decide the critical value and test to be chosen to validate any hypothesis.

Now, once you are clear on the concept of population, sample, and distribution you will be able to understand different kinds of test and the distribution types for which they’re used.

## Relationship between p-value, critical value and test statistic

The critical value is some extent beyond which we reject the null hypothesis. Alternatively, P-value is defined as the probability to the right of respective statistics (Z, T or chi).

The advantage of using p-value is that it calculates a probability estimate, that you can test at any desired level of significance by comparing this probability instantly with the significance level.

For e.g., assume Z-value for a selected experiment comes out to be 1.67 which is greater than the critical value at 5% which is 1.64. Now to check for a different significance level of 1% a new critical value is to be calculated.

However, if you calculate the p-value for 1.67 it comes to be 0.047. You can use this p-value to reject the hypothesis at 5% significance level since 0.047 < 0.05. But with an extra stringent significance level of 1%, the hypothesis can be accepted since 0.047 > 0.01.

The important point to observe here is that there isn’t any double calculation required.

## Z-test

In a z-test, the sample is assumed to be normally distributed. A z-score is calculated with population parameters resembling “population mean” and “population standard deviation” and it is used to validate the hypothesis that the sample drawn belongs to the same population.

**Null:** Sample mean is identical as the population mean

**Alternate:** Sample mean isn’t identical as the population mean

The statistics used for this hypothesis testing known as z-statistic, the score for which is calculated as:

z=\frac{(x-\mu)}{\sigma/\sqrt{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 or else reject the hypothesis.

## T-test

A t-test is used to compare the mean of two given samples. Like a z-test, a t-test also assumes the sample is normally distributed. A t-test is used when the population parameters (mean and standard deviation) are usually not known.

There are three variations of t-test:

- Independent samples t-test which compares mean for two groups
- Paired sample t-test which compares means from the identical group at different times
- One sample t-test which tests the mean of a single group against a known mean.

The statistic for this hypothesis testing is known as t-statistic, the score for which is calculated as:

t=\frac{x1-x2}{\sigma/\sqrt{n1}+ \sigma/\sqrt{n2}}where

x1 = mean of sample 1

x2 = mean of sample 2

n1 = size of sample 1

n2 = size of sample 2

There are a number of variations of t-test which you can find here.

## ANOVA

ANOVA also referred to as analysis of variance, is used to compare multiple (three or more) samples with a single test.

There are 2 main flavours of ANOVA

**One-way ANOVA**: It is used to compare the difference between the three or more samples/groups of a single independent variable.**MANOVA**: MANOVA allows us to test the effect of one or more independent variable on two or more dependent variables. In addition, MANOVA also can detect the difference in co-relation between dependent variables given the groups of independent variables.

The hypothesis being tested in ANOVA is

**Null:** All pairs of samples are identical i.e. all sample means are equal

**Alternate:** At least one pair of samples is considerably different

The statistics used to measure the significance, in this case, is known as F-statistics. The F value is calculated using the below formula:

where,

SSE = residual sum of squares

m = number of restrictions

k = number of independent variables

There are a number of tools out there such as SPSS, R packages, Excel and so forth. to perform ANOVA on a given sample.

## Chi-Square Test

Chi-square test is used to compare categorical variables. There are two kinds of chi-square test.

- The goodness of fit test, which determines if a sample matches the population.
- A chi-square fit test for 2 independent variables and is used to compare two variables in a contingency table to test if the data fits.

- Small chi-square value implies that data fits
- A high chi-square value implies that data doesn’t match.

The null and alternate hypothesis for chi-square is:

**Null:** Variable A and Variable B are independent

**Alternate:** Variable A and Variable B aren’t independent.

The statistic used to measure significance, in this case, is known as the chi-square statistic. The formula used for calculating the statistic is:

x2 = \sum_{[Or,c - Er,c]}2 / Er,cOr,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

**Note:** As one can see from the above examples, in all of the tests a statistic is being compared with an important value to accept or reject a hypothesis.

However, the statistic and approach to calculating it differ depending on the type of variable, the number of samples being analyzed and if the population parameters are known.

Thus relying upon such factors and appropriate test and the null hypothesis is chosen.