Population, sample and hypothesis testing

What is a hypothesis?

A hypothesis is an assumption that is neither proven nor disproven. In the research process, a hypothesis is made at the very beginning and the goal is to either reject or not reject the hypothesis. In order to reject or or not reject a hypothesis, data, e.g. from an experiment or a survey, are needed, which are then evaluated using a hypothesis test.

Usually, hypotheses are formulated starting from a literature review. Based on the literature review, you can then justify why you formulated the hypothesis in this way.

An example of a hypothesis could be: "Men earn more than women in the same job in Austira."


To test this hypothesis, you need data, e.g. from a survey, and a suitable hypothesis test such as the t-test or correlation analysis. Don't worry, DATAtab will help you choose the right hypothesis test.

How do I formulate a hypothesis?

In order to formulate a hypothesis, a research question must first be defined. A precisely formulated hypothesis about the population can then be derived from the research question, e.g. men earn more than women in the same job in Austria.

Formulate hypothesis

Hypotheses are not simple statements; they are formulated in such a way that they can be tested with collected data in the course of the research process.

To test a hypothesis, it is necessary to define exactly which variables are involved and how the variables are related. Hypotheses, then, are assumptions about the cause-and-effect relationships or the associations between variables.

What is a variable?

A variable is a property of an object or event that can take on different values. For example, the eye color is a variable, it is the property of the object eye and can take different values (blue, brown,...).

If you are researching in the social sciences, your variables may be:

  • Gender
  • Income
  • Attitude towards environmental protection

If you are researching in the medical field, your variables may be:

  • Body weight
  • Smoking status
  • Heart rate

What is the null and alternative hypothesis?

There are always two hypotheses that are exactly opposite to each other, or that claim the opposite. These opposite hypotheses are called null and alternative hypothesis and are abbreviated with H0 and H1.

Null hypothesis H0:

The null hypothesis assumes that there is no difference between two or more groups with respect to a characteristic.


The salary of men and women does not differ in Austria.

Alternative hypothesis H1:

Alternative hypotheses, on the other hand, assume that there is a difference between two or more groups.


The salary of men and women differs in Austria.

The hypothesis that you want to test or that you have derived from the theory usually states that there is an effect e.g. gender has an effect on salary. This hypothesis is called an alternative hypothesis.

The null hypothesis usually states that there is no effect e.g. gender has no effect on salary. In a hypothesis test, only the null hypothesis can be tested; the goal is to find out whether the null hypothesis is rejected or not.

Types of hypotheses

What types of hypotheses are available? The most common distinction is between difference and correlation hypotheses, as well as directional and non-directional hypotheses.

Differential and correlation hypotheses

Difference hypotheses are used when different groups are to be distinguished, e.g., the group of men and the group of women. Correlation hypotheses are used when the relationship or correlation between variables is to be tested, e.g., the relationship between age and height.

Difference hypotheses

Difference hypotheses test whether there is a difference between two or more groups.

Difference hypotheses

Examples of difference hypotheses are:

  • The "group" of men earn more than the "group" of women.
  • Smokers have a higher risk of heart attack than non-smokers
  • There is a difference between Germany, Austria and France in terms of hours worked per week.

Thus, one variable is always a categorical variable, e.g., gender (male, female), smoking status (smoker, nonsmoker), or country (Germany, Austria, and France); the other variable is at least ordinally scaled, e.g., salary, percent risk of heart attack, or hours worked per week.

Correlation hypotheses

Correlation hypotheses test correlations between two variables, for example height and body weight

Correlation hypotheses

Correlation hypotheses are, for example:

  • The taller a person is, the heavier he is.
  • The more horsepower a car has, the higher its fuel consumption.
  • The better the math grade, the higher the future salary.

As can be seen from the examples, correlation hypotheses often take the form "The more..., the higher/lower...". Thus, at least two ordinally scaled variables are being examined.

Directional and non-directional hypotheses

Hypotheses are divided into directional and non-directional or one-sided and two-sided hypotheses. If the hypothesis contains words like "better than" or "worse than", the hypothesis is usually directional.

directional hypotheses

In the case of a non-directional hypothesis, one often finds building blocks such as "there is a difference between" in the formulation, but it is not stated in which direction the difference lies.

  • With a non-directional hypothesis, the only thing of interest is whether there is a difference in a value between the groups under consideration.
  • In a directional hypothesis, what is of interest is whether one group has a higher or lower value than the other.
Directional and non-directional hypothesis test

Non-directional hypotheses

Non-directional hypotheses test whether there is a relationship or a difference, and it does not matter in which direction the relationship or difference goes. In the case of a difference hypothesis, this means there is a difference between two groups, but it does not say whether one of the groups has a higher value.

  • There is a difference between the salary of men and women (but it is not said who earns more!).
  • There is a difference in the risk of heart attack between smokers and non-smokers (but it is not said who has the higher risk!).

In regard to a correlation hypothesis, this means there is a relationship or correlation between two variables, but it is not said whether this relationship is positive or negative.

  • There is a correlation, between height and weight.
  • There is a correlation between horsepower and fuel consumption in cars.

In both cases it is not said whether this correlation is positive or negative!

Directional hypotheses

Directional hypotheses additionally indicate the direction of the relationship or the difference. In the case of the difference hypothesis a statement is made which group has a higher or lower value.

  • Men earn more than women
  • Smokers have a higher risk of heart attack than non-smokers

In the case of a correlation hypothesis, a statement is made as to whether the correlation is positive or negative.

  • The taller a person is the heavier he is
  • The more horsepower a car has, the higher its fuel economy
A one-sided or directional alternative hypothesis includes only values that differ in one direction from the value of the null hypothesis.

The p-value for directional hypotheses

Usually, statistical software always calculates the non-directional test and then also outputs the p-value for this.

To obtain the p-value for the directional hypothesis, it must first be checked whether the effect is in the right direction. Then the p-value must be divided by two. This is because the significance level is not split on two sides, but only on one side. More about this in the tutorial about the p-value.

If you select a directed alternative hypothesis in DATAtab for the calculated hypothesis test, the conversion is done automatically and you only need to read the result.

Step-by-step instructions for testing hypotheses

  • Literature research
  • Formulation of the hypothesis
  • Define scale level
  • Determine significance level
  • Determination of hypothesis type
  • Which hypothesis test is suitable for the scale level and hypothesis type?

Next tutorial about hypothesis testing

The next tutorial is about hypothesis testing. You will learn what hypothesis tests are, how to find the right one and how to interpret it.

Statistics made easy

  • many illustrative examples
  • ideal for exams and theses
  • statistics made easy on 412 pages
  • 5rd revised edition (April 2024)
  • Only 7.99 €
Free sample

"Super simple written"

"It could not be simpler"

"So many helpful examples"

Cite DATAtab: DATAtab Team (2024). DATAtab: Online Statistics Calculator. DATAtab e.U. Graz, Austria. URL

Contact & Support FAQ & About Us Privacy Policy Statistics Software