What is a Null Hypothesis?

The null hypothesis is like playing devil's advocate…

A person decides they want to prove that their favorite football team plays better in the rain than in sunny weather. One way to try to test their hypothesis is to propose the opposite idea: that the team plays better when it's bright than when it's raining. By asking the opposite question, and disproving it, she can lend weight to her argument. In the same way, the null hypothesis can test whether there is a statistically significant relationship between two things by positing first that there is not such a relationship.

In statistics, you use a null hypothesis as part of the process of proving whether two groups of variables are related in some way. Specifically, the null hypothesis is the default assumption that there is no relationship between two groups of variables. By disproving the null hypothesis, statisticians can strengthen the argument that there is a connection between two variables. Proving the null hypothesis provides evidence for the opposite.

There are two ways to use the null hypothesis. One is using the method of Ronald Fisher, a British statistician. In Fisher's method, investigators check for the statistical likelihood of events or variables being related. They reject the null hypothesis if the examined phenomena are unlikely to have occurred if there was no relationship between them. The method does not explicitly prove or disprove a relationship. It shows whether a connection is likely beyond reasonable levels of doubt, akin to the presumption of innocence in a trial.

The null hypothesis also plays a role in statistical hypothesis testing. This process compares two data sets. The null hypothesis states that there isn't a relationship between the data sets. An alternative hypothesis argues for a specific relationship between the sets.

Statistical hypothesis testing cannot find a definite answer as to whether a hypothesis is correct or not. Instead, it can provide a level of confidence. For example, a statistical hypothesis can determine that there is a 99% chance that a theory is correct, but it cannot prove it with a 100% likelihood.

Possibly the best-known example of a null hypothesis is the legal presumption of innocent until proven guilty. The null hypothesis for a court case is that the defendant is innocent. The prosecution must prove the alternative hypothesis that the defendant is guilty. They must show that the assumption of innocence is incorrect.

A historical example of a null hypothesis is the Medieval belief in a geocentric solar system, meaning a solar system where the Sun and other planets orbit around Earth. The alternative theory of heliocentrism (that Earth and other planets revolve around the Sun) was later proposed by Copernicus.

Some people argue that plants grow better when gardeners speak to them. The null hypothesis for this argument is that plant growth and health is unaffected by whether a gardener speaks to the plants. The proposed alternative hypothesis is that talking to plants does affect their growth.

In the world of finance, an investor could argue that stock prices are more likely to drop in the month of May than in other months. The null hypothesis for this idea is that the month of the year has no impact on whether stock prices rise or fall.

A null hypothesis is a starting point for testing a question statistically. By starting with the assumption that there isn't a relationship between datasets or particular events, statisticians can work to disprove that assumption.

To start, statisticians take a sample. For example, for a study examining the relationship between age and music preferences in the United States, the study might involve answers from 10,000 respondents. Instead of asking more than 300 million Americans about their taste, the test can use the smaller sample to get a manageable amount of data.

Because hypothesis testing relies on sampling, it's almost impossible to prove a hypothesis with certainty. That's why most statistical tests will state their conclusions with a level of certainty, like 99% or 95%.

The null hypothesis is an essential part of determining the level of certainty. If there is a correlation between age and music taste in the sample, the researchers must determine the likelihood of finding that level of correlation, still assuming the null hypothesis is true, based on the sample size. There is always the chance that the sample only included people for whom the correlation exists when among the whole American population there is no correlation.

You can generally make an assumption about whether the sample represents the population as a whole by looking at the size of the sample and the strength of the relationship between the variables. The likelihood of your sample showing a relationship even if the null hypothesis is true is represented by the "p-value."

Statisticians use z-tables to identify where a result falls within a distribution curve to come up with the relevant p-value. Usually, a p-value less than 0.05 indicates that your results are most likely representative of the population and not just a sampling error. If you have a p-value of .04, it indicates that a sample such as yours would only occur 4% of the time if the null hypothesis were true.

When testing hypotheses, there are two types of errors that statisticians encounter. A Type I error occurs when you reject the null hypothesis, even though the null hypothesis is true. A Type II error occurs when you accept the null hypothesis, even when the null hypothesis is false.

Type I errors are also called false positives. Type II errors are false negatives. When performing a hypothesis test based on a sample of a population, there is always the chance for error. That's why most statistical analyses don't provide concrete conclusions. Instead, they state a conclusion with a certain level of confidence, such as 99% or 95%.

Testers derive these levels of confidence from the p-value of their experiment. P-values indicate the odds of getting a certain result if the null hypothesis is true.

The alternative hypothesis is the hypothesis that researchers propose when looking for a relationship between two events or sets of data. The null hypothesis is the default assumption that the researchers are arguing against.

For example, researchers may want to show that people with long hair are more physically active. The null hypothesis is that hair length does not impact physical activity. The alternative hypothesis is that a relationship between hair length and physical activity exists.

Alternative hypotheses can be either one-sided or two-sided. A one-sided alternative hypothesis examines whether a sample differs in a single direction from the assumed value. For example, if a researcher wanted to know whether football teams from the south tend to score more than 20 points per game, they could use a one-sided hypothesis. This alternative hypothesis can test whether those teams score more than 20 regularly, but not if they score fewer than 20 points.

By contrast, two-sided alternative hypotheses can test in both directions. Researchers would use a two-sided alternative hypothesis to determine whether football teams from the south tend to have scores higher or lower than 20 points per game.

The alternative hypothesis must contradict the null hypothesis. Both hypotheses cannot both be true.

When writing null hypotheses, you should write a clear, measurable statement that you want to test against.

For example, "does eating fast food once a week correlate with obesity?" would not be a null hypothesis because it asks a question rather than making a statement. The proper null hypothesis is "eating fast food once a week has no effect on obesity."

Statisticians can use that null hypothesis to test the alternative hypothesis that regular fast food meals correlate with obesity.

For a medical trial testing a drug intended to reduce blood pressure, the alternative hypothesis would be "taking this drug reduces the patient's blood pressure.” The null hypothesis to test against is "taking this drug has no impact on the patient's blood pressure."

In sports, one could test whether soccer teams from Europe have higher rates of victory against teams from North America than against teams from South America. The alternative hypothesis is "soccer teams from Europe are more likely to beat North American opponents than South American opponents." The null hypothesis is "their opponent's continent of origin does not impact European soccer team's winning rate."

Each null hypothesis directly contradicts the alternative hypothesis and assumes that there is no relationship to be found. You cannot write a null hypothesis that can be true at the same time as the alternative hypothesis.

For example, if the alternative hypothesis is "college students drink 25% more alcohol than non-students," the null hypothesis cannot be "college students drink 10% more alcohol than non-students." The hypotheses are different but aren't mutually exclusive. That makes them poor hypotheses to test against each other.

Understanding null hypotheses is important for investors because statistical analysis is one method they can use to find and vet potential investments.

For example, an investor could believe that companies with an Earnings Per Share (EPS) of at least $30 perform better than businesses with an EPS under $30. The null hypothesis the investor could use to test this is that there is no difference in stock performance based on a company's EPS.

To test this hypothesis, the investor would start by finding sample data. They could choose a number of stocks to test and gather information about their EPS and their change in value over a certain number of years.

Next, the investor would look for correlation within the sample data. Does there appear to be a relationship between a stock's EPS and its performance? The investor may find that, within the sample, there is a positive correlation between EPS and performance, a negative correlation, or no apparent correlation at all.

The investor then finds the p-value of the level of correlation by looking it up on a z-table. Z-tables give p-values for different results based on the null hypothesis and the type of distribution that the data produces.

If the p-value is high, that means there is a high likelihood that a random sample would produce the result, even if the null hypothesis is true. That means that the null hypothesis is likely correct. If the p-value is low, for example, .05 or less, that means that the odds of receiving that result if the null hypothesis were true is low, lending weight to the investor's hypothesis that EPS and stock performance are related.