One– and Two–Tailed Tests:

So far, we have talked in general terms about the null and alternate hypotheses without getting into the specifics of the type of comparison that we wish to make in our tests. There are two main questions we might be asking when performing our tests:

  • Is our result greater (or less than) a certain value?
  • Is our result within (or outside) a certain range of values?

The first of these is what is known as a one–tailed test, while the second is known as a two–tailed test. This refers back to the normal distribution and our sample mean and standard deviation. Most analytical chemistry texts will present statistical tables for the various statistical significance tests in the most commonly used form for that test. On this page, you will learn the difference between one– and two–tailed tests, and what to do if the table you have isn't for the form that you need.

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The One–tailed Test:

In a one–tailed test, we are interested in seeing whether the test parameter calculated from the sample data is greater than (or less than) some critical value. For example, in the previous page we tested whether a sample mean was higher than an accepted true value: μ > μ0.

If we want to make this determination at the 95% confidence level, then we need to determine whether our sample data gives a result within the white area representing the upper 5% of possible values, or the blue area representing the lower 95% of possible values:


In this distribution, the shaded region shows the area represented by the null hypothesis, H0: μ = μ0. This actually implies μμ0, since the unshaded region shows μ > μ0. Because we were only interested in one side of the distribution, or one "tail", this type of test is called a one–sided or a one–tailed test.

Note that, if we wanted to know if the test mean was lower than the accepted value (μ < μ0) then the representation would be reversed i.e. the white area representing 5% of possible values would be on the left-end of the value axis. It would still be a one–tailed test, however.

A one–tailed test uses an alternate hypothesis that states either:

H1: μ > μ0

OR

H1: μ < μ0

but not both. If you want to test for both simultaneously, then you need to use a two–tailed test.

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The Two–tailed Test:

If we wanted to test whether two means were significantly different from one other, i.e. that they came from different populations, then we would use a two–tailed test. The probability distribution for a 90% confidence level, two-tailed test looks like this:


If two means, for example, came from the same population, then we would expect them to both lie within the shaded blue area representing 90% of the possible values centred on the population mean (i.e. μ±45%). If, on the other hand, the means were from different populations, then we would expect one of them to fall in the either of the white areas each representing 5% of the possible values – one above, and one below, the population mean.

Note that, if we wanted to use the same 95% confidence level we employed in the one–tailed test, we would need the white areas to each correspond to 2.5% of the possible values; that is, we would need to use critical values corrseponding to those for a one–tailed test at the 97.5% confidence level.

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Example:

As an example, let’s assume that we want to check if the pH of a stream has changed significantly in the past year. A water sample from the stream was analyzed using a pH electrode, where six samples were taken. It was found that the mean pH reading was 6.5 with standard deviation sold = 0.2. A year later, six more samples were analyzed, and the mean pH of these readings was 6.8 with standard deviation snew = 0.1.

We might be faced with a scenario in which a known source of contamination could increase the pH over time. In this case, we could use a one-tailed test to see if the stream indeed has a higher pH than one year ago. For this, we would use the alternate hypothesis HA: μold < μnew. A more likely scenario, however, is that the pH could have increased, decreased, or stayed the same. As a result, we would want to use a more rigorous two-tailed test for the hypothesis that:

H0: μold = μnew
HA: μoldμnew

In other words, we need to test for both:

HA: μold < μnew
HA: μold > μnew

We can calculate the required degrees of freedom and t-statistic as indicated previously:

ν = 7.35 ≈ 7 (since d.o.f. must be a whole number)

t = (6.8-6.5)/sqrt((0.1)^2/6 + (0.2)^2/6) = 3.29

We can either calculate the probability (p) of obtaining this value of t given our sample means and standard deviations, or we can look up the critical value tcrit from a table compiled for a two-tailed t-test at the desired confidence level. For example, the critical value tcrit at the 95% confidence level for ν = 7 is t7,95% = 2.36. Since in this case t is greater than t7,95%, we can reject the null hypothesis and conclude that the pH values are significantly different at the 95% level of confidence.

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Using Tables for One- and Two-Tailed Tests:

Most tables of critical t-values give you values for either a one- or two-tailed test, but not both. Because of this, you will need to know how to use one-tailed tables for two-tailed tests, and vice versa. The conversion is actually quite simple. As noted above, a on-tailed test at the 95% confidence level uses the same point on the value axis of the population distribution as a two-tailed test at the 90% confidence level. Similarly, a one-tailed test at the 90% confidence level uses the same criterion as a two-tailed test at the 80% confidence level:

Table you haveOperationTo get...
One-tailedTwo-tailed
Two-tailedOne-tailed

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