Definition

The Student’s t-test determines whether two populations express a significant or nonsignificant difference between population means. The Student’s t-test places emphasis on controlling for sample size. A significant difference, seen in Fig. 1, is distinguished from a nonsignificant difference, seen in Fig. 2, by the properties of the normal distributions characterized by the data.

Student’s t-Test, Fig. 1
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Significantly different populations. The dark- and light-gray lines represent the distributions of two different populations. Since these distributions have minimal overlap, they appear to be significantly different. (Parameters: dark-gray mean = 3, standard deviation = 1; light-gray mean = −3, standard deviation = 1)

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Student’s t-Test, Fig. 2
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Nonsignificantly different populations. The dark- and light-gray lines represent the distributions of two different populations. These distributions are nearly identical and, thus, are not significantly different. (Parameters: dark-gray mean = 0, standard deviation = 1; light-gray mean = 0.25, standard deviation = 1)

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Properties

Null Hypothesis

The null hypothesis for the Student’s t-test is that there is no difference in the means of two populations. Thus, rejection of the null hypothesis asserts a statistically significant difference between the population means.

Requirements

The Student’s t-test distinguishes between exactly two population sets. Typically, one control condition is compared to a test condition. Measurements of the two populations must be in the same units.

Since t-test calculations rely on mean and standard deviation values from a normal distribution, the Student’s t-test requires that both populations are reasonably approximated by a normal distribution. Further, the variance between the two populations must be roughly equal. If there exists unequal variance in populations, then the t-test for unequal variances, Wilcoxon Rank Sum Test, or Welch’s t-test may be used [Ruxton 2006].

t Score Calculation

The t score represents the difference between sample means divided by the standard error.

$$ {t = \frac{\text{Mean difference}}{{\text{Standard error}}}} $$

Standard error decreases as variance decreases and sample size increases. Accordingly, a lower standard error indicates more confidence in the answer. So, a high t score indicates that there is a significant difference in the means and a high confidence in the difference. To assess the significance of a t score, the score must be compared to the t distribution, discussed below.

t Distribution

Though similar to the bell shape of the normal distribution, the t distribution is characterized by a distinct Probability Distribution. The t distribution is parameterized by the degrees of freedom for the data. In Fig. 3, the normal- and t distributions are drawn in light- and dark-gray, respectively. The t distribution has a wider base than the normal distribution, indicating that a lower percentage of t scores lie near the mean than in a normal distribution.

Student’s t-Test, Fig. 3
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Normal distribution vs. t distribution. The normal- and t distributions are drawn in light- and dark-gray, respectively. (Parameters: normal distribution mean = 0, standard deviation = 1; t distribution degrees of freedom = 2)

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The t distribution is dependent on the degrees of freedom. In the case of the Student’s t-test, the degrees of freedom is the total sample size of both populations minus two. As the degrees of freedom increase, the t distribution increasingly favors the mean. Many textbooks and online sources contain tables where degrees of freedom and confidence intervals are used to look up threshold values for t scores. Once a threshold t score is determined, significance of the t score can be determined. If the t score is greater than the threshold, the difference between the populations is significant up to the selected confidence interval. Otherwise, there is no significant difference between the two populations.

Formalized t Score Calculation

Note: In practice, most statistical software and spreadsheet applications can be used to perform a student’s t-test.

To calculate the t score for two means μ1 and μ2 with variances s 1 and s 2, respectively, and sample sizes of n 1 and n 2, respectively, the formula is represented as:

$$ {t = \frac{{{\mu_{{1}}} - {\mu_{{2}}}}}{{s_p^{{2}}\sqrt {{\left( {\frac{{1}}{{{n_{{1}}}}} + \frac{{1}}{{{n_{{2}}}}}} \right)}} }}} $$

where the pooled variance, s p 2 is calculated as:

$$ {s_p^{{2}} = \frac{{\left( {{n_{{1}}} - {1}} \right)s_{{1}}^{{2}} + \left( {{n_{{2}}} - {1}} \right)s_{{2}}^{{2}}}}{{{n_{{1}}} + {n_2} - {2}}}} $$

Example

We performed a test to determine whether our new chemical significantly increased cell size. For testing purposes, we measured both a control population, C and a test population, T one hour after nontreatment and treatment, respectively. Our results for C and T are {35, 43, 40, 38, 36, 40} and {52, 47, 39, 43, 41, 48}, respectively.

From this dataset, we used our spreadsheet to calculate the t score as 2.73. Since there are 12 data points, the distribution has 10 degrees of freedom. From the t distribution table, we found that a 95% confidence interval with 10 degrees of freedom has a t threshold of 1.812, so we can say with 95% confidence that the difference between control and test was significant. Our trials require even more precision, so we look at the 99% confidence interval and find a threshold of 2.764. Unfortunately, our t score is less than 2.764, so we cannot say with 99% confidence that the difference is significant.