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For each set of paired samples there is a variance, the covariance of the paired samples. If the covariance is the standard deviation of the set, the standard deviation of the variance of the paired samples, then the d-bar is the average of the distances between the means of the paired samples.

More about the t-test for two dependent samples so you can understand in a better way the results delivered by the solver: A t-test for two paired samples is a hypothesis test that attempts to make a claim about the population means (\(\mu_1\) and \(\mu_2\)). More specifically, a t-test uses sample information to assess how plausible it is for difference \(\mu_1\) - \(\mu_2\) to be equal to zero. The test has two non-overlaping hypotheses, the null and the alternative hypothesis. The null hypothesis is a statement about the population parameter which indicates no effect, and the alternative hypothesis is the complementary hypothesis to the null hypothesis. The main properties of the t-test for two paired samples are:

When two methods of measurement are compared it is almost always wrong to present a scatter plot with correlation as a measure of agreement between the paired data. Highly correlated results often agree poorly, indeed large shifts in measurement scales may leave the correlation coefficient unaltered. It is therefore necessary to provide a measure of agreement. StatsDirect provides a plot of the difference against the mean for each pair of measurements. This plot also displays the overall mean difference bounded by the limits of agreement. A good review of this subject has been provided by Bland and Altman (Bland and Altman, 1986; Altman, 1991). (Source: www.statsdirect.com)

A t test compares the means of two groups. For example, compare whether systolic blood pressure differs between a control and treated group, between men and women, or any other two groups. Don't confuse t tests with correlation and regression. The t test compares one variable (perhaps blood pressure) between two groups. Use correlation and regression to see how two variables (perhaps blood pressure and heart rate) vary together. Also don't confuse t tests with ANOVA. The t tests (and related nonparametric tests) compare exactly two groups. ANOVA (and related nonparametric tests) compare three or more groups. Finally, don't confuse a t test with analyses of a contingency table (Fishers or chi-square test). Use a t test to compare a continuous variable (e.g., blood pressure, weight or enzyme activity). Use a contingency table to compare a categorical variable (e.g., pass vs. fail, viable vs. not viable).

Standard deviation in statistics, typically denoted by σ, is a measure of variation or dispersion (refers to a distribution's extent of stretching or squeezing) between values in a set of data. The lower the standard deviation, the closer the data points tend to be to the mean (or expected value), μ. Conversely, a higher standard deviation indicates a wider range of values. Similar to other mathematical and statistical concepts, there are many different situations in which standard deviation can be used, and thus many different equations. In addition to expressing population variability, the standard deviation is also often used to measure statistical results such as the margin of error. When used in this manner, standard deviation is often called the standard error of the mean, or standard error of the estimate with regard to a mean. The calculator above computes population standard deviation and sample standard deviation, as well as confidence interval approximations. (Source: www.calculator.net)