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Difference between Z-test and t-test: Z-test is used when sample size is large (n>50), or the population variance is known. t-test is used when sample size is small (n<50) and population variance is unknown. There is no universal constant at which the sample size is generally considered large enough to justify use of the plug-in test.
Comparison of the various grading methods in a normal distribution, including: standard deviations, cumulative percentages, percentile equivalents, z-scores, T-scores. In statistics, the standard score is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured.
The t-statistic is used in a t-test to determine whether to support or reject the null hypothesis. It is very similar to the z-score but with the difference that t-statistic is used when the sample size is small or the population standard deviation is
Student's t-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's t -distribution under the null hypothesis .
Thus the mean difference between the groups does not depend on whether we organize the data as pairs. Although the mean difference is the same for the paired and unpaired statistics, their statistical significance levels can be very different, because it is easy to overstate the variance of the unpaired statistic.
Simple back-of-the-envelope test takes the sample maximum and minimum and computes their z-score, or more properly t-statistic (number of sample standard deviations that a sample is above or below the sample mean), and compares it to the 68–95–99.7 rule: if one has a 3σ event (properly, a 3s event) and substantially fewer than 300 samples, or a 4s event and substantially fewer than 15,000 ...
The statistical errors, on the other hand, are independent, and their sum within the random sample is almost surely not zero. One can standardize statistical errors (especially of a normal distribution) in a z-score (or "standard score"), and standardize residuals in a t-statistic, or more generally studentized residuals.
The Student's t distribution plays a role in a number of widely used statistical analyses, including Student's t test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis.