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The normal distribution is used when the population distribution of data is assumed normal. It is characterized by the mean and the standard deviation of the data. The t statistic is an estimate of the standard error of the mean of the population or how well known is the mean based on the sample size.
Here's why. When you perform a t-test, you're usually trying to find evidence of a significant difference between population means (2-sample t) or between the population mean and a hypothesized value (1-sample t). The t-value measures the size of the difference relative to the variation in your sample data.
The main difference between a z-score and t-test is that the z-score assumes you do/don't know the actual value for the population standard deviation, whereas the t-test assumes you do/don't know the actual value for the population standard deviation.
Z-tests are statistical calculations that can be used to compare population means to a sample's. T-tests are calculations used to test a hypothesis, but they are most useful when we need to determine if there is a statistically significant difference between two independent sample groups.
Z score is a conversion of raw data to a standard score, when the conversion is based on the population mean and population standard deviation. T score is a conversion of raw data to the standard score when the conversion is based on the sample mean and sample standard deviation.
The T distribution is similar to the normal distribution, just with fatter tails. Both assume a normally distributed population. T distributions have higher Kurtis than normal distributions. The probability of getting values very far from the mean is larger with a T distribution than a normal distribution.
The mean of the distribution is equal to 0. The variance is equal to /( 2), if > 2. The variance is always greater than 1, although it is close to 1 when there are many degrees of freedom. With infinite degrees of freedom, the distribution is the same as the standard normal distribution.
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