When we calculate the standard deviation for a normally distributed series, we know that68% of values fall within 1 standard deviation of the mean,95% of values fall within 2 standard deviations of the mean, and99.7% of values fall within 3 SD of the mean.

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The central limit theorem essentially states that if you have a population with mean and standard deviation and you take sufficiently large random samples from that population, then the distribution of the sample meansx will be approximately normally distributed even if the population was not normally distributed.

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You should perform a Z test when you don't know the population variance or standard deviation.

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You should perform a T test only if you have larger than 30 observations in your sample.

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