T-Distribution and Z-Distribution Explained with Animations

Welcome to our exploration of the Normal Distribution, also known as the Z-Distribution!The normal distribution is characterized by its distinctive bell-shaped curve, which is perfectly symmetrical around its mean.This distribution has three key characteristics: it's symmetrical, has a mean of zero, and a standard deviation of one.Standard deviations, denoted by sigma, measure the spread of data from the mean. Let's mark these on our curve.One of the most important properties of the normal distribution is the empirical rule, also known as the 68-95-99.7 rule.Now, let's understand how to convert raw scores to Z-scores, which tell us how many standard deviations a value is from the mean.In our example, a raw score of 85, with a mean of 75 and standard deviation of 5, converts to a Z-score of positive 2, meaning it's two standard deviations above the mean.Now that we understand the normal distribution, let's explore the T-distribution and its unique characteristics.The T-distribution looks similar to the normal distribution, but has some key differences. Here's the normal distribution curve in blue.Let's see how the T-distribution changes with different degrees of freedom. First, with one degree of freedom, notice the much heavier tails.With five degrees of freedom, the distribution starts to look more like the normal distribution, but still has heavier tails.At fifteen degrees of freedom, we're getting closer to the normal distribution.Finally, at thirty degrees of freedom, the T-distribution is very close to the normal distribution.Let's examine the key characteristics that make the T-distribution unique.One of the most important features is the heavier tails. This means there's more probability in the extreme values compared to the normal distribution.The degrees of freedom in a T-distribution is determined by the sample size minus one.Now that we understand both distributions, let's compare when to use each one.The Z distribution and T distribution look similar, but they serve different purposes.The Z distribution is used when we have large samples of at least 30 and know the population standard deviation.The T distribution is more appropriate for small samples under 30 or when we don't know the population standard deviation.For a 95% confidence level, the Z distribution has critical values of plus or minus 1.96.The T distribution, accounting for additional uncertainty with small samples, has larger critical values, like plus or minus 2.13 for 14 degrees of freedom.In quality control, like measuring widget lengths with a large sample size, we use the Z distribution.For research studies with small sample sizes, like testing a new drug's effectiveness, we use the T distribution.Here's a simple decision flowchart to help you choose between the Z and T distributions.