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• Before we understand this concept, we need to explain what a transformation is.

• So, a transformation is a way in which we can alter every element of a distribution

• to get a new distribution with similar characteristics.

• For Normal Distributions we can use addition, subtraction, multiplication and division without

• changing the type of the distribution.

• For instance, if we add a constant to every element of a Normal distribution, the new

• distribution would still be Normal.

• Let’s discuss the four algebraic operations and see how each one affects the graph.

• If we add a constant, like 3, to the entire distribution, then we simply need to move

• the graph 3 places to the right.

• Similarly, if we subtract a number from every element, we would simply move our current

• graph to the left to get the new one.

• If we multiply the function by a constant it will widen that many times and if we divide

• every element by a number, the graph will shrink.

• However, if we multiply or divide by a number between 0 and 1, the opposing effects will

• occur.

• For example, dividing by a half, is the same as multiplying by 2, so the graph would expand,

• even though we are dividing.

• Alright!

• Now that you know what a transformation is, we can explain standardizing.

• Standardizing is a special kind of transformation in which we make the expected value equal

• to 0 and the variance equal to 1.

• The benefit of doing so, is that we can then use the cumulative distribution table from

• last lecture on any element in the set.

• The distribution we get after standardizing any Normal distribution, is called a “Standard

• Normal Distribution”.

• In addition to the “68, 95, 99.7” rule, there exists a table which summarizes the

• most commonly used values for the CDF of a Standard Normal Distribution.

• This table is known as the Standard Normal Distribution table or the “Z”- score table.

• Okay!

• So far, we learned what standardizing is and why it is convenient.

• What we haven’t talked about is how to do it.

• First, we wish to move the graph either to the left, or to the right until its mean equals

• 0.

• The way we would do that is by subtracting the meanmufrom every element.

• After this to make the standardization complete, we need to make sure the standard deviation

• is 1.

• To do so, we would have to divide every element of the newly obtained distribution by the

• value of the standard deviation, sigma.

• If we denote the Standard Normal Distribution with Z, then for any normally distributed

• variable Y, “Z equals Y minus mu, over sigma”.

• This equation expresses the transformation we use when standardizing.

• Amazing!

• Applying this single transformation for any Normal Distribution would result in a Standard

• Normal Distribution, which is convenient.

• Essentially, every element of the non-standardized distribution is represented in the new distribution

• by the number of standard deviations it is away from the mean.

• For instance, if some value y is 2.3 standard deviations away from the mean, its equivalent

• value “Z” would be equal to 2.3.

• Standardizing is incredibly useful when we have a Normal Distribution, however we cannot

• always anticipate that the data is spread out that way.

• A crucial fact to remember about the Normal distribution is that it requires a lot of

• data.

• If our sample is limited, we run the risk of outliers drastically affecting our analysis.

• In cases where we have less than 30 entries, we usually avoid assuming a Normal distribution.

• However, there exists a small sample size approximation of a Normal distribution called

• the Students’ T distribution and we are going to focus on it in our next lecture.

• Thanks for watching.

Before we understand this concept, we need to explain what a transformation is.

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# 標準正態分佈 (Standard Normal Distribution)

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林宜悉 發佈於 2021 年 01 月 14 日