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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