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  • Welcome back!

  • In this lecture we are going to introduce one of the most commonly found continuous

  • distributionsthe normal distribution.

  • For starters, we define a Normal Distribution using a capital letter N followed by the mean

  • and variance of the distribution.

  • We read the following notation asVariable “X” follows a Normal Distribution with

  • meanmuand variancesigmasquared”.

  • When dealing with actual data we would usually know the numerical values of mu and sigma

  • squared.

  • The normal distribution frequently appears in nature, as well as in life, in various

  • shapes of forms.

  • For example, the size of a full-grown male lion follows a normal distribution.

  • Many records suggest that the average lion weight between 150 and 250 kilograms, or 330

  • to 550 pounds.

  • Of course, there exist specimen which fall outside of this range.

  • Lions weighing less than 150, or more than 250 kilograms tend to be the exception rather

  • than the rule.

  • Such individuals serve as outliers in our set and the more data we gather, the lower

  • part of the data they represent.

  • Now that you know what types of events follow a Normal distribution, let us examine some

  • of its distinct characteristics.

  • For starters, the graph of a Normal Distribution is bell-shaped.

  • Therefore, the majority of the data is centred around the mean.

  • Thus, values further away from the mean are less likely to occur.

  • Furthermore, we can see that the graph is symmetric with regards to the mean.

  • That suggests values equally far away in opposing directions, would still be equally likely.

  • Let’s go back to the lion example from earlier.

  • If the mean is 400, symmetry suggests a lion is equally likely to weigh 350 pounds and

  • 450 pounds since both are 50 pounds away from that the mean.

  • Alright!

  • Instead of going through the complex algebraic simplifications in this lecture, we are simply

  • going to talk about the expected value and the variance.

  • The expected value for a Normal distribution equals its mean - “mu”, whereas its variance

  • sigmasquared is usually given when we define the distribution.

  • However, if it isn’t, we can deduce it from the expected value.

  • To do so we must apply the formula we showed earlier: “The variance of a variable is

  • equal to the expected value of the squared variable, minus the squared expected value

  • of the variable”.

  • Good job!

  • Another peculiarity of the Normal Distribution is the “68, 95, 99.7” law.

  • This law suggests that for any normally distributed event, 68% of all outcomes fall within 1 standard

  • deviation away from the mean, 95% fall within two standard deviations and 99.7 - within

  • 3.

  • The last part really emphasises the fact that outliers are extremely rare in Normal distributions.

  • It also suggests how much we know about a dataset only if we have the information that

  • it is normally distributed!

  • Fantastic work, everyone!

Welcome back!


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B1 中級

正規分布を理解する [統計学チュートリアル] (Understanding the Normal Distribution [Statistics Tutorial])

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    林宜悉 に公開 2021 年 01 月 14 日