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I heard that statement at the end of the lecture on Ted x by Arthur Benjamin on youtube at the end about why Probability and statistics should be at the top of the triangle before Calculus.

So what does it mean to say the two standard deviations from the means mean according to statistics and probability? Can anyone kindly give an example?

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    $\begingroup$ Your question doesn't match your title. Edit them accordingly for clarity. $\endgroup$
    – J.G.
    Nov 1, 2019 at 23:17
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    $\begingroup$ As far as "two standard deviations", you may want to read article on 68,95,99.7 rule. $95\%$ of all random data fitting a normal curve will be within two standard deviations of the mean. For more exotic distributions, you can come up with inequalities to match, but the normal distribution should be all you should care about as a first introduction. $\endgroup$
    – JMoravitz
    Nov 1, 2019 at 23:19
  • $\begingroup$ Is it similar to Bell curve? $\endgroup$
    – user35885
    Nov 1, 2019 at 23:23
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    $\begingroup$ Bell curve is normal curve $\endgroup$
    – JMoravitz
    Nov 1, 2019 at 23:26
  • $\begingroup$ Thank you very much. $\endgroup$
    – user35885
    Nov 1, 2019 at 23:28

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Standard deviation is a common way to measure the 'spread' of a probability distribution.

An informal definition is as follows: let $X$ be a 'nice' (it's square has a finite mean) random variable (or distribution), then the standard deviation of $X$, $\sigma_X$ is the root mean-squared of the difference between $X$ and its mean. That is, if $X$ has mean $\mu_X$, then $\sigma_X$ is the square root of the mean of $(X-\mu_X)^2$.

Standard deviation has nice properties for normal distributions in particular. Normal distributions come up a lot in probability and statistics, and are often used to approximate other non-normal distributions. One of the most notable properties is that if $X$ is normal, then about 68% of the distribution of $X$ lies between $\mu_X \pm \sigma_X$, about 95% lies between $\mu_X \pm 2\sigma_X$, and about 99.7% lies between $\mu_X \pm 3\sigma_X$.

One example of what this means is student grades. Grades distributions are often either approximately normal or adjusted to be normal (so I'm told), and thus if a student scored two standard deviations above the mean, then this means they scored in the top $\frac{1-0.95}{2}=$2.5% of students.

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