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Along with the knowledge of the shape of a given probability distribution - what practical information does the mean and standard deviation of a random variable provide?

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up vote 1 down vote accepted

The mean is measure of location and the standard deviation is a measure of scale.

If a random variable $X$ has mean $\mu$ and standard deviation $\sigma$ and you use define a new random variable as $Y=X+a$ then the mean of $Y$ is $\mu+a$. Similarly if you define $Z = bX$ then the standard deviation of $Z$ is $b\sigma$. More generally for a linear transformation, if $W=bX+a$ then the mean of $W$ is $b\mu +a$ and the standard deviation of $W$ is $b\sigma$.

There are other statistics with similar properties, such as the median and inter-quartile range.

If the mean and standard deviation exist then the distribution is not "too dispersed", and you can apply the central limit theorem.

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  • Mean tells you around which point is the distribution concentrated on.

  • Standard deviation tells you, hoe dispersed is the distribution around the mean. The greater the dispersion of values, the larger the standard deviation.

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To flesh out Kannappan Sampath's answer a little, you can use Chebyshev's inequality to assert that the probability that $X$ deviates from its mean $\mu$ by at least $\alpha$ standard deviations is at most $1/\alpha^2$, that is,

$$P\{X \leq \mu - \alpha\sigma\} + P\{X \geq \mu + \alpha\sigma\} = P\{|X-\mu| \geq \alpha\sigma\} \leq \frac{1}{\alpha^2}$$ where $\sigma$ denotes the standard deviation.

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