How to find the standard deviation of a probability distribution

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Standard deviation measures the amount of variation or spread of values around the mean in a probability distribution. It quantifies how much individual data points typically deviate from the average value, providing insight into the distribution's variability.

What does standard deviation represent in probability distributions?

Standard deviation represents the typical distance between data points and the mean of the distribution. A small standard deviation indicates that data points tend to cluster close to the mean, while a large standard deviation suggests greater spread or dispersion in the data.

The standard deviation helps describe how outcomes are spread around the expected value. In a normal distribution, approximately 68% of data lies within one standard deviation of the mean, around 95% within two standard deviations, and 99.7% within three standard deviations. This pattern is known as the empirical rule.

Understanding that standard deviation measures spread relative to the mean explains why the calculation involves finding distances from the mean, squaring them, averaging based on probabilities, and taking the square root. The standard deviation serves as a fundamental descriptor of distribution shape and variability.

How to calculate the standard deviation of a discrete probability distribution

To find the standard deviation of a discrete probability distribution, calculate the mean first, then the variance, and finally take the square root of the variance.

The formula for standard deviation of a discrete probability distribution is:

$\sigma =\sqrt{\sum (x-}$

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In this formula, $\sigma$ is the standard deviation, $x$ represents the possible values, $\mu$ is the mean, and $P(x)$ is the probability of each value $x$.

An equivalent formula that avoids direct deviation calculations is:

$\sigma =\sqrt{\sum x^{2}P(x)-{\mu }^2}$

Step 1: Calculate the mean

Compute the mean by multiplying each value by its probability and summing the products:

$\mu =\sum x\cdot P(x)$

Step 2: Calculate the variance

For each value $x$, find the squared deviation $(x - \mu)^2$, multiply by $P(x)$, and sum the results: 

${\sigma }^2=\sum (x-\mu {)}^{2}P(x)$

The alternative variance formula is:

${\sigma }^2=\sum x^{2}P(x)-{\mu }^2$

This approach weights each squared deviation by its probability of occurrence.

Step 3: Take the square root

The standard deviation is the square root of the variance:

$\sigma =\sqrt{{\sigma }^2}$

This process applies to any discrete distribution where probabilities sum to 1.

How to calculate the standard deviation of a continuous probability distribution

To find the standard deviation of a continuous probability distribution, follow the same conceptual process as the discrete case but replace summation with integration. Continuous distributions have infinite possible values, requiring calculus rather than arithmetic sums.

The formula for standard deviation of a continuous probability distribution is:

$\sigma =\sqrt{{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x-\mu {)}^{2}f(x)dx}$

In this formula, $f(x)$ is the probability density function (PDF) and $\mu$ is the mean.

The mean for a continuous distribution is calculated as:

$\mu ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}xf(x)dx$

An alternative formula for standard deviation is:

$\sigma =\sqrt{{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}^{2}f(x)dx-{\mu }^2}$

Differences between discrete and continuous calculations

The discrete case uses summation $\sum (x - \mu)^2 P(x)$, while the continuous case uses integration over the PDF because values are uncountable.

The probability mass function $P(x)$ in discrete distributions becomes the probability density function $f(x)$ in continuous distributions. Probabilities in continuous distributions are represented as areas under the curve, requiring integral calculus to compute.

Continuous distributions often require solving definite integrals analytically or numerically, whereas discrete distributions involve finite arithmetic sums.

Step 1: Calculate the mean

Integrate over the support of the distribution:

$\mu =\int xf(x)dx$

Step 2: Calculate the variance

Integrate the squared deviations weighted by the PDF:

${\sigma }^2=\int (x-\mu {)}^{2}f(x)dx$

The shortcut formula is:

${\sigma }^2=\int x^{2}f(x)dx-{\mu }^2$

Evaluate the limits of integration based on the support of $f(x)$.

Step 3: Take the square root

Obtain the standard deviation:

$\sigma =\sqrt{{\sigma }^2}$

For common distributions such as the normal distribution, closed-form expressions for standard deviation exist. The general integral formula applies universally to any continuous probability distribution with a defined variance.

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