Standard Deviation

The standard deviation of a probability distribution, just like the variance of a probability distribution, is a measurement of the deviation in that probability distribution. It allows one to quantify how much the outcomes of a probability experiment tend to differ from the expected value.

Standard deviation is often used in the calculation of other statistics such as the \(z\)-score and the coefficient of variation.

The standard deviation of a random variable \(X\), denoted as \(\sigma\) or \(\sigma(X)\), is defined as the square root of the variance.

The standard deviation of a random variable \(X\) is

\[ \sigma(X) = \sqrt{ \text{Var}[X] }.\]

Equivalently, the standard deviation of a random variable \(X\) is

\[\sigma=\sqrt{\sigma^2}.\]

What is the standard deviation of a fair six-sided die roll?

---

Let \(X\) be the random variable that represents the result of a fair six-sided die roll.

Recall from another example that \(\text{Var}[X]=\dfrac{35}{12}\). Then,

\[\sigma(X)=\sqrt{\dfrac{35}{12}}\approx 1.708.\ _\square\]

By the properties of variance, we have the following properties of standard deviation:

For random variable \(X\) and any constant \(c\), we have

\[\sigma(cX ) = \lvert c \rvert \big( \sigma(X) \big).\]

For random variable \(X\) and any constant \(c\), we have

\[\sigma(X + c ) = \sigma(X).\]

Let \(X_1, X_2, \ldots, X_k\) be pairwise independent random variables. Then

\[ \sigma(X_1 + X_2 + \ldots + X_k) = \sqrt{\text{Var}(X_1) + \text{Var}(X_2) + \cdots + \text{Var}(X_k)}.\]

The variance and standard deviation of a random variable intuitively measure the amount of spread, or dispersion of the random variable from the mean. This allows us to approach the following question: what is the probability that the value of the random variable is far from its expected value?

In the worked examples below, we will prove bounds on this probability for a general random variable using Chebyshev's inequality. In the special case of a random variable \(X\) with a normal distribution (or Gaussian distribution) \(N(\mu, \sigma)\), we have the following:

\[\begin{align} P(\mu - \sigma \leq X \leq \mu + \sigma) & \approx 0.6826 \\ P(\mu - 2\sigma \leq X \leq \mu + 2\sigma) & \approx 0.9544 \\ P(\mu - 3\sigma \leq X \leq \mu + 3\sigma) & \approx 0.9972. \end{align}\]

If we consider the probability density function of \(X\), this shows that the area under the curve in the interval \( ( \mu - \sigma, \mu + \sigma ) \) is approximately \(0.6826\), the area under the curve in the interval \( ( \mu - 2\sigma, \mu + 2\sigma ) \) is approximately \(0.9544\), and the area under the curve in the interval \( ( \mu - 3\sigma, \mu + 3\sigma ) \) is approximately \(0.9972\). These probabilities, derived from the area under the curve of a normal distribution, hold true regardless of the value of \(\mu\) or \(\sigma\).

For a random variable, the \(z\)-score, or standardized score, of a value is the number of standard deviations the value is from the mean of the data.

Let \(X\) be a random variable. If \(\mu\) is the expected value of \(X\), \(\sigma\) is the standard deviation of \(X\), and \(x\) is a value that \(X\) can take, then the \(z\)-score of \(x\) is

\[z=\frac{x-\mu}{\sigma}.\]

This gives a measure of how far the value is with respect to the mean value. If a point has value less than the expected value, the \(z\)-score is negative. If the point has value greater than the expected value, the \(z\)-score is positive.

While studying the spread of the random variable, we may want to take into account the magnitude of the expected value. The coefficient of variation is the ratio of the standard deviation to the expected value:

\[ CV(X) = \frac{\sigma}{\mu}. \]

The coefficient of variation is a measurement of the amount of deviation in a probability distribution relative to the expected value.

Prove Markov's inequality: For any nonnegative random variable \(X\) and positive constant \(a\),

\[P(X \geq a ) \leq \frac{E[X]}{a}.\]

---

Consider the event \(A = \{ s : X(s) \geq a \} \). Then

\[ \begin{align} E[X] &= \sum_s P(s)X(s) \\ &= \sum_{s \in A} P(s)X(s) + \sum_{s \not\in A} P(s)X(s) \\ & = \sum_{s \in A} P(s)X(s) \\ & \geq \sum_{s \in A} P(s)X(s) \qquad (\text{since } X(s) \geq 0 \text{ for all } s) \\ &\geq a \sum_{s \in A} P(s) \qquad (\text{since } X(s) \geq a \text{ for all } s \in A) \\ &= a P(A). \end{align}\]

Therefore, \(E[X] \geq a P( X \geq a) \), implying \( P( X \geq a) \leq \frac{E[X]}{a}.\) \(_\square\)

We now consider bounds for a general random variable (not necessarily nonnegative).

Prove Chebyshev's inequality: For a random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\) and for any positive constant \(a,\)

\[ P \big( \lvert X - \mu \rvert \geq a \sigma \big) \leq \frac{1}{a^2}.\]

---

Consider the random variable \( (X - \mu)^2\). This is a nonnegative random variable, so we can apply Markov's inequality to obtain

\[ \begin{align} P\big( (X - \mu) \geq a \sigma \big) &= P\big( (X - \mu)^2 \geq a^2 \sigma^2\big) \\\\ &\leq \frac{E\big[ (X - \mu)^2\big] }{a^2 \sigma^2} \\\\ &= \frac{\sigma^2 }{a^2 \sigma^2}\\\\ &= \frac{1}{a^2}. \end{align} \]

Therefore,

\[ P \big( \lvert X - \mu \rvert \geq a \sigma \big) \leq \frac{1}{a^2}.\ _\square\]