Standard Deviation

Standard deviation lets you understand how things are distributed within the range.

The formula to calculate a sample’s variance is $s^{2} = \frac{Σ x - x ˉ ^{2}}{n - 1}$

We have to take the sum of the squares of the difference because the underlying numbers may be negative, which the formula can’t handle.

You can then turn that sample variance into the sample’s “standard deviation” by taking the sqrt of it. $σ = σ^{2}$

Finding stddev from a probability distribution

Formula: $σ = \sum (x_{i} - μ)^{2} \cdot P (x_{i})$

Given this distribution of cat weights: 6.8, 8.2, 7.5, 9.4, 8.2

Mean = 8.02

would yield:

x	x-barx	(x-barx)^2
6.8	6.8-8.02 = -1.22	1.4884
7.5	-0.52	.2704
8.2	.18	.0324
8.2	.18	.0324
9.4	1.38	1.9044

If we then divide that last column by n-1 (4), we get 0.932 lbs^2 as the variance. sqrt = 0.9654 lb as the standard deviation.

The n-1 above is “degrees of freedom”, but I don’t know what that means yet.

Population variance and standard deviation are rare to see, but they are represented by lowercase sigma.

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A population’s variance is similar to sample variance, but it devides by $N$ , rather than $n - 1$ .

Population Variance

σ^{2} = \frac{\sum ( x - μ ) ^{2}}{N}

where $μ$ is the population mean, N is the total number of items in the population.

Sample Variance

s^{2} = \frac{Σ [( X - x ˉ ) ^{2} ]}{n - 1}

where $\overset{x}{ˉ}$ is the mean of the sample and $n$ is the number of items sampled.