Population standard deviation

### 20.1.4 Population standard deviation

Given a large population, rather than collecting all of the numbers it might be more feasible to get a smaller collection of numbers and try to extrapolate from that. For example, to get information about the ages of a large population, you might get the ages of a sample of 100 of the people and work with that.

If a list of numbers is a sample of data from a larger population, then the mean of the sample can be used to estimate the mean of the population. The standard deviation uses the mean to find the standard deviation of the sample, but since the mean of the sample is only an approximation to the mean of the entire population, the standard deviation of the sample does not provide an optimal estimate of the standard deviation of the population. An unbiased estimate of the standard deviation of the entire population is given by the population standard deviation; given a list L=[x1,…,xn] with mean µ, the population standard deviation is

s= |
√
(x1−µ)2+⋯+(xn−µ)2
n−1
| .

Note that

s2= |
n
n−1
| σ2.

where σ is the standard deviation of the sample.

The stddevp command finds the standard deviation.

stdDev is a synonym for stddevp, for TI compatibility. There is no population variance function; if needed, it can be computed by squaring the stddevp function.

- stddevp takes one mandatory argument and one optional argument:
  - L, a list or matrix of numbers.
  - W, a list or matrix of weights, the same size as L.
- stddevp(L ⟨,W⟩) returns the population standard deviation of the list or a list with the population standard deviations of the columns of the matrix.

#### Examples

stddev([1,2,3,4])

√ |
5
2

while:

stddevp([1,2,3,4])

√ |
15
3

A:=[0,1,2,3,4,5,6,7,8,9,10,11]:; stddevp(A,A)

√ |
66
3