TEST_STDEV - Standard Deviation Test

Mohamad

October 20, 2016 18:43

Calculates the p-value of the statistical test for the population standard deviation.

Syntax

TEST_STDEV(X, sigma, Return_type, Alpha)

X

is the input data sample (one/two-dimensional array of cells (e.g., rows or columns)).

sigma

is the assumed population standard deviation. If missing, the default value of one is assumed.

Return_type

is a switch to select the return output (1 = P-Value (default), 2 = Test Stats, 3 = Critical Value.

Alpha

is the statistical significance of the test (i.e., alpha). If missing or omitted, an alpha value of 5% is assumed.

The data sample may include missing values (e.g., #N/A).
The test hypothesis for the population standard deviation: $$H_{o}: \sigma =\sigma_o$$ $$H_{1}: \sigma \neq \sigma_o$$ Where:
- $H_{o}$ is the null hypothesis.
- $H_{1}$ is the alternate hypothesis.
- $\sigma_o$ is the assumed population standard deviation.
- $\sigma$ is the actual (real) population standard deviation.
For the case in which the underlying population distribution is normal, the sample standard deviation has a Chi-square sampling distribution: $$ \hat \sigma \sim \chi_{\nu=T-1}^2 $$ Where:
- $\hat \sigma $ is the sample standard deviation.
- $\chi_{\nu}^2()$ is the Chi-square probability distribution function.
- $\nu$ is the degrees of freedom for the Chi-square distribution.
- $T$ is the number of non-missing values in the sample data.
Using a given data sample, the sample data standard deviation is computed as: $$ \hat \sigma(x) = \sqrt{\frac{\sum_{t=1}^T(x_t-\bar x)^2}{T-1}}$$ Where:
- $\hat \sigma(x)$ is the sample standard deviation.
- $\bar x$ is the sample average.
- $T$ is the number of non-missing values in the data sample.
The underlying population distribution is assumed normal (gaussian).
This is a two-sides (i.e., two-tails) test, so the computed p-value should be compared with half of the significance level ($\alpha/2$).