# TEST_STDEV - Standard Deviation Test

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.
Method Description
1 P-Value.
2 Test Statistics (e.g., Z-score).
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.

## Remarks

1. The data sample may include missing values (e.g., #N/A).
2. 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.
3. 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.
4. 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.
5. The underlying population distribution is assumed normal (gaussian).
6. This is a two-sides (i.e., two-tails) test, so the computed p-value should be compared with half of the significance level ($\alpha/2$).