# 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 devation. 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 ($\alpha/2$).

## Examples

Example 1:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
A B
Date Data
1/1/2008 #N/A
1/2/2008 -2.83
1/3/2008 -0.95
1/4/2008 -0.88
1/5/2008 1.21
1/6/2008 -1.67
1/7/2008 0.83
1/8/2008 -0.27
1/9/2008 1.36
1/10/2008 -0.34
1/11/2008 0.48
1/12/2008 -2.83
1/13/2008 -0.95
1/14/2008 -0.88
1/15/2008 1.21
1/16/2008 -1.67

Formula Description (Result)
=STDEV($B$2:$B$17) Sample standard deviation (1.3725)
=TEST_STDEV($B$2:$B$17,1) p-value of the test when standard deviation = 1 (0.0232)