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:

 
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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)

Files Examples

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