Methods and formulas for 1 Variance

The standard deviation is the most common measure of dispersion, or how spread out the data are about the mean. The sample standard deviation is equal to the square root of the sample variance.

If the column contains x₁, x₂,..., x_N, with mean , then the standard deviation is given by:

Notation

Term	Description
xi	the ith observation in your sample
	the sample mean
S	the sample standard deviation
n	sample size

The variance measures how spread out the data are about their mean. The variance is equal to the standard deviation squared.

Formula

Notation

Term	Description
xi	ith observation
	mean of the observations
N	number of nonmissing observations

Confidence intervals

A 100(1 - α)% confidence interval for the population standard deviation is given by:

A 100(1 - α)% confidence interval for the population variance is given by:

Confidence bounds

When you specify a one-sided test, Minitab calculates a one-sided 100(1–α)% confidence bound, according to the direction of the alternative hypothesis.

• If you specify a "greater than" alternative hypothesis, a 100(1–α)% lower bound for the population standard deviation is given by:

  

  A 100(1–α)% lower bound for the population variance is given by:

  

• If you specify a "less than" alternative hypothesis, a 100(1–α)% upper bound for the population standard deviation is given by:

  

  A 100(1–α)% upper bound for the population variance is given by:

  

Notation

Use this method for any continuous data (normal or nonnormal). ¹

Confidence interval

A 100(1-α)% confidence interval for the population standard deviation is given by:

A 100(1-α)% confidence interval for the population variance is given by:

Confidence bounds

When you specify a one-sided test, Minitab calculates a one-sided 100(1–α)% confidence bound, according to the direction of the alternative hypothesis.

• If you specify a "greater than" alternative hypothesis, a 100(1–α)% lower bound for the population standard deviation is given by:

  

  An approximate 100(1- a)% lower bound for the population variance is given by:

  

• If you specify a "less than" alternative hypothesis, an approximate 100(1 – α)% upper bound for the population standard deviation is given by:

  

  An approximate 100(1- a)% upper bound for the population variance is given by:

  

Notation

Term	Description
α	1 – confidence level / 100
cα/2	n / (n – zα/2)
cα	n / (n – zα )
s2	observed value of the sample variance
zα/2	inverse cumulative probability of the standard normal distribution at 1 – α/2. If n is less than or equal to zα/2, Minitab does not calculate Bonett confidence intervals.
zα	inverse cumulative probability of the standard normal distribution at 1 – α. If n is less than or equal to zα , Minitab does not calculate Bonett confidence intervals.
se
	= estimated excess kurtosis
m	trimmed mean with trim proportion equal to ; m = sample mean when n is less than or equal to 5
σ	true value of the population standard deviation
σ2	true value of the population variance

Formula

The hypothesis test uses the following p-value equations for the respective alternative hypotheses:

H₁: σ² > σ₀²: p-value = P(Χ² ≥ x²)

H₁: σ² < σ₀²: p-value = P(Χ² ≤ x²)

H₁: σ² ≠ σ₀²: p-value = 2 × min{P(Χ² ≤ x²), P(Χ² ≥ x²)}

Notation

Term	Description
σ2	true value of the population variance
σ02	hypothesized value of the population variance
Χ2	follows a chi-square distribution with (n – 1) degrees of freedom when σ2 = σ02
x2	Term Description S2 observed value of the sample variance n sample size

Formula

The Bonett procedure is not associated with a test statistic. However, Minitab uses the rejection regions defined by the confidence limits to calculate a p-value.

For a two-sided hypothesis, the p-value is given by:

p = 2 × min(α_L, α_U)

• For a one-sided "less than" alternative hypothesis, the p-value is calculated as α_U after replacing α/2 with α in the notation.
• For the one-sided "greater than" alternative hypothesis, the p-value is calculated as α_L after replacing α/2 with α in the notation.

Notation

Term	Description
σ02	hypothesized variance
αL	smallest solution, α, of the equation
αU	smallest solution, α, of the equation
cα/2	n / (n – zα/2)
α	1 – confidence level / 100
s2	observed value of the sample variance
zα/2	inverse cumulative probability of the standard normal distribution at 1 – α/2. If n is less than or equal to zα/2, Minitab does not calculate Bonett confidence intervals.
se	Term Description = estimated excess kurtosis m trimmed mean with trim proportion equal to ; m = 0 when n is less than or equal to 5