# Comparison of noncentral and central distributions

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## Online calculator for critical values, cumulative probabilities, and critical noncentral parameters

### Input

### Results of critical statistic, cumulative probability, and critical noncentral parameter

*z* and noncentral distributions (χ^{2},*t*, and *F*)

### Noncentral χ^{2}

Let *Z*_{i},*i*=0,1,2,... denote a series of independent random variables of standard normal distribution.

will be a random variable of χ^{2} distribution with *df* degrees of freedom. For any given series of constants μ_{i},*i*=1,2,...,*df*,

will be a random variable of the respective noncentral χ^{2} distribution with the same *df* and the distinct noncetral parameter

It is different from the random variable of the respective central χ^{2} distribution with a central drift.

### Noncentral *t*

For any given constant μ_{0},

is a random variable of noncentral t-distribution with noncentrality parameter

*n**c**p*= μ_{0},

which is different from , the central t-distributed random variable drifted with the same mean.

If *df* on this display is set to (*Inf* in R) and noncentral parameter set to 0, a standard normal distribution will be plotted and critical *z* score calculated.

### Noncentral *F*

The noncentral parameter of *F* is only defined on its numerator. The noncentral *F* distributed

with noncentral parameter

- ncp=

is different from the central *F* distributed random variable plus the respective constant .

## Confidence interval of standardized effect size by noncentral parameters

Confidence interval of unstandardized effect size like difference of means (μ_{1} − μ_{2}) can be found in common statistics textbooks and software, while confidence intervals of standardized effect size, especially Cohen's and , rely on the calculation of confidence intervals of noncentral parameters (*ncp*).

A common method to find confident interval limits of *ncp* is to solve the critical *ncp* value for marginal extreme quantile. The *ncp* parameter of the black curve in the above diagram could be directly adopted. For example, can be 97.5% one-way confidence interval of *ncp* if observed *t*_{df = 4} = 5.1, while change quantile from .025 to .975, we shall find that the two-way interval (1.139, 8.968) can be of 95% confidence level.

### *T* test for mean difference of single group or two related groups

In case of single group, *M* (μ) denotes the sample (population) mean of single group , and *SD* (σ) denotes the sample (population) standard deviation. *N* is the sample size of the group. *T* test is used for the hypothesis on the difference between mean and a baseline μ_{baseline}. Usually, μ_{baseline} is zero, while not necessary. In case of two related groups, the single group is constructed by difference in each pair of samples, while *SD* (σ) denotes the sample (population) standard deviation of differences rather than within original two groups.

- and Cohen's is the point estimate of .

So,

- .

### *T* test for mean difference between two independent groups

*n*_{1} or *n*_{2} is sample size within the respective group.

- , wherein .

- and Cohen's is the point estimate of .

So,

- .

### One-way ANOVA test for mean difference across multiple independent groups

One-way ANOVA test applies noncentral F distribution. While with a given population standard deviation σ, the same test question applies noncentral chi-square distribution.

For each *j*-th sample within *i*-th group *X*_{i,j}, denote .

While,

So, both *ncp*(s) of *F* and χ^{2} equate

- .

In case of for *K* independent groups of same size, the total sample size is .

- .

*T*-test of pair of independent groups is a special case of one-way ANOVA. Note that noncentral parameter *n**c**p*_{F} of *F* is not comparable to the noncentral parameter *n**c**p*_{t} of the corresponding *t*. Actually, , and in the case.

### RMSEA of Structural Equation Model

*ncp* of χ^{2} reported by Structural Equation Model softwares is proportional to the population value of *R**M**S**E**A*^{2}, or the squared distance per *df* from population var-cov matrix to the model space.

## Power vs. Standardized Effect Size or *ncp*

### Power of *t* test for a given Cohen's δ

#### Example of one-group mean test

##### Input

##### Results

#### Two-related-group mean test

For two-related-group case, the difference scores between each pair of samples can apply one-group mean test interface. Usually μ_{null} is set to zero.

### Power of *F* test for a given Cohen's

Let's use denote the population of Cohen's *f*^{2}, specially

in one-way ANOVA of *K* groups setup with within-group sample size *n* and within-group population mean respectively. The noncentral parameter of the corresponding *F* or χ^{2} distribution is .

### Power of SEM close-fit test for a given RMSEA

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## External links

- Noncentral
*t*-distribution on Wikipedia - Noncentral χ
^{2}distribution on Wikipedia - Noncentral
*F*-distribution on Wikipedia - Confidence interval of Effect Size on Wikipedia