MedicalStat

Decimals:

## The Standard Normal Distribution:$$X \text{~} N(0,1)$$

### The z-value and the p-value to be entered in the table below are defined to satisfy the following:

$$p = 2 \ \cdot P(Z > |z|) = 2 \ \cdot \int_{|z|}^{\infty} f(x) \: dx$$

### Where $$f(x)$$ is the standard normal distribution with mean 0 and standard deviation 1:

$$f(x) = \frac{1}{\sqrt{2\pi}} \ \cdot \text{e}^{-\frac{1}{2} x^2}$$

### For example:

$$0.05 = 2 \ \cdot 0.025 = 2 \ \cdot P(Z > 1.96) = 2 \ \cdot \int_{1.96}^{\infty} f(x) \: dx$$
Enter either z or p to get the other
Standard Normal distribution X ~ N(0, 1)
z-values p-values

## The Normal Distribution:$$X \text{~} N(μ , σ)$$

### If you have a general normal distribution with mean μ and standard deviation σ, and need the area under the curve from z1 to z2 then the p value is defined as follows:

$$p = P(z_1 < Z < z_2) = P(Z < z_2) - P(Z < z_1) = \int_{z_1}^{z_2} f(x) \: dx$$

### Where $$f(x)$$ is (in this case) the normal distribution with mean μ and standard deviation σ :

$$f(x) = \frac{1}{\sqrt{2\pi \sigma^2}} \ \cdot \text{e}^{-\frac{(x - \mu)^2}{2 \sigma^2}}$$

### If you need the lower bound to be minus infinity ( $$-\infty$$ ) or the upper bound to be infinity ($$\infty$$), in other words if you need one of the areas

$$\int_{-\infty}^{z_2} f(x) dx \:\: \color{blue}{\text{or}} \:\: \color{black}{\int_{z_1}^{\infty} f(x) dx}$$

### you can can type either "-INF" as z1 or "INF" as z2.

Two of the four inputs must be μ and σ in order to get the fifth
Normal distribution X ~ N(μ, σ)
Mean (μ) SD (σ) z1 z2 p-value

## The F Distribution

### The number p1 is the smallest of the two following numbers:

$$p = \int_{f_1}^\infty{f(x)} \: dx \: \: \: \: \: \: \text{and} \:\:\:\:\:\: 1-p$$

### The number p2 is the smallest of the two following numbers:

$$p = \int_{f_2}^\infty f(x) \: dx \:\:\:\:\:\: \text{and} \:\:\:\:\:\: 1-p$$

### The overall p-value in an F-test checking for same variances $$(SD_{1})^2 = (SD_{2})^2$$ , taking into account both f1 and f2, is the following number:

$$p_{total} = p_1 + p_2$$

### The function $$f(x)$$ is the probability density function of the F-distribution defined as:

$$f(x) = \frac{df_1^{\frac{df_1}{2}} \cdot df_2 ^\frac{df_2}{2} \cdot \Gamma\left(\frac{df_1 + df_2}{2}\right) \cdot x^{\frac{df_1}{2}-1}}{\Gamma\left(\frac{df_1}{2}\right) \cdot \Gamma\left(\frac{df_2}{2}\right) \cdot \left(df_2 + df_1 \cdot x\right)^{\frac{df_1 + df_2}{2}}}$$

### For more details, go to chapter "Formulas" in the menu.

Enter the four inputs SD1, SD2, DF1 and DF2 to get the remaining five
F-distribution
SD1 SD2 DF1 DF2     f1         f2         p1         p2     ptotal

### If you only need to find a specific p-value corresponding to a particular f-value (or the other way around), in other words an f and p value satisfying that

$$p = P(X > f) = \int_{f}^\infty{f(x)} \: dx ,$$

### (notice: the p-values in this table are not automatically the smallest of the two numbers

$$p = \int_{f}^\infty{f(x)} \: \: \color{blue}{\text{and}} \: \: \color{black}{1-p}$$
Two of the three inputs MUST be DF1 and DF2 in order to get the fourth.
F-distribution
DF1 DF2     f         p

## The T-Distribution

### The t-value, the DF-number and the corresponding p-value are defined as follows:

$$p = 2 \ \cdot P(T > |t|) = 2 \ \cdot \int_{|t|}^{\infty} f(x) \: dx$$

### Where $$f(x)$$ is the probability density function of the T-distribution with DF degrees of freedom:

$$f(x) = \frac{\Gamma{\left( \frac{df+1}{2}\right)}}{\sqrt{\pi \cdot df} \cdot \Gamma{\left(\frac{df}{2}\right)} \cdot \left( 1 + \frac{x^2}{df}\right)^{(df+1)/2}}$$

### For more details, go to chapter "Formulas" in the menu.

One of the two inputs MUST be DF to get the third
T-distribution
t-value DF p-value

### The p-value will then be the integral:

$$p = P(t_1 < T < t_2) = \int_{t_1}^{t_2} f(x) \: dx$$

### The p-values would then in those cases be:

$$p = P(-\infty < T < t_2) = \int_{-\infty}^{t_2} f(x) \: dx \:\: \color{blue}{\text{or}} \:\: \color{black}{p = P( t_1 < T < \infty) = \int_{t_1}^{\infty} f(x) \: dx}$$
One of the three inputs MUST be DF in order to get the fourth
T-Distribution
DF t1 t2 p-value

## The Chi-Square-Distribution ($$\chi^2$$)

### The χ2 - value, the DF-number and the corresponding p-value are defined as follows:

$$p = P(X > \chi^2) = \int_{ \chi^2 }^{\infty} f(x) \: dx$$

### Where $$f(x)$$ is the probability density function of the chi-square-distribution with DF degrees of freedom:

$$f(x) = \frac{x^{\frac{1}{2} \cdot df - 1} \cdot \text{e}^{-\frac{1}{2} \cdot x}}{2^{\frac{1}{2} \cdot df} \cdot \Gamma\left( \frac{df}{2}\right)}$$

### For more details, go to chapter "Formulas" in the menu.

One of the two inputs MUST be DF to get the third
Chi-Square-Distribution
$$\chi^2$$ value DF p-value