Tables
Decimals:
The Standard Normal Distribution: \( N(0,1) \) The zvalue and the pvalue to be entered in the table below are defined to satisfy the following:$$ p = 2 \ \cdot P(Z > z) = 2 \ \cdot \int_{z}^{\infty} \phi(z) \: dz $$Where \( \phi(z) \) is the standard normal distribution with mean 0 and standard deviation 1:$$ \phi(z) = \frac{1}{\sqrt{2\pi}} \ \cdot \text{e}^{\frac{1}{2} z^2} $$In other words: the pvalue is two times the area under the bell shaped curve from z to infinity. The number z is the numerical (positive) value of z. If, for example, the zvalue is z = 1.43, then the number to be entered into the table is z = 1.43 = 1.43. And if z = 2.27, then z = 2.27 = 2.27.For example:$$ 0.05 = 2 \ \cdot 0.025 = 2 \ \cdot P(Z > 1.96) = 2 \ \cdot \int_{1.96}^{\infty} \phi(z) \: dz $$The pvalue is the twosided (or twotailed) area under the bell curve; one area from each "tail". The two areas are the same, since the function is symmetric. If you only need (or only have) half of the pvalue, the table can still be used; just enter the z value and get the p and ½p values. Or enter either p or ½p to get the other two 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 bell shaped curve from x_{1} to x_{2} then the p value is defined as follows:$$ p = P(x_1 < X < x_2) = P(X < x_2)  P(X < x_1) = \int_{x_1}^{x_2} f(x) \: dx $$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 following areas;$$ p = \int_{\infty}^{x_2} f(x) \: dx \:\: \color{blue}{\text{or}} \:\: \color{black}{p = \int_{x_1}^{\infty} f(x) \: dx} $$then you can type either "INF" as the x_{1} value or "INF" as the x_{2} value.


The F Distribution An ftest is performed to test whether the variances of two samples can be assumed equal. In other words whether the null hypothesis is rejected or not rejected. The null hypothesis states that H0: Var_{1} = Var_{2} , where Var_{1} = (SD_{1})^{2} is the variance of the first sample and Var_{2} = (SD_{2})^{2} is the variance of the second sample. Consequently, \( SD_1 = \sqrt{Var_1} = \sqrt{(SD_1)^2}\) is the standard deviation of the first sample. And \( SD_2 = \sqrt{Var_2} = \sqrt{(SD_2)^2}\) is the standard deviation of the second sample involved in the ftest. Another way of stating the H0 hypothesis is that the ratio of the variances is 1:$$ H_0 : \frac{(SD_1)^2}{(SD_2)^2} = 1 \:\: \text{(or)} \:\: H_0 : \frac{(SD_2)^2}{(SD_1)^2} = 1 $$The f_{1}value, the f_{2} value and the corresponding p_{1} and p_{2} values are defined as follows:$$ f_1 = \frac{(SD_1)^2}{(SD_2)^2} , f_2 = \frac{(SD_2)^2}{(SD_1)^2} = \frac{1}{f_1} $$The number p_{1} is the smallest of the two following numbers:$$ p = \int_{f_1}^\infty{f(x)} \: dx \: \: \: \: \: \: \text{and} \:\:\:\:\:\: 1p $$The number p_{2} is the smallest of the two following numbers:$$ p = \int_{f_2}^\infty f(x) \: dx \:\:\:\:\:\: \text{and} \:\:\:\:\:\: 1p $$The overall pvalue in an Ftest, taking into consideration both f_{1} and f_{2}, is therefore the following number:$$ p_{total} = p_1 + p_2 $$The ftest starts by assuming that H0 is in fact true, namely that Var_{1} = Var_{2}. The overall pvalue p_{total}
(a number between 0 and 1) is the probability of getting as big a deviation from the hypothetical fvalue of 1, or an even bigger deviation, as the one being observed.
For the start assumption was that H0 is true; meaning that both fvalues should be equal to 1, or at least close to 1, in correspondance with the assumption.
The function \( f(x) \) is the probability density function of the Fdistribution defined as:$$ f(x) = \frac{df_1^{\frac{df_1}{2}} \cdot df_2 ^\frac{df_2}{2} \cdot \Gamma\left(\frac{df_1 + df_2}{2}\right) \cdot x^{\frac{df_1}{2}1}}{\Gamma\left(\frac{df_1}{2}\right) \cdot \Gamma\left(\frac{df_2}{2}\right) \cdot \left(df_2 + df_1 \cdot x\right)^{\frac{df_1 + df_2}{2}}} , $$where DF_{1} = N_{1}  1 is the degree of freedom of the first sample and DF_{2} = N_{2}  1 is the degree of freedom of the second sample. For more details, go to page "Formulas" in the menu.

Fdistribution table in the case of known standard deviations  

SD_{1}  SD_{2}  DF_{1}  DF_{2}  f_{1}  f_{2}  p_{1}  p_{2}  p_{total} 
Use the following table if you have the variances Var_{1} = (SD_{1})^{2} and
Var_{2} = (SD_{2})^{2} from the two samples:
Fdistribution table in the case of known variances  

(SD_{1})^{2}  (SD_{2})^{2}  DF_{1}  DF_{2}  f_{1}  f_{2}  p_{1}  p_{2}  p_{total} 
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If you only need to find a specific pvalue corresponding to a particular fvalue (or the other way around), in other words an f and a p value satisfying that
$$ p = P(X > f) = \int_{f}^\infty{f(x)} \: dx , $$you can then instead use the table below over fdistribution values given DF_{1} and DF_{2} and either p or f:
Fdistribution table  

DF_{1}  DF_{2}  f  p 
The TDistribution
The tvalue, the DFnumber and the corresponding pvalue 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 Tdistribution with DF degrees of freedom:
$$ f(x) = \frac{\Gamma{\left( \frac{df+1}{2}\right)}}{\sqrt{\pi \cdot df} \cdot \Gamma{\left(\frac{df}{2}\right)} \cdot \left( 1 + \frac{x^2}{df}\right)^{(df+1)/2}} $$t is the numerical (positive) value of the tvalue.
You can input either the twosided (twotailed) pvalue in order to get the t value and the onesided ½p value. Or you can input the onesided ½p value
and get the t value and the pvalue. Or input the t value and get p as well as ½p.
For more details, go to the page "Formulas" in the menu.
Tdistribution table  

tvalue  DF  p (twosided)  \( \frac{1}{2} \) p (onesided) 
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If you are looking for the areal under the bell curve of the probability density function of the TDistribution between two tvalues t_{1} and t_{2}, you can use the table below instead:
The pvalue will then be the integral:
$$ p = P(t_1 < T < t_2) = \int_{t_1}^{t_2} f(x) \: dx , $$where \( f(x) \) is the probability density function of the Tdistribution with DF degrees of freedom (see above)
If t_{1} is \(\infty\) (minus infinity) you can write "inf" as input under t_{1}.
If t_{2} is \(\infty\) (infinity) you can write "inf" as input under t_{2}.
The pvalues would then in those cases be:
$$ p = P(\infty < T < t_2) = \int_{\infty}^{t_2} f(x) \: dx \:\: \color{blue}{\text{or}} \:\: \color{black}{p = P( t_1 < T < \infty) = \int_{t_1}^{\infty} f(x) \: dx} $$TDistribution table  

DF  t_{1}  t_{2}  pvalue  
The ChiSquareDistribution (\(\chi^2\))
The χ^{2}  value, the DFnumber and the corresponding pvalue 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 chisquaredistribution with DF degrees of freedom:
$$ f(x) = \frac{x^{\frac{1}{2} \cdot df  1} \cdot \text{e}^{\frac{1}{2} \cdot x}}{2^{\frac{1}{2} \cdot df} \cdot \Gamma\left( \frac{df}{2}\right)} $$In other words the pvalue is the area under the curve from χ^{2} to infinity. For more details, go to the page "Formulas" in the menu.
ChiSquareDistribution table  

\( \chi^2 \) value  DF  pvalue 