## t-Values Calculator

- This tool computes
*t*-values for*Student's t*hypothesis testing. - To use the tool, just submit a
*p*-value and its degrees of freedom, υ. - This tool comes handy when you don't have around statistical
*t*-tables or are working with*p*and*t*values or degrees of freedom not available from such tables. Avoid pausing a problem for annoying linear interpolation workarounds!

**About***t*-values

A*t*-value is used for null hypothesis (*H*) significance testing and as follows._{0}We first define

*H*, a critical_{0}*p*-value (*p*), and the degrees of freedom υ. With_{critic}*p*and υ, a critical_{critic}*t*-value (*t*) is determined and then compared against a sample-calculated_{critic}*t*-value (*t*), such that_{sample}*H*is rejected if_{0}*t*_{sample}> t_{critic}*H*is not rejected if_{0}*t*_{sample}< t_{critic}

Statistical tables can be used to find a

*t*. These tables are compiled by solving the Inverse Cumulative Distribution Frequency (ICDF) of the_{critic}*Student's t Distribution*. Solving the ICDF is done through numerical approximation algorithms (Hill, 1970; Shaw, 2005; Shaw 2006). These are precisely the types of algorithms used by our tool to generate a*t*._{critic}If

*H*is rejected, the experiment was conclusive at the critical confidence level selected, meaning the data is sufficiently inconsistent with_{0}*H*. By contrast, if_{0}*H*is not rejected, the experiment is inconclusive at the chosen confidence level, so the data is sufficiently consistent with_{0}*H*._{0}Evidently,

*t*suggests that the observed data is so sufficiently inconsistent with_{sample}> t_{critic}*H*that it may be rejected. On the other hand,_{0}*t*does not prove that_{sample}< t_{critic}*H*can be accepted. It only means that it cannot be rejected with the sample data examined. We can do better by formulating an alternative hypothesis known to be true,_{0}*H*so if_{1}*H*is rejected, we could accept_{0}*H*._{1}However, to accept

*H*, we must consistently reject_{1}*H*. This can be done by increasing the statistical power of the test, by increasing_{0}*p*then decreasing_{critic}*t*. Thus, if one end rejecting_{critic}*H*more often, there is a greater chance of safely accepting_{0}*H*(Trochim, 2006)._{1}**Calculating***t*-values for*t*-testing Hypotheses

Given a*p*-value and υ,*t*-values can be computed from the Inverse Cumulative Distribution (ICDF) of the*Student's t Distribution*by the following procedure (Wikipedia, 2017a; 2017b; 2017c).We first define the Probability Distribution Function (PDF)

(1)

Integrating (1), we obtain the Cumulative Distribution Function (CDF)

(2)

Inverting it, the Quantile Function or Inverse CDF is obtained.

(3)

where

- Γ stands for a gamma function.
_{2}*F*_{1}stands for a hypergeometric function with two parameters in its numerator and one in its denominator.*I*^{-1}is the inverse of the regularized form of an incomplete beta function.

In (3),

*n = υ*and*u = p*.Abramowitz and Stegun (1972; see equations 26.7.3, 26.7.4, and 26.7.5), Hill (1970), Shaw (2005, 2006), and Culham (2004) have described these functions extensively.

For the reverse calculations, computing

*p*-values from*t*-values, you may want to use our p-values Calculator.In addition, our t,p & Effect Size Estimator tool allows you to easily move back-and-forth between calculations, then between CDF and ICDF transforms.

- Anyone that need to compute
*t*-values for*t*-test hypothesis testing.

- Calculate a
*t*for a one-tailed test at the 95% level of confidence and 15 degrees of freedom._{critic}

Hint:*p*_{critic, two-tailed}=*2p*_{critic, one-tailed} - Repeat previous exercise, this time for a two-tailed test and at the same confidence level. Compare results.
- From 25 pairs of measurements from a population you find a Pearson correlation coefficient equal to
*r*= 0.466, where at the 95% confidence levels (two-tailed) the following hypotheses are formulated:

Can we reject*H*: "The correlations are not statistically different from zero."_{0}*H*: "A significant correlation exits in the population."_{1}*H*?_{0}

Hint. Calculate first*t*from_{sample}*r*. - In the above case: Compute a range of
*t*values for safely accepting_{critic}*H*._{1}

- Abramowitz, M. and Stegun, I. A. (1972).
Handbook of Mathematical Functions.

p. 948-949. National Bureau of Standards. See also full pdf. - Culham, J. R. (2004).Factorial, Gamma and Beta Functions.
- Hill, G. W. (1970). Algorithm 396: Student's t-Quantiles. Communications of the ACM 13(10):619-620. ACM Press, October, 1970.
- Shaw, W. T. (2006). Sampling Student's
T distribution - Use of the inverse cumulative distribution function.

Computational Finance Notes 2011-12, JCF Volume 9/Number 4, Summer 2006. Journal of Computational Finance. 9 (4): 37-73. - Shaw, W. T. (2005). New Methods for Simulating the Student t-Distribution - Direct Use of the Inverse Cumulative Distribution. The Mathematical Institutem University of Oxford, Eprints Archive.
- Trochim, W.M.K. (2006). Statistical Power.
- Wikipedia (2017a). p-value.
- Wikipedia (2017b). Student's t-distribution.
- Wikipedia (2017c). Quantile function.

#### Feedback

Contact us for any suggestion or question regarding this tool.