13.6 Chapter 13 Formulas

Ranking Data Order the data from smallest to largest. The smallest value gets a rank of 1. The next smallest gets a rank of 2, etc. If there are any values that tie, then each of the tied values gets the average of the corresponding ranks.	Sign Test $H_{0}:$ Median $= MD_{0}$ $H_{1}:$ Median $\neq MD_{0}$ p-value uses binomial distribution with $p = 0.5$ and $n$ is the sample size not including ties with the median or differences of 0. For a two-tailed test, the test statistic, $x$ , is the smaller of the plus or minus signs. If $x$ is the test statistic, the p-value for a two-tailed test is $2* \text{P}(X \leq x)$ . For a right-tailed test, the test statistic, $x$ , is the number of plus signs. For a left-tailed test, the test statistic, $x$ , is the number of minus signs. The p-value for a one-tailed test is the $\text{P}(X \geq x)$ for a right-tailed test, or $\text{P}(X \leq x)$ for a left-tailed test.
Wilcoxon Signed-Rank Test $n$ is the sample size not including a difference of 0. When $n < 30$ , use test statistic $w_{s}$ , which is the absolute value of the smaller of the sum of ranks. CV uses table in Figure 13-5. If critical value is not in tables then use an online calculator: http://www.socscistatistics.com/tests/signedranks. When $n \geq 30$ , use z-test statistic: $z = \frac{\left(w_{s} - \left(\dfrac{n (n+1)}{4}\right) \right)}{\sqrt{\left( \dfrac{n(n+1)(2n+1)}{24} \right)}} \nonumber$	Mann-Whitney U Test When $n_{1} \leq 20$ and $n_{2} \leq 20$ : $U_{1} = R_{1} - \frac{n_{1} \left(n_{1}+1\right)}{2}, U_{2} = R_{2} - \frac{n_{2} \left(n_{2}+1\right)}{2}$ . $U = \text{Min} \left(U_{1}, U_{2}\right)$ CV uses table in Figures 13-8 or 13-9. If critical value is not in tables then use an online calculator: https://www.socscistatistics.com/tests/mannwhitney/default.aspx. When $n_{1} > 20$ and $n_{2} > 20$ use z-test statistic: $z = \frac{\left(U - \left(\dfrac{n_{1} \cdot n_{2}}{2}\right) \right)}{\sqrt{\left( \dfrac{n_{1} \cdot n_{2} \left(n_{1} + n_{2} + 1\right)}{12} \right)}} \nonumber$

Ranking Data Order the data from smallest to largest. The smallest value gets a rank of 1. The next smallest gets a rank of 2, etc. If there are any values that tie, then each of the tied values gets the average of the corresponding ranks.

Sign Test

H_{0}:

Median

= MD_{0}

H_{1}:

Median

\neq MD_{0}

p-value uses binomial distribution with

p = 0.5

and

n

is the sample size not including ties with the median or differences of 0. For a two-tailed test, the test statistic,

x

, is the smaller of the plus or minus signs. If

x

is the test statistic, the p-value for a two-tailed test is

2* \text{P}(X \leq x)

. For a right-tailed test, the test statistic,

x

, is the number of plus signs. For a left-tailed test, the test statistic,

x

, is the number of minus signs. The p-value for a one-tailed test is the

\text{P}(X \geq x)

for a right-tailed test, or

\text{P}(X \leq x)

for a left-tailed test.

Wilcoxon Signed-Rank Test

n

is the sample size not including a difference of 0. When

n < 30

, use test statistic

w_{s}

, which is the absolute value of the smaller of the sum of ranks. CV uses table in Figure 13-5. If critical value is not in tables then use an online calculator: http://www.socscistatistics.com/tests/signedranks. When

n \geq 30

, use z-test statistic:

$z = \frac{\left(w_{s} - \left(\dfrac{n (n+1)}{4}\right) \right)}{\sqrt{\left( \dfrac{n(n+1)(2n+1)}{24} \right)}} \nonumber$

Mann-Whitney U Test When

n_{1} \leq 20

and

n_{2} \leq 20

U_{1} = R_{1} - \frac{n_{1} \left(n_{1}+1\right)}{2}, U_{2} = R_{2} - \frac{n_{2} \left(n_{2}+1\right)}{2}

U = \text{Min} \left(U_{1}, U_{2}\right)

CV uses table in Figures 13-8 or 13-9. If critical value is not in tables then use an online calculator: https://www.socscistatistics.com/tests/mannwhitney/default.aspx. When

n_{1} > 20

and

n_{2} > 20

use z-test statistic:

$z = \frac{\left(U - \left(\dfrac{n_{1} \cdot n_{2}}{2}\right) \right)}{\sqrt{\left( \dfrac{n_{1} \cdot n_{2} \left(n_{1} + n_{2} + 1\right)}{12} \right)}} \nonumber$

“For instance, a race of hyperintelligent pan‐dimensional beings once built themselves a gigantic supercomputer called Deep Thought to calculate once and for all the Answer to the Ultimate Question of Life, the Universe, and Everything. For seven and a half million years, Deep Thought computed and calculated, and in the end announced that the answer was in fact Forty‐two - and so another, even bigger, computer had to be built to find out what the actual question was.”

(Adams, 2002)

Adapted from Mostly Harmless Statistics by Rachel Webb (Portland State University, https://mostlyharmlessstat.wixsite.com/webpage), © Rachel Webb, licensed under CC BY-SA 4.0. Changes were made. License: CC-BY-SA-4.0.