Take a little break, but then put your foot to the pedal– last chapter!
Summary of Formulas/Tests (Chapter 13 Nonparametric Statistics)
1. Sign Test
Claims involving matched pairs of sample data:
H0: There is no difference. (The median of the differences is equal to 0.)
H1: There is difference. (The median of the difference is not equal to 0.)
Claims involving nominal data:
H0: p = 0.5 (The proportion of girls is equal to 0.5.)
H1: p ≠ 0.5
Claims about the median of a single population:
H0: Median is equal to 98.6℉.
H1: Median is less than 98.6℉.
Test statistic:
For 𝑛 ≤ 25: x (the number of times the less frequent sign occurs)
For 𝑛 > 25: 𝑧 =

Notation:
𝑛
2
(𝑥+𝑜.5)−( )
√𝑛
2
X: the number of times the less frequent sign occurs
n: the total number of positive and negative signs combined
Critical values:
For 𝑛 ≤ 25: critical x values are found in Table A-7
For 𝑛 > 25: critical z values are found in Table A-2.
Requirements:
1. The sample data have been randomly selected.
2. There is no requirement that the sample data come from a population with a
particular distribution, such as a normal distribution.
2. Wilcoxon Signed-Ranks Test for Matched Pairs
H0: The matched pairs have differences that come from a population with a median
equal to zero.
H1: The matched pairs have differences that come from a population with a nonzero
median.
Test statistic:
1. If 𝑛 ≤ 30, the test statistic is T.
2.
If 𝑛 > 30, the test statistic is 𝑧
𝑇−
=
√
𝑛(𝑛+1)
4
𝑛(𝑛+1)(2𝑛+1)
24
Notation:
T: the smaller of the following two sums:
1. The sum of the absolute values of the negative ranks of the nonzero
differences d
2. The sum of the positive ranks of the nonzero differences d
Critical values:
n is the number of pairs of data for which the difference d is not 0.
If 𝑛 ≤ 30, the critical T value is found in Table A-8. (Note: Reject the null hypothesis
if the test statistic T is less than or equal to the critical value found in A-8).
If 𝑛 > 30, the critical z values are found in Table A-2.
Requirements:
1. The data consist of matched pairs that have been randomly selected.
2. The population of differences (found from the pairs of data) has a
distribution that is approximately symmetric. (There is no requirement that
the data have a normal distribution.)
Note: Sign Test vs. Wilcoxon Signed-Ranks Test for Matched Pairs
Both sign test and Wilcoxon signed-ranks test are nonparametric tests. Sign Test uses
plus and minus signs to test different claims, including: claims involving matched
pairs of sample data, claims involving nominal data and claims about the median of
a single population. The Wilcoxon signed-Ranks test uses ranks of sample data
consisting of matched pairs. It is used to test null hypothesis that the population of
differences has a median of zero.
The sign test can also be used with matched pairs, but the sign test uses only the
signs of the differences. By using ranks, the Wilcoxon signed-ranks test takes the
magnitudes of the differences into account. Because the Wilcoxon signed-ranks test
incorporates and uses more information than the sign test, it tends to yield
conclusions that better reflect the true nature of the data.
3. Wilcoxon Rank-Sum Test for two independent samples
H0: The two samples come from populations with equal medians.
H1: The two samples come from populations with different medians.
Test statistic:
𝑛1 (𝑛1 + 𝑛2 + 1)
𝜇𝑅 =
2
𝑛1 𝑛2 (𝑛1 + 𝑛2 + 1)
𝜎𝑅 = √
12
𝑧=
𝑅 − 𝜇𝑅
𝜎𝑅
Critical values: Critical values can be found in Table A-2.

Notation:
𝑛1 : size of sample 1
𝑛2 : size of sample 2
𝑅1 : sum of ranks for sample 1
𝑅2 : sum of ranks for sample 2
𝑅: same as 𝑅1
𝜇𝑅 : mean of the sample R values that are expected when the two populations have
equal medians
𝜎𝑟 : standard deviation of the sample R values that is expected with two populations
having equal medians
Requirements:
1. There are two independent samples of randomly selected data.
2. Each of the two samples has more than 10 values.
3. There is no requirement that the two populations have a normal distribution or
any other particular distribution.
4. Kruskal-Wallis Test
H0: The samples come from populations with equal medians.
H1: The samples come from populations with medians that are not all equal.
Test statistic:
12
𝑅12 𝑅22
𝑅𝑘2
𝐻=
( +
+ ⋯ + ) − 3(𝑁 + 1)
𝑁(𝑁 + 1) 𝑛1 𝑛2
𝑛𝑘
Critical Values: df = k – 1 where k is the number of different samples. Use Table A-4.

Notation:
N: total number of observations in all samples combined
K: number of samples
𝑅1 : sum of ranks for sample 1
𝑛1 : number of observations in sample 1
Requirements:
1. We have at least three independent samples, all of which are randomly
selected.
2. Each sample has at least five observations.
3. There is no requirement that the populations have a normal distribution or any
other particular distribution.
Note: Both Wilcoxon Rank-Sum Test and Kruskal-Wallis Test need to temporarily
combine all samples into one big sample and assign a rank to each sample
value.
5. Rank correlation
H0: 𝜌𝑠 = 0 (no correlation)
H1: 𝜌𝑠 ≠ 0 (correlation)
Test statistic:
Not ties: 𝑟𝑠 = 1 −
Ties: 𝑟𝑠 =
6 ∑ 𝑑²
𝑛(𝑛2 −1)
𝑛 ∑ 𝑥𝑦−(∑ 𝑥)(∑ 𝑦)
√𝑛(∑ 𝑥²)−(∑ 𝑥)²√𝑛(∑ 𝑦²)−(∑ 𝑦)²
Critical values:
1. If 𝑛 ≤ 30, critical values are found in Table A-9.
2. If 𝑛 > 30, critical values are found by using formula 𝑟𝑠 =

±𝑧
√𝑛−1
where z
corresponds to the significance level.
Notation:
𝑟𝑠 : rank correlation coefficient for sample paired data
𝜌𝑠 : rank correlation coefficient for all the population data
𝑛: number of pairs of sample data
𝑑: difference between ranks for the two values within a pair
Requirements:
1. The sample paired data have been randomly selected.
2. There is no requirement that the sample pairs of data have a bivariate normal
distribution. There is no requirement of a normal distribution for any population.