The document discusses several non-parametric statistical tests used in psychology: i) The chi-squared test compares observed and expected categorical data to test hypotheses. It is used to test goodness of fit and independence. ii) The Wilcoxon signed-rank test compares two matched samples to test the location of a population or compare two populations. It assumes dependent samples, independence, continuous dependent variables, and ordinal measurement. iii) The Mann-Whitney U test compares two independent samples to test if they come from the same population. It assumes a continuous or ordinal dependent variable, a dichotomous independent variable, independence of observations, and the same or different shape of distributions between groups.

1. STATISTICS
IN PSYCHOLOGY
FROM
ROLL NO -

2. SUBMITTED TO-
DR.EKTA MAM
SUBMITTED BY- RITIKA

3. A chi-squared test (symbolically represented as χ2) is basically a data
analysis on the basis of observations of a random set of variables.
Usually, it is a comparison of two statistical data sets. This test was
introduced by Karl Pearson in 1900 for categorical data analysis and
distribution. So it was mentioned as Pearson’s chi-squared test.
The chi-square test is used to estimate how likely the observations that
are made would be, by considering the assumption of the null
hypothesis as true.
A hypothesis is a consideration that a given condition or statement
might be true, which we can test afterwards. Chi-squared tests are
usually created from a sum of squared falsities or errors over the
sample variance.
CHI
SQUARE

4. Where
C= Degrees of freedom
O = Observed Value
E = Expected Value

5. Applications of Chi-
square Distribution:

6. Chi-square distribution has a large number of applications in
Statistics, some of which are enumerated below:
i) To test if the hypothetical value of the population variance is
𝜎 2 = 𝜎 02
ii) To test the ‘goodness of fit’.
iii) To test the independence of attributes.
iv) To test the homogeneity of independent estimates of the
population variance.
v) To combine various probabilities obtained from independent
experiments to give a single test of significance.
vi) To test the homogeneity of independent estimates of the
population correlation coefficient.

7. WILCOXON SIGNED
RANK TEST

8. The Wilcoxon signed-rank test is a non-parametric statistical
hypothesis test used either to test the location of a population
based on a sample of data, or to compare the locations of two
populations using two matched samples. The one-sample
version serves a purpose similar to that of the one-sample
Student's t-test.

9. W = Test Statistic
𝑵𝒓 = Sample size, excluding pairs where x1=x2
Sgn = Sign function
𝒙𝟏, ⅈ,𝒙𝟐,i = corresponding ranked pairs from two distributions
𝑹𝒊 = rank i

10. Assumptions of the Wilcoxon Sign Test

11. The Wilcoxon Sign test makes four important assumptions:
1. Dependent samples – the two samples need to be dependent observations of the
cases. The Wilcoxon sign test assess for differences between a before and after
measurement, while accounting for individual differences in the baseline.
2. Independence – The Wilcoxon sign test assumes independence, meaning that the
paired observations are randomly and independently drawn.

12. 3. Continuous dependent variable – Although the Wilcoxon signed rank test ranks the
differences according to their size and is therefore a non-parametric test, it assumes that
the measurements are continuous in theoretical nature. To account for the fact that in
most cases the dependent variable is binominal distributed, a continuity correction is
applied.
4. Ordinal level of measurement – The Wilcoxon sign test needs both dependent
measurements to be at least of ordinal scale. This is necessary to ensure that the two
values can be compared, and for each pair, it can be said if one value is greater, equal, or
less than the other.

13. MannWhitney
U Test

14. The Mann-Whitney U Test assesses whether two sampled groups are likely to derive from the
same population, and essentially asks; do these two populations have the same shape with
regards to their data? In other words, we want evidence as to whether the groups are drawn
from populations with different levels of a variable of interest. It follows that the hypotheses
in a Mann-Whitney U Test are:
•The null hypothesis (H0) is that the two populations are equal.
•The alternative hypothesis (H1) is that the two populations are not equal.

15. Assumptions of the Mann
Whitney U test

16. •Assumption #1: You have one dependent variable that is measured at
the continuous or ordinal level. Examples of continuous variables include revision time (measured in
hours), intelligence (measured using IQ score), exam performance (measured from 0 to 100), weight
(measured in kg), and so forth. Examples of ordinal variables include Likert items (e.g., a 7-point
scale from "strongly agree" through to "strongly disagree"), amongst other ways of ranking categories
(e.g., a 5-point scale explaining how much a customer liked a product, ranging from "Not very much"
to "Yes, a lot").
•Assumption #2: You have one independent variable that consists of two
categorical, independent groups (i.e., a dichotomous variable). Example independent variables that
meet this criterion include gender (two groups: "males" or "females"), employment status (two groups:
"employed" or "unemployed"), transport type (two groups: "bus" or "car"), smoker (two groups: "yes"
or "no"), trial (two groups: "intervention" or "control"), and so forth.

17. •Assumption #3: You should have independence of observations, which means that there is no
relationship between the observations in each group of the independent variable or between the groups
themselves. For example, there must be different participants in each group with no participant being in
more than one group. This is more of a study design issue than something you can test for, but it is an
important assumption of the Mann-Whitney U test. If your study fails this assumption, you will need to
use another statistical test instead of the Mann-Whitney U test (e.g., a Wilcoxon signed-rank test).
•Assumption #4: You must determine whether the distribution of scores for both groups of your
independent variable (e.g., the distribution of scores for "males" and the distribution of scores for
"females" for the independent variable, "gender") have the same shape or a different shape. This will
determine how you interpret the results of the Mann-Whitney U test. Since this is a critical assumption
of the Mann-Whitney U test, and will affect how to work your way through this guide, we discuss this
further in the next section.