Statistics Help Guide
Becki Paynich
I.
II.
A Few Important Things:
A.
Math is the least important factor in statistics. Knowing which statistics
test to employ based on sample, level of measurement, and what you are trying to
understand is the most important part in statistics. However, you need to
understand how the math works to better understand why you pick any given test.
B.
Statistics is not brain surgery. Lighten up. You’ll do fine.
C.
This is a help guide, not a comprehensive treatment of statistics. Please
refer to a good Statistics textbook for a deeper understanding of statistics.
Levels of Measurement: Knowing the level of measurement is extremely
important in statistics. Most tests of significance are explicit about the level of
measurement they require. If the requirements for the level of measurement are not
adhered to, results lose validity. For example, Pearson’s r assumes that variables are at
least interval level. If a Pearson’s r is employed on nominal or ordinal level variables,
results cannot be directly interpreted (pretty much your results are meaningless).
A.
Nominal—a variable that is measured in categories that cannot be ranked.
Usually nominal variables are collapsed into the smallest number of categories
possible (0,1) so that they may be used in analysis. A variable in this form
(dichotomous) can be used in virtually any type of analysis. For example: the
variable RACE may be originally separated into 12 different categories. These 12
might be collapsed into 2 (most likely being Caucasian=1 and Non-Caucasian=0).
The important thing to remember is that collapsing categories into larger ones can
hide important relationships between them. For example, the difference between
African-Americans and Native Americans on a given variable may be important
but if they are both collapsed into the category of Non-Caucasian, this
relationship will be obscured. Another thing to keep in mind is that a dichotomous
or dummy variable (one with only two categories as in above example) is the most
powerful form, statistics wise, for a nominal variable to be in.
B.
Ordinal—a variable that can be ranked but the exact difference between
categories cannot be identified. For example: The categories of Strongly Agree,
Agree, No Opinion, Disagree, and Strongly Disagree, can be differentiated by
order but one does not know the exact mathematical difference between two
respondents who answer Strongly Agree and Agree respectively. Another
example would be categorical ranges. Income, for example, can be grouped into
the following ranges: $0-$10,000; $10,0001-$20,000; and $20,000+. While
certainly, this variable is measured with numbers, one cannot tell the exact
difference between a respondent who selected the 0-10,000 range and a
respondent that selected the 10,001-20,000 range. In theory, they could by $1.00
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apart or closer to $10,000 apart in income. A final form of ordinal variable which
is actually treated as interval-level is when one adds together multiple ordinal
level responses. For example, I may want to make an index on how well a student
likes the course materials by combining his/her responses to questions about the
books, the web page, and the class notes. Thus, if a student answered Strongly
Agree to questions about these three items, and the score for Strongly Agree is 5,
the student’s total score for this new index is 15.
Ordinal variables can also be and often are collapsed into a smaller number of
categories. For example: A variable measuring rank may have 8 or more original
categories that can be collapsed into the categories of administration=1 and nonadministration=0. Or, there could be three categories with a middle category
being middle-management. The problem again becomes sacrificing important
relationships that may be obscured by the collapse for ease in analysis. The best
rule of thumb is to run analyses with all the categories in their smaller form first
to identify important relationships between them and then run analyses again after
collapsing the variables to identify important relationships across larger ones.
** When you create nominal and ordinal level questions, make sure they are mutually exclusive
(no respondent can fit into more than one category) and exhaustive (you have listed all
possibilities). Also, when designing numerical ranges, make sure that the intervals have the same
width. In the following example, notice that the width for each category is $10,000:
Question: What is your income range?
1 = 0-10,000
2 = Above 10,000 to 20,000
3 = Above 20,000 to 30,000
C.
Interval—a variable or scale that uses numbers to rank order. This is the best type
of variable to have as you can keep it in interval form or turn it into an ordinal or
nominal level variable and do virtually any type of analysis. Furthermore, the
exact difference between two respondents can be identified. For example: with the
variable age, when respondents report their actual age, one can mathematically
determine the differences between the ages of respondents. If however, the
variable is grouped into age ranges (0-5, 6-10, 11-15, 16-20…) then the variable
is ordinal and not interval scale. Interval level variables can be collapsed into
ordinal or even nominal categories but this is usually avoided as most statistical
tests designed for interval level variables are more powerful and provide more
understanding of the nature of the relationships than those tests designed for
ordinal or nominal variables.
D.
Ratio—a ratio scale variable is identical to interval in almost every aspect
except for it has an absolute zero. Age and Income have absolute zeros. If a
variable can be considered on a ratio scale, then it is also interval scale. **Many
researchers in criminal justice ignore the difference between interval and ratio as
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the tests of significance that we use for ratio and interval variables are the
same.**
**Make sure to code variables in the most logical order.** For example, if you are trying to
measure the frequency of smoking, use the highest numbers for the most amount of smoking and
the lowest numbers for the least amount of smoking. For example:
Question: How often do you smoke?
0 = Never
1 = Less than once per week
2 = At least once per week
3 = 2 or 3 times per week
4 = Daily
III.
Numerical Measures:
A.
Measures of Central Tendency
1.
Mode-the most prevalent variable (for example, in a data set of
pets: 25 cats, 13 dogs, 12 hamsters, 47 fish. Fish is the modal category
with 47 being the mode). The only measure of central tendency for
nominal-level data is the mode. The mode can also be used with ordinal
and interval-level data.
2.
Median-the mid point in a value. An equal number of the sample
will be above the median as will be below it. In a sample of
n =101: 50 of the scores will be above the median, and 50 will be
below the median). The median is most often used for interval-level data.
When ranked from lowest to highest, the 51st score will be the median. If
your n is an even number, take the average between the two middle scores.
3.
Mean-the average of a value. The mean is usually not computed on
ordinal and nominal-level data because it has little explanatory value.
However, many statistical tests require that the mean be computed for the
formula. The mean is most appropriate for interval-level data.
B.
Measures of Variability
1.
Range—the difference between the largest and smallest values.
2.
Variance—the sum of the squared deviations of n measurements
from their mean divided by (n – 1).
3.
Standard Deviation—the positive square root of the variance. One
must compute the variance first to get the standard deviation because the
sum of deviations from the mean always equals zero.
4.
Typically, standard deviations are used in the form of standard
errors from the mean in a normal curve. In a normal distribution, the
interval from one standard error below the mean to one standard error
above the mean contains approximately 68% of the measurements. The
interval between two standard errors below and above the mean contains
approximately 95% of the measurements.
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5.
IV.
V.
In a perfect normal curve, the mode, median, and mean are all the
same number.
Probability:
A.
Classic theory of probability—the chance of a particular outcome
occurring is determined by the ratio of the number of favorable outcomes
(successes) to the total number of outcomes. This theory only pertains to
outcomes that are mutually exclusive (disjoint).
B.
Relative Frequency Theory—is that if an experiment is repeated an
extremely large number of times and a particular outcome occurs a percentage of
the time, then that particular percentage is close to the probability of that
outcome.
C.
Independent Events—outcomes not affected by other outcomes.
D.
Dependent Events—outcomes affected by other outcomes.
E.
Multiplication Rule
1.
Joint Occurrence—to compute the probability of two or more
independent events all occurring, multiply their probabilities.
F.
Addition Rule—to determine the probability of at least one successful
event occurring, add their probabilities.
G.
Remember to account for probabilities with and with replacement. For
example, when picking cards out of a deck, the probability of choosing on the first
try the Queen of Hearts is 1/52. However, if you don’t put back the first card
before choosing for the second time, your probability increases to 1/51 because
there are only 51 cards left to choose from.
H.
Probability Distribution is nothing more than a visual representation of the
probabilities of success for given outcomes.
I.
Discrete variables are those that do not have possible outcomes in
between values. Thus, a coin toss can only result in either a heads or tails
outcome. Continuous variables are those that have possible outcomes between
values. For example, it is absolute possible to be 31.26 years old.
Sampling:
A.
Independent Random Sampling: Most statistical tests are based on the
premise that samples are independently and randomly selected. If in fact a sample
was purposive, statistical analysis cannot be generalized outside the sample.
Random sampling is done so that inferential statistics can be interpreted outside
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the sample. Statistical tests can still be employed in samples that have been drawn
through non-random techniques, however, their interpretation must be confined to
the sample at hand, and limitations of the sampling design must be addressed in
the results and methods discussions of your research.
B.
VI.
Central Limit Theorem—the larger the number of observations (the bigger
your n), the more likely the sample distribution will approximate a normal curve.
Principles of Testing:
A.
Research hypothesis—can be stated in a certain direction or without
direction. Directional hypotheses are tested with one-tailed tests of significance.
Non-directional hypotheses are tested with two-tailed tests of significance.
B.
Null hypothesis—is essentially the statement that two values are not
statistically related. If a test of a research hypothesis is not significant, then the
null hypothesis cannot be rejected.
C.
Type I error: (alpha) is when a researcher rejects the null hypothesis when
in fact it is true. That is, stating that two variables are significantly related when in
fact they are not.
D.
Type II error: (beta) is when a researcher fails to reject the null hypothesis when
in fact the null hypothesis is false. That is, stating that two variables are not
significantly related when in fact they are.
VII.
Univariate Inferential Tests: (See also Appendix A for quick reference
guide)Essentially, univariate tests are only looking at one variable and how the
scores for that variable differ between sample and population groups, two
samples, and within groups at different points in time. Many univariate tests can
be employed on proportions when that is all that is available. This is not a
comprehensive listing of all univariate tests but can the most commonly used are
briefly discussed below.
A.
Steps in testing:
1.
State the assumptions
2.
State the Null and Research Hypotheses.
3.
Decide on a significance level for the test, Determine the test to be used.
4.
Compute the value of a test statistic.
5.
Compare the test statistic with the critical value to determine
whether or not the test statistic falls in the region of acceptance or the
region of rejection.
B.
One-sample z-test:
1.
Requirements: normally distributed population. Population
variance is known.
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2.
3.
4.
Test for population mean
One-tailed or two-tailed.
This test essentially tells you if your sample mean is statistically
different from the larger population mean.
C.
One-sample t-test:
1.
Requirements: normally distributed population. Population
variance is known.
2.
Test for population mean
3.
One-tailed or two-tailed.
4.
This test essentially tells you if your sample mean is statistically
different from the larger population mean.
D.
Two-sample t and z-tests for comparing two means:
1.
Requirements: two normally distributed but independent
populations. Population variance can be known or unknown. Different
formulas depending on whether the variance is known.
E.
Paired Difference t-test:
1.
Requirements: a set of paired observations from a normal
population. This test is usually employed to compare “before” and “after”
scores. Also employed in twin studies.
F.
Chi-Square for population distributions:
1.
Requirements: assumption of normal distribution
2.
This test basically compares frequencies from a known sample
with frequencies from an assumed population. There are many different
types of chi-square tests which test different things—consistency,
distribution, goodness of fit, independence…make sure you are employing
the right chi-square for your needs.
VIII. Bivariate Relationships: Bivariate simply means analysis between two
variables. Before beginning this section it is important to note that there is a difference
between measures of association and tests of significance. A Pearson’s r (a measure of
association) will give you are statistic between -1 and +1. -1 is a perfect negative
correlation, +1 is a perfect positive correlation. 0 is no correlation. When SPSS provides
output, it has to run a separate test to tell you if the relationship is significant. Thus, if
doing stats by hand, one has to complete two different formulas—one to get the
correlation and one to get the significance.
A.
Measures of Association: (See also Appendix A for quick reference guide)
Nominal Variables
1.
Lambda-a PRE (proportionate reduction in error) approach that
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2.
3.
essentially tells one how well the prediction in one variable is affected
with the knowledge of another. Lambda ranges from 0-1. Thus the
direction of relationship between two variables cannot be determined, only
the strength can be determined with Lambda. This measure of association
is used for nominal level variables and is based on modal categories.
Phi-a correlation coefficient used to estimate association in a 2 x 2
table.
Cramer’s V- a correlation coefficient to estimate association in
tables larger than 2 x 2.
Ordinal Variables
1.
Gamma-the most frequently used PRE measure of ordered crosstabular association. It ranges from -1 to +1 thus the direction and the
strength of the relationship can be determined with gamma. Gamma is
used for ordinal level variables. Gamma is based on
concordant and discordant pairs. Gamma does not account for tied pairs.
2.
Kendall’s tau b- like gamma, is a PRE measure based on
concordant and discordant pairs, but accounts for tied pairs on the
dependent variable. Kendall’s tau b ranges from -1 to +1. Use Kendall’s
tau b for ordinal variables that when they are arranged in tables, the
number of rows and columns is equal.
3.
Kendall’s tau c- same as Kendall’s tau b but used for when the
number of rows and columns are not equal.
4.
Somers’s dxy – measure of association that accounts for tied pairs
on both the independent and dependent variable. Somers’s dxy will
have different values depending on which variable is X (independent) and
which variable is Y(dependent).
Interval Variables
1.
Spearman’s rho- measure of association for interval level data that
is less sensitive to outliers than Pearson’s r. Pearson’s r is usually the
preferred test statistic so if outliers are not a problem, use Pearson’s r.
2.
Pearson’s r – measure of association that requires interval level
variables. Pearson’s is great and used often because it holds strong
interpretive power. However, it is sensitive to outliers. In cases where your
data holds outliers that cannot be taken out of the analysis, use Spearman’s
rho which is less sensitive to outliers.
B.
Tests of Significance (Again, see also Appendix A)
1.
Chi-Square- used for nominal and sometimes ordinal level data.
The important thing to remember here is that if you have too many
categories, the data will be spread out which can be problematic for this
test. Chi-square assumes that there are at least 5 cases in each box in the
table. To avoid this problem, categories can be collapsed (see discussion
under nominal variables in levels of measurement). Chi-square can tell
you if the observed frequency is statistically significant relative to what
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2.
3.
IX.
we might expect by chance. Again, be careful that you are using the right
chi-square formula as there are several uses for chi-square.
Bivariate Regression- regression can be used for as little as two
variables. However, the assumptions of linear regression must be
followed, and the causal order of X and Y must be specified correctly. If
the causal order is not specified, try running the regression while flipping
around X and Y. Perhaps a reciprocating relationship is present. A more
detailed discussion or regression is provided below under multivariate
linear regression.
Logistic regression-Logistic regression is used for when the
dependent variable is dichotomous (only two categories: yes/no,
black/white…etc.) Logistic regression is interpreted slightly different than
linear regression in that the coefficients refer to the likelihood or
probability that X causes a change in Y.
Multivariate Relationships:
A.
Linear Regression Assumptions: There are several assumptions that must
be followed when employing linear regression. As a researcher, if you violate any
of these assumptions, you must provide a discussion of the limitations produced
by the violated assumptions. Some violations are more serious and problematic
than others. These assumptions are listed and briefly discussed below. Many of
them apply to logistic and bivariate equations discussed above. They are in no
particular order.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
The relationship between the independent variables (X) and the
dependent variable (Y) is linear.
Errors are random and normally distributed.
The mean of error is zero.
Errors are not correlated with X.
No autocorrelation between the disturbances.
No covariance between error and X.
The number of observations must be greater than the number of
parameters to be estimated (n must be larger than the number of X’s in the
equation).
There is variability in both the X and Y values. (X & Y should be
interval level but this assumption violation is not a huge deal if Y is
interval and at least one X is interval). X can be dichotomous as well but
the coefficient cannot be directly interpreted, must be restated into
probabilities.
No specification bias of error.
No perfect multicollinearity.
The direction of the hypothesis is properly stated (X impacts Y in
this manner)
X and Y are normally distributed.
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X.
B.
Multivariate regression assumes that the variables included in the
equation are meaningful and make theoretical sense. If one throws every variable
possible into the equation, the r2 will be inflated. To avoid this, only include those
variables which make theoretical sense and use the Adjusted r2 if there is a
significant difference between r2 and adjusted r2.
C.
The output from SPSS will provide a table with the standardized and
unstandardized beta coefficients. The standardized coefficients reflect the relative
importance of any given X in the equation. For standardized beta coefficients, the
largest one signifies the variable with biggest impact on Y. The unstandardized
coefficients will tell you the change in Y with a one unit increase in X. There is
also a column (sig) which will tell you if that X variable is statistically and
significantly related to Y.
D.
Regression is an extremely powerful and useful tool in statistics. Please
refer to class notes and stats books if you have any questions about regression.
Common Mistakes:
A.
Forgetting to convert between standard deviation and variance. Some
formulas require standard deviation, others require variance. Remember that one
must compute the variance first and then take its square root to get the standard
deviation because the sum of deviations from the mean always equals zero.
B.
Misstating one-tailed and two-tailed hypotheses. If your hypotheses
suggest a direction—greater than or less than (for example individuals with more
education will have a higher income), then the hypothesis is one-tailed. If your
hypothesis only says that there will be a difference in values—not equal (for
example, education has an effect on income), then your hypothesis is two-tailed.
Make sure that the appropriate test is employed based on how your hypothesis is
stated.
C.
Failing to split the alpha level for two-tailed tests.
D.
Misreading the standard normal (z) table.
E.
Using n instead of n – 1 degrees of freedom in one-sample t-tests. Make
sure to use the appropriate formula to figure out the degrees of freedom, it differs
across tests.
F.
Confusing confidence level with confidence interval. The confidence level
is the significance level of the test or the likelihood of obtaining a given result by
chance. The confidence interval is a range of values between the lowest and
highest values that the estimated parameter could take at a given confidence level.
G.
Confusing interval width with margin of error. Interval width is the
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distance between the margins of error. For example: +/-3 margin of error has an
interval width of 6.
XI.
H.
Confusing statistics with parameters. Parameters are characteristics of the
population that you usually don’t know. Statistics are characteristics of samples
that you are usually able to compute. Essentially you compute statistics in order to
estimate parameters.
I.
Confusing the addition rule with the multiplication rule.
J.
Forgetting the addition rule applies to mutually exclusive outcomes.
K.
Using a “percentage” instead of a “proportion” in a formula.
What do I do with a data set?
A.
There are no hard and fast rules because every research project is
different. However, there are a few basic steps in data analysis:
1.
Run frequencies of every variable. This will give you a feel for the
data and will alert you to any problems or anomalies that need to be
addressed.
2.
Run a correlation matrix of important variables. Use Kendall’s tau
b for this preliminary analysis. I know that Kendall’s tau b is for
ordinal level data and not all variables will be ordinal, however, it is a
good starting point to begin to see what relationships look like between
variables. Also, the matrix might help you identify degrees of
multicollinearity which will justify combining variables into scales and
indices. Once you review the matrix you may get some concrete ideas
about how to proceed with the analyses. Do not use this matrix as a form
of hypotheses testing unless both variables you are looking at are ordinal
level.
3.
Construct scales, indices, and interaction variables. Usually,
questions/items which measure similar things are combined into a scale by
adding each item together. This step will eliminate multicollinearity
problems. Keep in mind that the items you add together must be
theoretically or empirically related to each other. Also, each item must be
measured the same way and in the same direction. Also, you may have to
place weight on items which hold more predictive power. Make sure to
run an alpha reliability analysis. Alpha should be at least a .60 to justify
the scale. However, if it is close and you have sound theoretical arguments
why the item(s) should remain together in scale form, it is OK to keep
them together. But, you should analyze items separately to identify which
variables/relationships were most important.
4.
Begin hypotheses testing. At this stage, you should be careful
about the assumptions you violate because you are going to use your
analyses as tests of hypotheses rather than exploratory analyses. Choose
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5.
the appropriate tests based on the requirements and assumptions. Recode
variables in scales and indices (be sure to run reliability analyses). Be
aware that you may have to tests hypotheses in different ways with
different tests depending on the data set. For example, experience may be
measured in several ways and the measures may be at different levels.
You may not be able to combine these measures into a scale or index
which means you will have to analyze each measure of experience
separately.
Discuss your analyses. Make sure to include limitations and
alternative explanations for your results.
APPENDIX A
Tests of Significance
Information Obtained: Tests of significance can only tell you whether or not something is
statistically significant such as:
 Whether or not two variables are dependent upon one another (chi-square)
 Whether or not the distribution of one variable is not likely based on random chance (chisquare)
 Whether or not a sample mean is significantly different from a populations mean (T-tests
and Z-tests for comparing a sample to a population)
 Whether or not two sample means are significantly different from one another (T-tests
and Z-tests for comparing 2 samples)
 Whether or not 3 or more groups or samples are statistically significant from each other
(ANOVA)
Level of
measurement
Why Use?
Step 1
Step 2
T-Tests (Univariate)
The test variable is either measured in the nominal (when you compare
proportions) or interval (when you compare means)
 To see if a population (based on information from a sample) is
significantly different from a population.
 To see if two samples are significantly different.
State your Assumptions
Random Sampling (Independent random sampling if comparing two
samples)
Level of Measurement _________
Normal sampling distribution
State the Null and Research Hypotheses
H0: The population (based on the sample) is not
significantly different from the overall population.
Or
The two samples are not significantly different.
H1: The population (based on the sample) is
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significantly different from the overall population.
Or
The two samples are significantly different.
Step 3
Step 4
Step 5
Special
Instructions
Level of
measurement
Why Use?
Step 1
Step 2
*You may also state the research hypothesis in a certain direction if you
know the direction. This is called a one-tailed test.*
Select the Sampling Distribution (T Distribution) and Establish
the Critical Region
1. Pick your alpha level (For ex: .05)
2. Determine if this is a one or two-tailed test
3. Determine the degrees of freedom
Df= n-1 (for comparing a sample to a population)
Df= n1+ n2 – 2 (for comparing two samples)
4. Look up the critical T in back of book based on if it is a one or
two tailed test and the alpha.
Compute the Test Statistic (Solve the formula to get the obtained T)
Interpret your results.
Anything beyond the critical T on both tails for two-tailed test, and
beyond one tail for a one-tailed test you reject the null hypothesis
(The area beyond T is considered the “critical” region.
Anything smaller than the critical T for a one-tailed test and within
the region between the critical Ts (plus and minus) means you fail to
reject the null hypothesis
 T can be used when the n is smaller than 100. It can also be used
when n is larger than 100.
 When comparing samples (Ch.8) be sure to compute ALL
formulas needed.
Z –Tests (Univariate)
The test variable is either measured in the nominal (when you compare
proportions) or interval (when you compare means)
 To see if a population (based on information from a sample) is
significantly different from a population.
 To see if two samples are significantly different.
State your Assumptions
Random Sampling (Independent random sampling if comparing two
samples)
Level of Measurement _________
Normal sampling distribution
State the Null and Research Hypotheses
H0: The population (based on the sample) is not
significantly different from the overall population.
Or
The two samples are not significantly different.
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H1: The population (based on the sample) is
significantly different from the overall population.
Or
The two samples are significantly different.
Step 3
Step 4
Step 5
Special
Instructions
Level of
measurement
Why Use?
Step 1
Step 2
Step 3
*You may also state the research hypothesis in a certain direction if
you know the direction. This is called a one-tailed test.*
Select the Sampling Distribution (Z Distribution) and Establish
the Critical Region
1. Pick your alpha level (For ex: .05)
2. Determine if this is a one or two-tailed test
3. Look up the critical Z in back of book based on if it is a one or
two tailed test and the alpha.
Compute the Test Statistic (Solve the formula to get the obtained
Z)
Interpret your results.
Anything beyond the critical Z on both tails for two-tailed test, and
beyond one tail for a one-tailed test you reject the null hypothesis
(The area beyond Z is considered the “critical” region.
Anything smaller than the critical Z for a one-tailed test and within
the region between the critical Zs (plus and minus) means you fail
to reject the null hypothesis
 Z can only be used when the n is larger than 100.
 When comparing samples be sure to compute ALL formulas
needed.
ANOVA (Univariate)
The test variable is interval (you are comparing means). The groups
are based on Nominal or Ordinal categories.
 To see if 3 or more groups or samples are significantly
different.
State your Assumptions
Independent Random Sampling
Level of Measurement _________
Normal sampling distribution
Population variances are equal.
State the Null and Research Hypotheses
H0: The population means (based on the samples) are not
significantly different from one another.
H1: The population means (based on the samples) are
significantly different from one another.
Select the Sampling Distribution (F Distribution) and Establish
the Critical Region
1. Pick your alpha level (For ex: .05)
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Step 4
Step 5
Special
Instructions
2. Determine the degrees of freedom between and within.
Dfb = k-1
Dfw = N-k
3. Look up the critical F in back of book based on the
degrees of freedom and the alpha.
Compute the Test Statistic (Solve the formula to get the obtained
F)
Interpret your results.
Anything equal to or larger than the critical F means you reject the
null hypothesis.
Anything smaller than the critical F means you fail to reject the
null hypothesis
 ANOVA should be used when the n’s are comparable in
number.
 When using ANOVA be sure to compute ALL formulas
needed.
 With ANOVA, there is no such thing as a one or two-tailed
test. The F ratio is based solely on the differences between
groups.
Level of measurement
Why Use?
Step 1
Step 2
Step 3
Step 4
Chi-Square (Univariate and Bivariate)
Nominal and Ordinal
 To see if two variables are dependent.
 To see if distribution of one variable is not likely
based on random chance.
State your Assumptions
Random Sampling
Level of Measurement _________
State the Null and Research Hypotheses
H0: The two variables are independent.
Or
The distribution of ____ is random.
H1: The two variables are dependent.
Or
The distribution of ____ is not random.
Select the Sampling Distribution (chi-square) and
Establish the Critical Region
4. Pick your alpha level (For ex: .05)
5. Calculate degrees of freedom:
Df = (r-1)(c-1) for two variable tests.
Df = (k-1) for one variable tests
6. Look up the critical chi-square in the back of the
book.
Compute the Test Statistic (Solve the formula to get
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Step 5
Special Instructions
the obtained chi-square)
Interpret your results.
Anything equal to or above the critical chi-square
means you reject the Null hypothesis. Anything
smaller than the critical chi-square means you fail to
reject the null hypothesis
 Don’t use on anything bigger than a
4 x 4 table.
 If any cell has a frequency of less than 5, do not
use
Measures of Association (Bivariate)
Information Obtained: Measures of Association will tell you the strength of a relationship but
will not tell you whether or not it is statistically significant. {For example, the relationship
between eating apples and getting cancer may achieve statistical significance (because it is
consistent) but it can either be a strong relationship (it only takes a few apples to give you
cancer) or a weak one (it takes many apples—like millions—over a lifetime to increase the
probability of getting cancer.)}
Tests which measure the association between two variables are based on the following
assumptions:


Random Sampling
Normal Sampling Distribution
The level of measurement of the variables will determine which test you choose.
Nominal Level Variables
Test
Phi
Cramer’s V
Lambda
Formula
Formula is based on ChiSquare statistic
Formula is based on ChiSquare statistic
PRE measure (means your
prediction error is reduced
when you have knowledge
of the independent variable)
Special Instructions
Use for 2x2 tables. Ranges
0 to +1
Use for larger than 2x2
tables. Ranges 0 to +1.
Use for any type of table
except for when the row
marginals are vastly
different. Ranges from 0 to
+1. Multiply Lambda by
100 and get the exact
reduction in error by
knowing the independent
variable.
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Ordinal Level Variables
Test
Gamma
Formula
PRE measure based on
concordant and discordant
pairs (Gamma does not
account for tied pairs)
Kendall’s tau-b
(use when the number of
rows is equal to the number
of columns)
PRE measure based on
concordant and discordant
pairs. This test accounts for
tied pairs (thus a little more
accurate than Gamma)
Kendall’s tau-c
(use when the number of
rows is NOT equal to the
number of columns)
PRE measure based on
concordant and discordant
pairs. This test accounts for
tied pairs (thus a little more
accurate than Gamma)
Somers dxy
This test accounts for tied
pairs on both the
independent and dependent
variable. With this formula,
you must specify the
independent and dependent
variables correctly.
Special Instructions
Ranges from -1 to +1 (-1 is
a perfect negative
relationship and +1 is a
perfect positive
relationship. 0 means no
relationship.
Ranges from -1 to +1 (-1 is
a perfect negative
relationship and +1 is a
perfect positive
relationship. 0 means no
relationship.
Ranges from -1 to +1 (-1 is
a perfect negative
relationship and +1 is a
perfect positive
relationship. 0 means no
relationship.
Ranges from -1 to +1 (-1 is
a perfect negative
relationship and +1 is a
perfect positive
relationship. 0 means no
relationship.
Interval and Ratio Level Variables
Test
Spearman’s rho
(Can be used for ordinal
level variables that are
“continuous”—scales with
actual scores—in some
cases)
Pearson’s r
Formula
Use this formula when you
have outliers. This test is
less sensitive to outliers.
Special Instructions
Ranges from 0 to +1. Once
you square the obtained
Spearman’s rho, you make
multiply that number by
100 and interpret it as the
percentage of reduced error.
This formula is the most
Ranges from -1 to +1 (-1 is
often used but is sensitive to a perfect negative
outliers
relationship and +1 is a
perfect positive
relationship. 0 means no
relationship. If you square
this number it can be
interpreted to represent the
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percentage of variation of
the dependent variable that
the independent variable
explains.
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