1. STATISTICAL ANALYSIS

2.  Statistical analysis means investigating trends,
patterns, and relationships using quantitative data. It is
an important research tool used by scientists,
governments, businesses, and other organizations.
 To draw valid conclusions, statistical analysis requires
careful planning from the very start of the research
process. You need to specify your hypotheses and make
decisions about your research design, sample size, and
sampling procedure.
 After collecting data from your sample, you can organize
and summarize the data using descriptive statistics.
Then, you can use inferential statistics to formally test
hypotheses and make estimates about the population.
Finally, you can interpret and generalize your findings.

3. STEP 1: WRITE YOUR HYPOTHESES AND PLAN
YOUR RESEARCH DESIGN
 The goal of research is often to investigate a
relationship between variables within a population.
You start with a prediction, and use statistical
analysis to test that prediction.
 A statistical hypothesis is a formal way of writing a
prediction about a population. Every research
prediction is rephrased into null and alternative
hypotheses that can be tested using sample data.
 While the null hypothesis always predicts no effect
or no relationship between variables, the alternative
hypothesis states your research prediction of an
effect or relationship.

4.  Example: Statistical hypotheses to test an
effectNull hypothesis: A 5-minute meditation
exercise will have no effect on math test scores in
teenagers.
 Alternative hypothesis: A 5-minute meditation
exercise will improve math test scores in teenagers.

5. PLANNING YOUR RESEARCH DESIGN
A research design is your overall strategy for data
collection and analysis. It determines the statistical tests
you can use to test your hypothesis later on.
 In an experimental design, you can assess a cause-
and-effect relationship (e.g., the effect of meditation on
test scores) using statistical tests of comparison or
regression.
 In a correlational design, you can explore relationships
between variables (e.g., parental income and GPA)
without any assumption of causality using correlation
coefficients and significance tests.
 In a descriptive design, you can study the
characteristics of a population or phenomenon (e.g., the
prevalence of anxiety in U.S. college students) using
statistical tests to draw inferences from sample data.

6. Your research design also concerns whether you’ll
compare participants at the group level or individual
level, or both.
 In a between-subjects design, you compare the
group-level outcomes of participants who have
been exposed to different treatments (e.g., those
who performed a meditation exercise vs those who
didn’t).
 In a within-subjects design, you compare
repeated measures from participants who have
participated in all treatments of a study (e.g., scores
from before and after performing a meditation
exercise).

7. MEASURING VARIABLES
When planning a research design, you
should operationalize your variables and decide
exactly how you will measure them.
For statistical analysis, it’s important to consider
the level of measurement of your variables, which
tells you what kind of data they contain:
 Categorical data represents groupings. These may
be nominal (e.g., gender) or ordinal (e.g. level of
language ability).
 Quantitative data represents amounts. These may
be on an interval scale (e.g. test score) or a ratio
scale (e.g. age).

8.  Many variables can be measured at different levels of
precision. For example, age data can be quantitative (8
years old) or categorical (young). If a variable is coded
numerically (e.g., level of agreement from 1–5), it
doesn’t automatically mean that it’s quantitative instead
of categorical.
 Identifying the measurement level is important for
choosing appropriate statistics and hypothesis tests. For
example, you can calculate a mean score with
quantitative data, but not with categorical data.
 In a research study, along with measures of your
variables of interest, you’ll often collect data on relevant
participant characteristics.

9. STEP 2: COLLECT DATA FROM A SAMPLE
Sampling for statistical analysis
There are two main approaches to selecting a
sample.
 Probability sampling: every member of the
population has a chance of being selected for the
study through random selection.
 Non-probability sampling: some members of the
population are more likely than others to be
selected for the study because of criteria such as
convenience or voluntary self-selection.

10. Create an appropriate sampling procedure
Based on the resources available for your research,
decide on how you’ll recruit participants.
 Will you have resources to advertise your study
widely, including outside of your university setting?
 Will you have the means to recruit a diverse sample
that represents a broad population?
 Do you have time to contact and follow up with
members of hard-to-reach groups?

11. Calculate sufficient sample size
 Significance level (alpha): the risk of rejecting a
true null hypothesis that you are willing to take,
usually set at 5%.
 Statistical power: the probability of your study
detecting an effect of a certain size if there is one,
usually 80% or higher.
 Expected effect size: a standardized indication of
how large the expected result of your study will be,
usually based on other similar studies.
 Population standard deviation: an estimate of the
population parameter based on a previous study or
a pilot study of your own.

12. STEP 3: SUMMARIZE YOUR DATA WITH
DESCRIPTIVE STATISTICS
Inspect your data
There are various ways to inspect your data,
including the following:
 Organizing data from each variable in frequency
distribution tables.
 Displaying data from a key variable in a bar
chart to view the distribution of responses.
 Visualizing the relationship between two variables
using a scatter plot.

13.  By visualizing your data in tables and graphs, you
can assess whether your data follow a skewed or
normal distribution and whether there are any
outliers or missing data.
 A normal distribution means that your data are
symmetrically distributed around a center where
most values lie, with the values tapering off at the
tail ends.

15.  In contrast, a skewed distribution is asymmetric
and has more values on one end than the other.
The shape of the distribution is important to keep in
mind because only some descriptive statistics
should be used with skewed distributions.
 Extreme outliers can also produce misleading
statistics, so you may need a systematic approach
to dealing with these values.

16. CALCULATE MEASURES OF CENTRAL
TENDENCY
 Measures of central tendency describe where most
of the values in a data set lie. Three main measures
of central tendency are often reported:
 Mode: the most popular response or value in the
data set.
 Median: the value in the exact middle of the data
set when ordered from low to high.
 Mean: the sum of all values divided by the number
of values.

17. CALCULATE MEASURES OF VARIABILITY
Measures of variability tell you how spread out the
values in a data set are. Four main measures of
variability are often reported:
 Range: the highest value minus the lowest value of
the data set.
 Interquartile range: the range of the middle half of
the data set.
 Standard deviation: the average distance between
each value in your data set and the mean.
 Variance: the square of the standard deviation.

18. STEP 4: TEST HYPOTHESES OR MAKE ESTIMATES
WITH INFERENTIAL STATISTICS
 A number that describes a sample is called
a statistic, while a number describing a population
is called a parameter. Using inferential statistics,
you can make conclusions about population
parameters based on sample statistics.

19. Researchers often use two main methods
(simultaneously) to make inferences in statistics.
 Estimation: calculating population parameters
based on sample statistics.
 Hypothesis testing: a formal process for testing
research predictions about the population using
samples.

20. ESTIMATION
You can make two types of estimates of population
parameters from sample statistics:
 A point estimate: a value that represents your best
guess of the exact parameter.
 An interval estimate: a range of values that
represent your best guess of where the parameter
lies.

21.  If your aim is to infer and report population characteristics
from sample data, it’s best to use both point and interval
estimates in your paper.
 You can consider a sample statistic a point estimate for the
population parameter when you have a representative sample
(e.g., in a wide public opinion poll, the proportion of a sample
that supports the current government is taken as the
population proportion of government supporters).
 There’s always error involved in estimation, so you should
also provide a confidence interval as an interval estimate to
show the variability around a point estimate.
 A confidence interval uses the standard error and the z score
from the standard normal distribution to convey where you’d
generally expect to find the population parameter most of the
time.

22. HYPOTHESIS TESTING
 Using data from a sample, you can test
hypotheses about relationships between variables
in the population. Hypothesis testing starts with the
assumption that the null hypothesis is true in the
population, and you use statistical tests to assess
whether the null hypothesis can be rejected or not.

23. Statistical tests determine where your sample data
would lie on an expected distribution of sample data if
the null hypothesis were true. These tests give two
main outputs:
 A test statistic tells you how much your data differs
from the null hypothesis of the test.
 A p value tells you the likelihood of obtaining your
results if the null hypothesis is actually true in the
population.

24. Statistical tests come in three main varieties:
 Comparison tests assess group differences in
outcomes.
 Regression tests assess cause-and-effect
relationships between variables.
 Correlation tests assess relationships between
variables without assuming causation.

25. Parametric tests
 Parametric tests make powerful inferences about
the population based on sample data. But to use
them, some assumptions must be met, and only
some types of variables can be used. If your data
violate these assumptions, you can perform
appropriate data transformations or use alternative
non-parametric tests instead.

26.  A regression models the extent to which changes
in a predictor variable results in changes in
outcome variable(s).
 A simple linear regression includes one predictor
variable and one outcome variable.
 A multiple linear regression includes two or more
predictor variables and one outcome variable.

27. REGRESSION MODELS
 Regression models describe the relationship
between variables by fitting a line to the observed
data. Linear regression models use a straight line,
while logistic and nonlinear regression models use
a curved line. Regression allows you to estimate
how a dependent variable changes as the
independent variable(s) change.

28. Simple linear regression is used to estimate the
relationship between two quantitative variables.
You can use simple linear regression when you want
to know:
 How strong the relationship is between two
variables (e.g. the relationship between rainfall and
soil erosion).
 The value of the dependent variable at a certain
value of the independent variable (e.g. the amount
of soil erosion at a certain level of rainfall).

29. ASSUMPTIONS OF SIMPLE LINEAR REGRESSION
 Simple linear regression is a parametric test, meaning
that it makes certain assumptions about the data. These
assumptions are:
 Homogeneity of variance (homoscedasticity): the
size of the error in our prediction doesn’t change
significantly across the values of the independent
variable.
 Independence of observations: the observations in
the dataset were collected using statistically valid
sampling methods, and there are no hidden
relationships among observations.
 Normality: The data follows a normal distribution.

30. Comparison tests usually compare the means of
groups. These may be the means of different groups
within a sample (e.g., a treatment and control group),
the means of one sample group taken at different
times (e.g., pretest and posttest scores), or a sample
mean and a population mean.
 A t test is for exactly 1 or 2 groups when the sample
is small (30 or less).
 A z test is for exactly 1 or 2 groups when the
sample is large.
 An ANOVA is for 3 or more groups.

31. STEP 5: INTERPRET YOUR RESULTS
Statistical significance
 In hypothesis testing, statistical significance is the
main criteria for forming conclusions. You compare
your p value to a set significance level (usually
0.05) to decide whether your results are statistically
significant or non-significant.
 Statistically significant results are considered
unlikely to have arisen solely due to chance. There
is only a very low chance of such a result occurring
if the null hypothesis is true in the population.


32. Effect size
 A statistically significant result doesn’t necessarily
mean that there are important real life applications
or clinical outcomes for a finding.
 In contrast, the effect size indicates the practical
significance of your results. It’s important to report
effect sizes along with your inferential statistics for
a complete picture of your results. You should also
report interval estimates of effect sizes if you’re
writing an APA style paper.

33. Decision errors
 Type I and Type II errors are mistakes made in
research conclusions. A Type I error means
rejecting the null hypothesis when it’s actually true,
while a Type II error means failing to reject the null
hypothesis when it’s false.
 You can aim to minimize the risk of these errors by
selecting an optimal significance level and ensuring
high power. However, there’s a trade-off between
the two errors, so a fine balance is necessary.

34. FREQUENTIST VERSUS BAYESIAN STATISTICS
 Traditionally, frequentist statistics emphasizes null
hypothesis significance testing and always starts
with the assumption of a true null hypothesis.
 However, Bayesian statistics has grown in
popularity as an alternative approach in the last few
decades. In this approach, you use previous
research to continually update your hypotheses
based on your expectations and observations.
 Bayes factor compares the relative strength of
evidence for the null versus the alternative
hypothesis rather than making a conclusion about
rejecting the null hypothesis or not.

35. Linear regression makes one additional assumption:
 The relationship between the independent and
dependent variable is linear: the line of best fit
through the data points is a straight line (rather than
a curve or some sort of grouping factor).

36.  Multiple linear regression is used to estimate the
relationship between two or more independent
variables and one dependent variable. You can
use multiple linear regression when you want to
know:
 How strong the relationship is between two or more
independent variables and one dependent variable
(e.g. how rainfall, temperature, and amount of
fertilizer added affect crop growth).
 The value of the dependent variable at a certain
value of the independent variables (e.g. the
expected yield of a crop at certain levels of rainfall,
temperature, and fertilizer addition).

37. ASSUMPTIONS OF MULTIPLE LINEAR
REGRESSION
Multiple linear regression makes all of the same
assumptions as simple linear regression:
 Homogeneity of variance (homoscedasticity):
the size of the error in our prediction doesn’t
change significantly across the values of the
independent variable.
 Independence of observations: the observations
in the dataset were collected using statistically valid
methods, and there are no hidden relationships
among variables.

38. In multiple linear regression, it is possible that some
of the independent variables are actually correlated
with one another, so it is important to check these
before developing the regression model. If two
independent variables are too highly correlated (r2 >
~0.6), then only one of them should be used in the
regression model.
 Normality: The data follows a normal distribution.
 Linearity: the line of best fit through the data points
is a straight line, rather than a curve or some sort of
grouping factor.