Review Of Descriptive Statistics
Types of Data Analysis
• Quantitative Methods
o Testing theories using numbers
• Qualitative Methods
o Testing data using language
The Research Process
• Initial Observation
o Find something that needs explaining
Observe real world
Read existing research
• Generating Theory and Hypothesis
A hypothesized general principle that explains known findings about a topic and
from which new hypotheses can be generated
A prediction from a theory
The act of disproving a theory or hypothesis
• Collecting Data to Test Theory
o Purpose of this stage:
1) Decide What to measure and 2) How to measure it.
o The What = Variables
o Independent Variable (Predictor Variable):
Variable thought to be the cause of some effect. The variable manipulated in
o Dependent Variable (Outcome Variable):
Variable that is affected by changes in an independent variable.
o Categorical Variable
Variables that are made up of categories.
You must clearly fall into one category.
There is no overlap between categories.
o Continuous Variable
Variables that give a score for each participant and the score can take on any
value. Levels of Measurement
o Two categories (also called dichotomous).
o Example: Status - Dead or Alive
o More than two categories
o Example: music preference - rock, pop or jazz
o Same as nominal, but the categories have a logical order
o Example: Letter Grade - A, B, C, D, F; Position in a Race
o Equal intervals on the variable represent equal differences in the property being
measured (the difference between 5 and 7 is the same as 10 and 12).
o Example: 7-point Likert scale variables
o Same as interval, but the ratios must make sense; the measure must have meaning.
o A score of 10 on an exam must mean that the person has understood twice as much
material as someone who scored a 5.
o Example: Age, Reaction time
• How do we know if we are measuring something accurately?
o Measurement Error:
The discrepancy between the actual value you are trying to measure and the
number used to represent that value.
• Minimizing Error
Whether an instrument measures what it set out to measure
• The instrument measures what it claims to measure through comparison to
• Evidence that the content of a test corresponds to the content of the
construct it was designed to cover
• Evidence that the results allow to inferences to real-world conditions
Whether a measure produces the same results in same conditions
To Be Valid, An Instrument Must Be Reliable!
• The ability of a measure to produce consistent results when the same
entities are tested at two different points in time Research Design
• The How = The way the variable Is Measured
• Correlational Research
o Observe what is occurring naturally, without directly interfering.
• Cross-Sectional Research
o Data comes from people at different age points with different people representing each
• Experimental Research
o One or more variables are systematically manipulated to see their effect on an outcome
o Leads to cause and effect statements.
Methods of Data Collection
• Independent Groups (Between-Subject/Between-Group):
o Different entities (participants, people) in the different conditions
o Problem: Increased Error
• Repeated Measure (Within-Subject)
o The same entities take part in all experimental conditions
o Problem: Practice effects & participant fatigue
o Randomize which condition is 1st
Types of Variation
• Systematic Variation
o Differences in performance created by a specific experimental manipulation
• Unsystematic Variation
o Differences in performance created by unknown factors (age, gender, IQ, time of
day, measurement error, condition order, etc.)
• We want to minimize unsystematic variation.
• Randomization can help with this.
• Descriptive Statistics
o Statistical procedures used to DESCRIBE the population (or sample) that is being
• Inferential Statistics
o Statistical procedures used to make INFERENCES or predictions about a population
from observations and analyses of a sample
• Concerned with:
o The shape of the distribution of scores, the center of the distribution, the dispersion
of scores, and the deviance from the center. 1. Distribution Of Scores
o We can look at the distribution of scores in two ways
1. Frequency Chart
Great for identifying missing data and impossible values
2. Bar Chart (Categorical Data) or Histogram (Continuous Data)
The Normal Distribution
• Characteristics: Bell shaped with the majority of scores in the center of the distribution
• There are 2 ways in which a distribution can deviate from normal:
1. Lack of Symmetry (Skew)
2. Lack of Pointiness (kurtosis)
• Lack of Symmetry (Skew)
o The most frequent scores are clustered near one end of the distribution.
• Lack of Pointness (Kurtosis)
o The degree to which scores cluster to the end of distributions (the tails) or
towards the center. 2. The Centre of the Distribution
• Measures of Central Tendency
The score that occurs most frequently.
Appropriate for CATEGORICAL (binary, nominal, and ordinal) data.
• Variables can be bimodal or multimodal
The score in the middle of the distribution.
Appropriate for ordinal, interval and ratio data.
• Relatively unaffected by extreme scores and skew
• Most scores are not represented
The sum of all scores, divided by the number of
Appropriate for interval and ratio data.
• It uses all scores and is the most stable
across different samples.
• Affected by extreme scores and skew.
• It can only be used with interval and ratio data.
o The distance between the lowest and highest scores in a distribution.
o Highly influenced by extreme scores
o Relevant to ordinal, interval and ratio data
• Interquartile Range:
o A range that focuses on the middle 50% of a distribution
o Eliminates the issues surrounding extreme scores
o Applicable to interval and ratio data.
• Deviance: FORMULA o The spread of scores can be calculated by determining how
different each score is from the center ofiance the distribution (mean).
o This approach uses all scores.
o This applies to interval and ratio data.
• Sum of Squares (SS) / Total Dispersion: FORMULA
o To determine the total dispersion, the deviance
scores are summed. SS
o Each score is squared in order to eliminate
If this step is kipped, it will equal 0.
The size of the SS is dependent on the number of scores in the data,
making it impossible to compare it across samples that differ in size.
• Variance (s ) / Average Dispersion / Mean Squared
o Represents the average error between the
mean and the observation.
o The calculate the sample variation (not the
population), divide the SS/Total Dispersion by
the number of observations minus 1.
The variance is in squared united, not
the units of the original variable.
• Standard Deviation (s): FORMULA
o The square root of the variance (Represent the
average dispersion, in the same units as the
o A small Standard Deviation (relative to the
value of the mean itself) indicated that the data
points are close to the mean OR
o A large Standard Deviation (relative to the value
of the mean itself) indicates that the data points
are distant from the mean
• The Sum Of Squares, Variance, and Standard Deviation represent the same thing:
o The “Fit” of the mean to the data
o The variability within the data
o How well the mean represents the data
o The amount of error present
• Going Beyond the Data
o Another way to think about frequency distributions is not in terms of how often scores actually occurred, but how likely it is that a score would occur (probabilities).
o A probability value can range from 0 (it will not happen) to 1 (it will definitely happen).
o Probabilities can also be viewed in terms of percentages. A probability of .25
indicates that there is a 25% likelihood of obtaining that score.
• Using the histogram (or frequency chart), it is possible to calculate the probability of scores
• MONTHLY RENT EXAMPLE:
o Question: How likely is it that someone is paying 750$ or more in rent?
o There are 76 participants where 7 report paying 750$ or more for their monthly rent.
o By dividing 7 by 76, we obtain a probability of 0.092 (9.2%) of paying more than 750$
or more in monthly rent
• Another way to calculate the probability of obtaining a particular score is to use a probability
= NORMAL = ZDistribution
(M = 0, SD = 1)
o Probability distributions (there are many others: t-distribution, chi-square, etc.) are all
defined by an equation that enables us to calculate precisely the probability of
obtaining a given score.
• To determine the probability of obtaining a given score using theFORMULA
standard normal distribution:
1. Centre the data around zero: X − X
Subtract the mean from the score. z =
2. Make standard deviation equal to 1: s
Divide the resulting value by the sample’s original standard
3. Resulting value (z-score):
Expresses the score in terms of how many standard deviations it is away from the
To determine the respective probability associated with that score, check table.
Preparing Data for Analysis
• Before you can run analyses, your data set needs to be properly prepared.
• One of the most important aspects of this preparation is the data cleaning process.
• Data cleaning involves a series of specific steps.
• In most cases, researchers will spend the majority of their time cleaning their data and
only a small portion on the actual analyses.
Data Cleaning Procedures
1. Impossible and Out-of-Range Values
• Participants (or the individual who is manually entering the data) may sometimes
select values that are not within the actual range of possible answers for a given question.
o Examine the frequency table for the variables in question & identify responses
that are invalid
o Correct Error:
If you can access the original data, or the error is an obvious typo,
change the value to what it should have been
o Remove Value:
If you have no way of knowing what the original value should have
been, remove it and code it as missing.
2. Response Sets
• Some participants do not complete the questionnaires properly. It is important to
ensure, as much as possible, that these participants are not included in the analyses.
o Examine the raw data and look for suspicious patterns. This is often very
difficult to detect.
o Some researchers include trick questions to ensure the questionnaire is
o Remove Participant:
If you can identify a response set like this, exclude that participant from
3. Missing Values
• In social science research, it is VERY COMMON to have missing data.
• Participants will have missing data for a number of reasons (measurement error,
• It is essential to deal with it and report how it was managed.
o Examine the frequency tables for all variables, the number of missing values
will be indicated.
• Dealing with missing data is a highly controversial and heavily disputed practice.
• In general, the appropriate method is determined based on whether the missing data
is missing at random, or whether there is an underlying pattern behind the missing
• A common practice is to run a “Missing Values Analysis” to determine the pattern of
the missing data, and then impute (generate) new data to replace the missing
o Demographic Variables:
If demographic information is missing, you cannot modify it.
Assign it a missing values code in SPSS (usually 99) and leave it as
o Nominal or Ordinal Variables:
If the missing values represent more than 5% of the missing data on a particular variable, a new category (i.e. “Not Disclosed”) can be created
to represent those scores.
If the missing values represent less than 5% they can be left as
missing, like the demographic variables.
o Interval or Ratio Variables:
Missing values on interval and ratio variables can be replaced with the
mean score for that particular variable.
This is called Mean Replacement
• In statistics, OUTLIER scores are observation points that are very distant from the
other observations (INTERVAL AND RATIO DATA).
• Outlier scores can occur for a number of reasons but may be due to measurement
• Since these scores can drastically bias results, it is important to manage them.
Using histograms or boxplots
• Outlier scores exist at different levels within a given data set.
o Univariate Outliers:
Unusual scores in the observed variables.
o Bivariate Outliers:
Unusual scores that result from the
combination of two variables. FORMULA
o Multivariate Outliers: REARRANGED ZSCORE
Unusual scores in a combination of X = ( X−x ) +XsX z=
multiple variables. z= s
• Options: s
Substituting the outlier score with the next closest value that is not an
5. Check for Assumptions
• The analyses in this course are PARAMETRIC TESTS that are based on a normal
distribution of data.
• The assumptions of parametric tests include:
• The normal distribution is relevant to the following:
o Parameter estimates
o Confidence intervals
o Null hypothesis testing
• Does this assumption mean that all of our variables must have a normal
• When does the assumption of Normality matter?
o IT MATTERS WITH SMALL SAMPLES: The central limit theorem (the distribution of means will increasingly
approximate a normal distribution as the size of samples increases)
allows us to forget about this in larger samples.
o There are a number of different ways to detect normality within a particular
1. EYEBALL TEST: visually inspecting the histogram.
2. SKEWNESS AND KURTOSIS: by assessing the skewness and kurtosis
• By dividing the skewness and kurtosis values by their respective
standard errors, you create a standardized score (this is also called the
skewness and kurtosis ratios) that can be interpreted like a z-score.
o If the value exceeds the cut-off value, you can conclude you
have non-normal data.
For small samples of less than 50, use the Z-Score cut-off
associated with 95% (+/- 1.96).
For medium-sized samples (50 – 300), use the Z-Score
cut-off associated with 99% (+/- 3.29).
3. FORMAL NORMALITY TESTS: by conducting a hypothesis test on the
shape of the distribution (not reliable with large samples).
o Change Tests: The easiest solution to non-normal data is to use tests that
do not require normal data.
o Bootstrapping: Estimates the properties of the sampling distribution,
based on the sample data (will do this 1000s of times).
o Transformation: Use mathematical functions to transform data into a
distribution that is normal.
• The outcome variable is linearly related to the predictors.
o The amount of change between scores on two variables is constant for the
entire range of scores for the variables.
• If a relationship is nonlinear, the statistics that assume it is linear will
underestimate the strength of the relationship, or fail to detect the existence of a
III. Homoscedasticity/Homogeneity Of Variance
• When testing several groups of participants, samples should come from
populations with the same variance.
• In correlational designs, the variance of the outcome variable should be stable at
all levels of the predictor variable.
• Impacts parameter estimates and null
hypothesis significance testing.
• Assessing Homogeneity of Variance
o LEVENE’S TESTS Tests if variances in different groups are the same.
Significant = PROBLEM! Variances not equal.
Non-Significant = Variances are equal.
• The errors in your model should not be related to each other.
• Example: A statistics test is given to students to assess their
comprehension. Any errors executed by the students on the test will be the
result of lack of knowledge, lack of question comprehension, or a
performance error. These errors are not associated with anyone else’s
performance on the test, they are independent. Two students copy off of
each other. These students’ errors are now associated with each other and
no longer truly independent.
• If this assumption is violated: Confidence intervals and significance tests
will be invalid.
•These assumptions must be met before proceeding with analyses.
• Do these items measure the same thing?
o A coefficient of internal consistency: Cronbach’s Alpha (α)
Cronbach's alpha will generally increase as the intercorrelations among
test items increase, and is thus known as an internal consistency
estimate of reliability of test scores.
Because intercorrelations among test items are maximized when all
items measure the same construct, Cronbach's alpha is widely believed
to indirectly indicate the degree to which a set of items measures a
single unidimensional latent construct.
0.9 – 1.0 = Excellent Fit
0.7 – 0.9 = Good Fit
0.6 – 0.7 = Acceptable Fit
0.5 – 0.6 = Poor Fit
0.0 – 0.5 = Unacceptable Fit
• Note: Cronbach’s Alpha will increase if you add items
Building Statistical Models
• Scientists build models of real-world processes in an attempt to predict how these processes
operate under certain conditions.
• Since we cannot access the real-world processes directly, we can make inferences based on
• We want our model to be as accurate so that the predictions are valid, therefore we must build
a statistical model based on the data we collect.
• The degree to which a statistical model represents the data collected is known as the fit of the
o GOOD FIT: An excellent representation of the real-world. Since this model represents
reality so closely, it can be used to confidently make predictions about reality. o MODERATE FIT: Some similarities to reality, but some important decisions. The
predictions from this model may be inaccurate.
o POOR FIT: The model is completely different to the real-world situation. Predictions from
this model will likely be completely inaccurate.
The Equation Underlying All Models
OUTCOME = (MODEL) + error
• INTEPRETATION: The data we observe is predicted from the model we choose to fit the data,
• Model: Models vary in their complexity and the equation will change depending on the study
design, the type of data, and the purpose of the model. Models are made of variables and
o VARIABLES: Measured constructs that vary across the sample (the data we have
o PARAMETERS: Estimated from the data (rather than being measured) and are
constants that represent relationships between the variables in the model.
Examples: The mean and median estimate the centre of the distribution.
Correlation and regression coefficients estimate the relationship between two
• Building Model Equations
• Testing Model Fit
o We can use the Sum of Squared Error and the Mean Squared Error to assess the fit
of a model.
Large values, relative to the model, indicate a lack of fit.
Small values, relative to the model, indicate good fit.
o If our model is the mean, the Sum of Squared Error = Sum of Squares and the Mean
Squared Error = Variance.
In more complex models, the sum of squared error and mean squared error
represent the same thing, but is measured differently.
• Populations and Samples
o POPULATION: The collection of units (people, plants, cities, etc.) that we want to
generalize a set of findings or a statistical model.
o SAMPLE: A smaller (but hopefully representative) collection of units from a
population used to answer questions about that population. • The Standard Error
o The standard deviation tell us about how well the mean represents our sample data.
o If we want to use our sample mean as a parameter estimate for the population mean,
we need to know how well it represents the population values.
o Sampling Variation:
Samples will vary because they contain different members of the population.
o Sampling Distribution:
If you were to plot the means of all of the samples, it would result in a nice
o The standard error is a measure of how representative a sample is likely to be of the
o A large standard error (relative to the sample mean) means there is a lot of variability
between the mean of different samples. The mean may not be representative of the
o A small standard error (relative to the sample mean) means there is less variability
and it is likely representative of the population mean.
o When we estimate the parameters in a model, we use data from our sample to infer the
true value in the population.
o This estimate can change across samples (we have just seen this with the mean). The
standard error gives us an idea of how much these estimates differ.
o We can also use the standard error to calculate boundaries (CONFIDENCE
INTERVALS) that we believe the true population value will fall.
o Calculating Confidence Intervals (Mean):
• The sampling distribution will be normal.
• We use the standard normal distributions and the z-score formula to
determine the values.
• Replace the standard deviation with the standard error since we
estimating the population. Null Hypothesis Significance Testing
• Based on 2 main principles:
1. Computing probabilities to evaluate evidence
o You calculate the probability of something occurring within the research context. If it
is unlikely that it would occur (less than 5%), it would be strong evidence to back up
2. Competing Hypotheses
o Alternative Hypothesis:
The hypothesis that comes from the theory and states that an effect will be
• H1: Tutorial attendance has a positive effect on course success.
o Null Hypothesis:
The hypothesis that comes from the theory and states that an effect will be