Intro statistics

Introduction to Statistics
February 21, 2006

Statistics and Research Design
• Statistics: Theory and method of
analyzing quantitative data from samples
of observations … to help make
decisions about hypothesized relations.
– Tools used in research design
• Research Design: Plan and structure of
the investigation so as to answer the
research questions (or hypotheses)

Statistics and Research Design
• Analogy:
– Research design is the blueprint of the study.
– In quantitative designs, statistical design and
procedures are the craft and tools used to conduct
quantitative studies.
– The logic of hypothesis testing is the decision-
making process that links statistical design to
research design.

Statistics
• There are two types of statistics
– Descriptive Statistics: involve tabulating,
depicting, and describing data
– Inferential Statistics: predicts or estimates
characteristics of a population from a
knowledge of the characteristics of only a
sample of the population

Descriptive Statistics
Scales of Measurement
– Nominal
• No numerical or quantitative properties. A way to
classify groups or categories.
• Gender: Male and Female
• Major: RC or PH
– Ordinal
• Used to rank and order the levels of the variable
being studied. No particular value is placed
between the numbers in the rating scale.
• Movie Ratings: 4 Stars, 3 Stars, 2 Stars, and 1 Star

Scales of Measurement Cont.
– Interval
• Difference between the numbers on the scale is meaningful
and intervals are equal in size. NO absolute zero.
• Allows for comparisons between things being measured
• Temperatures on a thermometer: The difference between 60
and 70 is the same as the difference between 90 and 100. You
cannot say that 70 degrees is twice as hot as 35 degrees, it is
only 35 degrees warmer.
– Ratio
• Scales that do have an absolute zero point than indicated the
absence of the variable being studied. Can form ratios.
• Weight: 100 pounds is ½ of 200.
• Time

• Frequency Distributions
– In tables, the frequency distribution is
constructed by summarizing data in terms of
the number or frequency of observations in
each category, score, or score interval
– In graphs, the data can be concisely
summarized into bar graphs, histograms, or
frequency polygons

– Normal Curve – Bimodal Curve

– Positively Skewed – Negatively
Skewed

• Measures of Central Tendency
– Mode
• The most frequently occurring score
• 3 3 3 4 4 4 5 5 5 6 6 6 6: Mode is 6
• 3 3 3 4 4 4 5 5 6 6 7 7 8: Mode is 3 and 4
– Median
• The score that divides a group of scores in half with 50% falling
above and 50% falling below the median.
• 3 3 3 5 8 8 8: The median is 5
• 3 3 5 6: The median is 4 (Average of two middle numbers)
– Mean
• Preferred whenever possible and is the only measure of central
tendency that is used in advanced statistical calculations:
– More reliable and accurate
– Better suited to arithmetic calculations
• Basically, and average of all scores. Add up all scores and divide
by total number of scores.
• 2 3 4 6 10: Mean is 5 (25/5)

• Measures of Central Tendency
– Your Turn!
– Mode
• Example: 2 3 4 4 4 6 8 9 10 11 11
– Median
• Example: 2 3 4 4 4 6 8 9 10 11 11
– Mean
• Example: 2 3 4 4 4 6 8 9 10 11 11

• Measures of Variability (Dispersion)
– Range
• Calculated by subtracting the lowest score from the
highest score.
• Used only for Ordinal, Interval, and Ratio scales as the
data must be ordered
– Example: 2 3 4 6 8 11 24 (Range is 22)
– Variance
• The extent to which individual scores in a distribution of
scores differ from one another
– Standard Deviation
• The square root of the variance
• Most widely used measure to describe the dispersion
among a set of observations in a distribution.

• Standard Scores: Z-Scores and T-Scores
– Z-Scores
• Most widely used standard score in statistics
– It is the number of standard deviations above or below the
mean.
• A Z score of 1.5 means that the score is 1.5 standard
deviations above the mean; a Z score of -1.5 means that
the score is 1.5 standard deviations below the mean
• Always have the same meaning in all distributions
• To find a percentile rank, first convert to a Z score and
then find percentile rank off a normal-curve table

• Standard Scores: Z-Scores and T-Scores
– T-Scores
• Most commonly used standard score for
reporting performance
• May be converted from Z-scores and are always
rounded to two figures; therefore, eliminating
decimals
• Always reported in positive numbers
• The mean is always 50 and the standard
deviation is always 10.
– A T-score of 70 is 2 SDs above the mean
– A T-score of 20 is 3 SDs below the mean

• Correlation or Covariation
– A correlation coefficient is a statistical summary of
the degree or magnitude and direction of the
relationship or association between two variables
– It is possible to have a negative or positive
correlation
• Linear Regression
– The purpose of a regression equation is to make
predictions on a new sample of observations from
the findings on a previous sample

Inferential Statistics: Sampling
• Sampling relates to the degree to which those
surveyed are representative of a specific
population
• The sample frame is the set of people who have
the chance to respond to the survey
• A question related to external validity is the
degree to which the sample frame corresponds
to the population to which the researcher wants
to apply the results (Fowler, 1988)

Sampling
• Two basic types: probability and non-
probability
• Probability sampling can include random
sampling, stratified random sampling, and
cluster sampling
• Non-probability sampling can include quota
sampling, haphazard sampling, and
convenience sampling

Random Sampling
• Every unit has an equal chance of
selection
• Although it is relatively simple, members
of specific subgroups may not be
included in appropriate proportions

Stratified Random Sampling
• The population is grouped according to
meaningful characteristics or strata
• This method is more likely to reflect the
general population, and subgroup
analysis is possible
• However, it can be time consuming and
costly

Systematic Sampling
• Every xth unit is selected
– (e.g., every other person entering the Swamp at
Gate 1 was selected)
• The method is convenient and close to random
sampling if the starting point is randomly
chosen
• Recurring patterns can occur and should be
examined

Cluster/Multistage Sampling
• Natural groups are sampled and then
their members are sampled
• This method is convenient and can use
existing units

Convenience Sampling
• This method uses readily available groups or
units of individuals
• It is practical and easy to use
• However, it may produce a biased sample
• Convenience sampling can be perfectly
acceptable if the purpose of the research is to
test a hypothesis that certain variables are
related to one another

Snowball Sampling
• Previously identified members identify
others
• This method is useful when a list of
potential names is difficult to obtain
• However, it may produce a biased
sample

Quota Sampling
• The population is divided into subgroups
and the sample is selected based on the
proportions of the subgroups necessary to
represent the population
• This method depends on reliable data
about the proportions in the population

Statistics & Parameters
– A parameter is a value, usually unknown (and
which therefore has to be estimated), used to
represent a certain population characteristic. For
example, the population mean is a parameter that is
often used to indicate the average value of a
quantity
– A statistic is a quantity that is calculated from a
sample of data. It is used to give information about
unknown values in the corresponding population.
For example, the average of the data in a sample is
used to give information about the overall average
in the population from which that sample was
drawn.

Sampling Distribution
• The sampling distribution describes
probabilities associated with a statistic
when a random sample is drawn from a
population

Response Rates
• Whatever the sampling technique, response
rates and non-response bias must be
considered
http://content.apa.org/journals/pro/32/3/248.html
• Lowered response rates introduce bias into
the sample
• In cases of low response rates, people who
respond to the survey are likely to be
systematically different from people who do
not respond to the sample

Response Rates
• In mail surveys, the results of non-response bias can be
examined by comparing those who respond early with
those who respond after follow up
• Most government-sponsored surveys require response
rates of 75%
• For mail surveys, post-cards, follow-up letters, and
telephone calls are used to increase the response rates
(Fowler, 1988)
• According to Babbie (1989), a response rate of 70% is
very good, 60% is good, and 50% is adequate

Inferential Statistics
• Interval Estimate
– A range or band within which the parameter is
thought to lie, instead of a single point or value as
the estimate of the parameter

• Sampling Distributions
– The sampling distribution of the mean is a
frequency distribution, not of observations, but of
means of samples, each based on n observations.
– The standard error of the mean is used as an
estimate of the magnitude of sampling error. It is
the standard deviation of the sampling distribution
of the sample means.

• Confidence Intervals
– Same as the percentage of cases in a normal
distribution that lie within 1, 2, or 3 standard
deviations from the mean
• Central Limit Theorem
– States that the distribution of samples (means,
medians, variances, and most other statistical
measures) approaches a normal distribution as the
sample size, n, increases
• Hypothesis Testing – will cover next.

• Types of Statistical Analysis - Descriptive
– Quantify the degree of relationship between
variables
– Parametric tests are used to test hypotheses with
stringent assumptions about observations
• e.g., t-test, ANOVA
– Nonparametric tests are used with data in a nominal
or ordinal scale
• e.g., Chi-Square, Mann-Whitney U, Wilcoxon

• Types of Statistical Analysis - Inferential
– Allow generalization about populations using data
from samples
– Non-parametric
• Non-parametric tests do not require any assumptions about
normal distribution, but are generally less sensitive than
parametric tests.
• The test for nominal data is the Chi-Square test
• The tests for ordinal data are the Kolmogorov-Smirnov
test, the Mann-Whitney U test, and the Wilcoxon
Matched-Pairs Signed-Ranks test
– Parametric
• The tests for interval and ratio data include the t-test,
ANOVA, ANCOVA, and Post-Hoc ANOVA tests

Intro statistics

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