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# Research Statistics (Psych 2040) Chapter Summaries

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University of Guelph

Psychology

PSYC 2040

David Stanley

Winter

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Research Statistics Chapter Summaries
Chapter 1: Statistics, Computers and Statistical Packages
Basic Definitions
Samples and Populations
Samples refers to some subset of a population and a population is some large set of
numbers
o There are 3 types of populations
Finite population: has a definite number of individuals
Infinite population: has no limit on the number of individuals
Theoretical population: is simply an equation
The difference between a random sample and a non-random sample
o Random sample is one in which every observation in the population has an equal
opportunity of being selected
Inferential statistics are all based on the assumption of random sampling
o A non-random sample is one in which this is not the case
Statistics and Parameters
Statistic: is a number that describes a sample
o Ex. the mean, standard deviation and the variance
Parameter: is a number that describes a population 2
o µ is the mean, σ is the standard deviation and σ is the variance
Difference between the parameter and the statistic
o We are usually unable to assess the population
o Instead we have to make do with a sample from the population and use the
information we obtain on the sample to estimate the corresponding value of the
population
We have a statistic and wish to estimate the parameter
Unbiased and Biased Estimates
The difference between biased and unbiased estimates of parameters
A statistic is said to be unbiased if the mean of all possible values of that statistic is equal
to the parameter
o The mean is a statistic that is unbiased
o This is because the mean of the means of all possible samples from the population
equals the population mean
A statistic is said to be biased if the mean of all possible values is not equal to the
population value
o The variance is said to be biased because the mean of the variance of all possible
samples from a population is less than the population variance Types of Statistics
Statistics of Location
Statistics serve to locate the sample on the number line, which is a representation of all
possible numerical values ranging from minus infinity to plus infinity
Any statistic that helps you locate where the sample is on this number line is a statistic of
location
One class of such statistics are the 3 measures of central tendency
o Median
The median is that value such that 50% of the values are greater than that
value and 50% are less
It is the value that is at the centre of all the values
o Mode
The mode is that value that occurs most frequently and the most popular
one
If 2 values had the same highest frequency, the distribution of the scores
are called bimodal and if more it is called multi-modal
o Mean
The mean is that value such that the sum of the deviations of the scores
from their mean and add to 0
It is identified as ̅
The formula is
̅
Statistics of Scale
Statistics of scale describe how much differentiation there is in a sample
o Sometimes called statistics of variation or dispersion
If all the numbers are close together, the scale is small and if big, the scale is large
They give an indication of the amount of variability in a sample
There are a few statistics of scale
o Range
The simplest measure of scale is range
It is defined as the difference between the highest and the lowest value
However that makes it a problem measure since it only takes 2 values into
account
Measures that use all of the numbers in the sample would be expected to
give much more stable answers
o Semi-interquartile range
A type of range statistic that uses more information in the distribution is
the semi-interquartile range, also known as the quartile deviation
Is it defined as the difference between the 75 and the 25 percentile
divided by 2
This makes it more stable than the range
o Absolute deviations
Mean absolute deviation Mean absolute deviation is computed by summing the absolute
deviations and dividing by the mean of deviations
This mean would give an indication of the relative size of the
deviations of the values from the mean
Median absolute deviations
The median absolute deviation is simply the median of the
absolute deviations
This tells us that roughly 50% of the values differ more than this
much from the mean, while 50% of the values deviate less than
this
o Variance and standard deviation
Variance is the squared deviations of the values from the mean and then
calculate the mean of these squared deviations
2
It is generally identified as S
The formula is biased formula is
o If you wished to describe the variance or the standard
deviation of the sample, you would use the biased estimate
( ̅
The formula for the unbiased estimate of the population variance is
defined as
o If you wished to used your statistic to estimate the
population variance or the standard deviation of the sample,
you would use the unbiased estimate
( ̅
The square root of the variance is referred to as the standard deviation
and it is identified as S
Statistic of Shape
A statistic of shape tells us how the values are distributed along the line, whether they are
symmetrically distributed around the mean or skewed to one end of the other
A standard score is identified by the letter Z and it is defined as
̅
o S is the biased estimate of the variance and Z values are a transformation of the
original X values, such that the mean of the Z is 0 and the variance is 1
The statistic of shape are:
o Skewness
Skewness is a measure of asymmetry of the distribution of numbers
It is identified as1g and the formula is
( ̅ )
Cubing a large deviation yields a large number and retains the sign of the
deviation, thus if 1 is Positive it indicates that there is a long tail running out to the right
(the larger values)
Negative, the tail runs out to the left (smaller values)
Is 0, it means the distribution is symmetrical
o Kurtosis
Kurtosis is a measure of the presence of extreme values in the distribution
Is it defined by the notation 2
Statistics of Association
The statistics of association are association, correlation and regression
Types of Parameters
For each type of statistic, there are the corresponding parameters
We expect that there will be a sampling distribution of sample means around the
population mean
Standard Error and Statistical Inference
The standard deviation of the sampling distribution is referred to as the standard error
o It is defined as
̅
√
o It is the standard deviation of all possible means for a population
Normal distribution
o If a population is normally distributed, the sampling distribution of means will
also be normally distributed, and will have a mean equal to the mean in the
population (µ) and a standard deviation equal to √ .
o 95% of the standard normal distribution falls between -1.96 and +1.96 where 5%
falls out of that range
The central limit theorem
o States that the distribution of means will tend to be normal regardless of the shape
of the distribution in the population provided that the population variance is finite,
and sample size is large
o A sample size of 30 is often considered reasonable
Overview of SPSS 14
The SPSS system is a general purpose program that permits you to permits you to
perform many statistical tests and to conduct graphs of the results
This system is made up of three major components, a data editor, a syntax editor and a
viewer
o The data editor permits you to type in the data or to inspect it if it already exists
o You can enter the data and variable names into the table directly or create a data
file (text file) and open it in the data editor
Types of SPSS files
There are 3 types of files in SPSS
o .SAV file produced by the data editor o .SPS file that is produced by syntax editor and consists of the instructions given to
the computer, and serves as a handy reminder of precisely what operations were
executed
Inputting Data
Data can be typed into the Data Editor directly
o Each row represent a different individual participant
o Each column represents a different variable
Data can be typed as an text file
Running Jobs
The standard way of running jobs is to first of all make sure the data are in the Data
Editor, then click Analyse on the Menu bar and choose the program you want
Inputting ASCII (Text files) into SPSS14
Types of ASCII files
There are 2 types of ASCII data files that can be considered
o A Delimited file is one that separates the data by some form of a delimiter, and in
SPSS, you are provided with a number of options such as tab, space, etc…
Ex. 1 2 3 4
Follows the same rules as inputting data, rows = participant, column =
variables
o A fixed width file contains the numbers in given fields
Steps in Using SPSS Frequencies
Many of the statistics can be computed using the SPSS frequencies program which can be
run by:
o Opening the data/inputting the data
o Click on Analyze and pick Descriptive statistics
o Click on frequencies
o Click the statistics you want from the window, such as mean, median, mode,
Skewness, kurtosis, standard deviation, variance and range Chapter 2: The T-test
We have the information about the means of two samples and want to know whether they
are different from each other
These two samples can be the experimental condition or the control condition
We can use a t-test to do this and try to conclude that the null hypothesis is false if the t-
statistic is less than .05
General Rationale
Inferences Involving a Single Sample Mean
How likely is it that a sample with a given mean (statistic) could originate from a
population with a given mean (parameter)?
The T-test and Comparison of Independent Means
A t-test is performed to compare two independent means, and it is found that the two
sample variance are themselves more variable than reasonably can be attributed to
chance, it is necessary to compute the Welch estimate

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