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Chapter

PSYC 2040 Chapter Notes -Median Absolute Deviation, Average Absolute Deviation, Variance


Department
Psychology
Course Code
PSYC 2040
Professor
David Stanley

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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
o µ is the mean, σ is the standard deviation and σ2 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

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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 75th and the 25th percentile
divided by 2
This makes it more stable than the range
o Absolute deviations
Mean absolute deviation

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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
It is generally identified as S2
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 as g1 and the formula is
  

Cubing a large deviation yields a large number and retains the sign of the
deviation, thus if g1 is
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