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Psychology
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PSYC 3250
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Jeffrey Spence
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Chapter 2

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Psychology

PSYC 3250

Jeffrey Spence

Winter

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Chapter 2: The Basic Statistic of Measurement
The Role of Mathematics in Assessment
Scales of Measurement
- what is measurement?
o measurement - set of rules for assigning numbers to represent objects, traits, attributes, or behaviours
o psychological tests – measuring devices, involve rules for assigning numbers to an individual’s performance that
are interpreted as reflecting characteristics of the individual
o scale of measurement – units of measurement have this mathematical property
o scale – system or scheme for assigning values or scores to the characteristic being measured
o Stevens
proposed a taxonomy that specified 4 scales of measurement
• nominal, ordinal, interval, and ratio
- nominal scales
o nominal scales
simplest
classify people or objects into categories, classes, or sets
can assign numbers simply to identify or label categories
• categories are not ordered in a meaningful manner
• assignment of numbers is completely arbitrary
• serve only as names for the categories
• numbers should not be added, subtracted, ranked, or otherwise manipulated
- ordinal scales
o ordinal scales - rank people or objects according to the amount of a characteristic they display or possess
o enable use to quantify the variables under examination and provide substantially more information than nominal
scales
o provides quantitative information
does not ensure that the intervals between the ranks are consistent
o indicate the rank-order position among individuals or objects, but they do not indicate the extent by which they
differ
o somewhat limited in both the measurement information they provide and the statistical procedures that can be
applied
o examples – percentile rank, age equivalents, and grade equivalents
- interval scales
o interval scales - rank people or objects like an ordinal scale but on a scale with equal units
o equal scale unit – difference between adjacent units on the scale is equivalent
o many psychological tests are designed to produce interval-level scores
can be manipulated using common mathematical operations, whereas less scales cannot
o most statistical procedures can be used with interval scale data
o limitation
no true zero
• zero does not reflect the total absence of the attribute
ratios are not meaningful
o behavioural variables
intelligence or personality characteristics – don’t know where the true zero lies
o physical characteristics
zero point is well defined
o derived by first locating the midpoint of a variable and then measuring outward in each direction, above and
below, as far as we can establish scores with reasonable accuracy
o psychology
interval scale scores are most commonly seen in the form of standard scores
- ratio scales o ratio scales - have the properties of interval scales plus a true zero point that reflects the complete absence of the
characteristic being measured
o not meaningful or interpretable with other scales
o few ratios in psychological measurement
- hierarchy among the scales
o nominal scales
least sophisticated and providing the least information
allowed to assign a number to a person that associates the person with a set or category
other useful quantitative properties are missing
o ratio scales
most sophisticated and providing the most information
have all the positive properties of nominal scales with addition of the ability to rank people according to
the amount of a characteristic they possess
have all the positive properties of an interval scale with the addition of an absolute zero point
• allows one to form meaningful ratios between scores
o interval scales
have all the positive properties of ordinal scales and incorporate equal scale units
• allows one to make relative statements regarding scores
The Description of Test Scores
- to interpret or describe test scores meaningfully you need to have a frame of reference
- often the frame of reference is how other people performed on the test
- distributions
o distribution - set of scores
o represented – tables and graphs
o grouped frequency
possible scores are “combined” or “grouped” into class intervals that encompass a range of possible
scores
o frequency graphs
popular and provide a visual representation of a distribution
x-axis – scores
y-axis – frequency of scores
o symmetrical distributions
o skewed distributions – positively or negatively
- measures of central tendency
o many distributions tend to concentrate around a center
o 3 measures of central tendency
mean, median, and mode
o mean
mean – arithmetic average of a distribution
MEAN = SUM OF SCORES/ NUMBERS OF SCORES
useful in providing a sense of the central tendency of the groups of scores
• 1. mean is meaningful for distributions containing interval and ratio level scores
• 2. mean of a sample is a good estimate of the mean for the population from which the same was
drawn
• 3. essential to the definition and calculation of other descriptive statistics that are useful in the
context of measurement
undesirable characteristic
• sensitive to unbalanced extreme scores
• influence of an extreme score decreases as the total number of scores in the distribution
increases
Box: Population Parameters and Sample Statistics • population – complete group of people, objects, or other things of interest
• sample – subset of the larger population that is though to be representative of the population
• population values – parameters and are represented by Greek symbols
o mu (u) – indicate a population mean
o sigma (o) – indicate a population standard deviation
• statistic – value that is calculated based on a sample and represented by Roman letters
o sample statistics can provide information about the corresponding population parameters
o median
median – score or potential score that divides a distribution in half
calculated for distributions containing ratio, interval, or ordinal level scores
• not appropriate for nominal level score
useful and versatile measure of central tendency that is particularly useful for many common descriptive
purposes
o mode
mode – most frequently occurring score in a distribution
advantage
• can be used with nominal, ordinal, interval, and ratio data
limitations
• 1. some distributions have two scores that are equal in frequency and higher than other scores
o bimodal distribution
mode is ineffective as a measure of central tendency
• 2. mode is not a stable measure of central tendency
o choosing between the mean, median, and mode
which measure of central tendency is most useful and appropriate?
• mean
o essential when calculating other useful statistics
o considerable utility as a measure of central tendency
• median
o most versatile and useful measure of central

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