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Chapter 2

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PSYC 3250
Jeffrey Spence

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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