Ψlogical
Testing

Chapter 2…
…Norms &
Basic Stats

Office keeping 💼

• Project:
  • Settle on construct (name)
    within group
  • Find 2-3 empirical articles
    (each team member)
  • Complete Assignment #1
    in Lab Thursday
    • Construct complexity
      (beyond simple label)

A primary function of tests…

…is to translate observations into numbers

How we do this:

Measurement – assignment of numbers to objects according to rules

Note

The terms measurement and scaling are widely considered synonymous, however, we’re mostly going to try to reserve use of the term scale to refer to item aggregations

Use of numbers…

…is typically accomplished via statistical manipulation

• Descriptive
• Inferential

Summaries, typically contrasting:

• one group of numbers relative to another group
• one number relative to a larger group of numbers

• common descriptives:
  • mean, median, mode
  • standard deviation, variance
  • z-score

Logical deductions about broader group(s):

• do sample differences imply population differences?
• are sample associations reflective of population effects?
• realm of p-values (e.g., p < .05)

• common inferentials:
  • t-test
  • ANOVA
  • Regression

Fundamental properties of numbers

• Numbers are used to represent an individual’s level of standing along a psychological attribute
  • However – Not all numbers retain their inherent characteristics with Psychological measurement
• Different rules of assignment collectively define different “scales of measurement”
  • NOIR (Stevens, 1946)

Note

HOW we assign numbers to objects is very important, because if the numbers do not retain their inherent properties, then perhaps it’s not appropriate to perform mathematical/ statistical operations on those numbers…

Property Zero¹

Numbers can reflect sameness versus differentness

• Numbers here are merely labels
• Differences in kind / sort / ilk, not in amount
• Things within a category (i.e., with same number) are identical in terms of the relevant feature

1. Book doesn’t explicitly mention but important for understanding Stevens (1946)

Property I

Numbers can reflect magnitude of differences between things

• Numbers here are again used as labels, but sequence is also being reflected
• “How much”?

Property II

Differences between numbers hold equal intervals

• moving from “4” to “7” has the same meaning as moving from “10” to “13”
• numbering doesn’t “increase” until underlying Psychological trait “moves” the same distance

Property III

Zero reserved for only very special scenario

• complete absence of the attribute
• If zero is absolute, then we can compare scores in terms of ratios (e.g., twice as long)

Scales of measurement (Stevens, 1946)

• Nominal
• Ordinal
• Interval
• Ratio

To measure something, we must acknowledge units of measurement (e.g., weight: # of lbs; temp = Fº)

Question for class: What is the unit of measurement in psychology?

Each test has its own “unit” of measurement (one “point” on an IQ test is not same as one “point” on a math test)

Most often for us, the unit of measurement is a statistical index called the standard deviation

Stats stuff!!

Frequency distributions

Tell us:

1. location of scores
2. spread of scores

Formally, how frequently is each value encountered within a sample

Location & Spread

Location:

• Mean
• Median
• Mode

Spread:

• Range
• Standard deviation
• Variance

In-class data collection:

Collect data (number of pets?) and demonstrate:

1. data
2. dataset
3. frequency distribution
4. location
5. spread

Importance of frequency distributions

• We use “distributional” information (location and spread) to generate relative scores
  • Relative to other scorers in the distribution
  • Commonly expressed as either %^ile or standard score

Percentiles

• percent of distribution that scores at or below a raw score value

• proceed from low (e.g., 1) to high (e.g., 99) as travel from left to right within distribution

Book makes an odd distinction between “percentile rank” and “percentiles” – this is unnecessarily confusing. Treat them as equivalent (e.g., both “percentile rank” and “percentile” should be interpreted as referring to the same thing).

Standard Scores

• Most commonly rescaled z-scores
• z-score: how many standard deviations away from the mean a raw score is
• trick to understanding z-scores:
  • raw score (\(X\))
  • relative score (\(Z\))

Z as reported score

• \(Z\) scores below distribution’s mean always negative
• If delivering feedback, probably not wise to keep that scale
• \(Z\)’s therefore typically transformed (p. 49)
  • IQ score of 100 is actually a \(Z\) of 0
  • \(Z\) of 0 is ACT score of 20

Tip

Percentiles are never rescaled (probably less confusing)

Standard Normal Distribution

• Defined by mathematical formula
• Nothing distributed “normally”
  • very many things distributed “approximately normally”
• Tells us relationship between z-score and percentiles within this mathematically ideal distribution

Note

“Relationship” detailed in book’s Appendix I

Norms

Can refer to:

• frequency distribution itself (aka normative group), or
• common values within that distribution
  • raw score at 50^th percentile

More generally, the sample retained for obtaining relative scores

Two types of tests

Norm-referenced:

• How the test-taker performs relative to the comparison group (e.g., norm)
• Feedback most typically percentile and scale score (transformed \(Z\))

Criterion-referenced:

• How the test-taker performs relative to an objective standard
• Feedback often raw scores and might also include pass or fail designation

References

Stevens, S. S. (1946). On the theory of scales of measurement. Science, 103(2684), 677–680.