Variance:

  • Represents the amount of variability in the data
  • Three main methods of representing variability o Range (smallest and largest data points) o Standard deviation o Variance

Correlation:

  • Standardised representation of the associations between two variables
  • Can range from -1.0 to 1.0
  • Coefficient of determination is the correlation squared

Composite variables:

  • Test scores tend to be based on the sum of two or more items
  • g. the Beck Depression inventory consists of 21 items:

o Items are scored on an ordinal scale from 0-3 o Thus the range of scores possible is 0-63

  • Such ‘sum scores’ are known as ‘composite scores’.

Composite variables and variance:

  • The variance of a composite score is a function of:

o The variance associated with the individual items o The correlation amongst the items

  • A positive correlation between two variables increases the variance associated with the composite derived from those two variables.

o i.e. as correlation increases (positive), the magnitude of the corresponding composite score variance also increases.

Binary items- Variance:

  • Most frequently encountered in achievement type tests, e.g. dichotomous items such as exam or intelligent test items
  • All other things equal, having more variance is typically better than having less variance with respect to some important psychometric characteristics
  • Calculating the variance of a composite based on dichotomous items works the same way as demonstrated for ordinal/continuous items

Interpreting test scores:

Most scores produced by psychological tests are not clearly interpretable in their own right

  • Two of the most common psychological test score interpretations are (1) relative and (2) abstract
  • Relative interpretations are based on the analysis of data
  • Abstract interpretations are based on theoretically relevant characteristics of the body of research which supports the test scores as valid indicators of a psychological construct

Relative interpretations:

  • From the relative perspective, to interpret an individual’s score we need to:

o Make reference to an entire distribution of scores on the test and o Identify where the individual falls in the distribution

  • In practical terms this involves knowing the mean distribution of scores and the standard deviation of the distribution of scores

Raw scores- limitations:

  • Although knowledge of the mean and standard deviation associated with a group of scores allows us to interpret a particular score in a relative way, it is not a precise method unless you are a human calculator.
  • Instead raw scores can be converted into standardised scores which incorporate information about the mean and standard deviation- z-scores.

Z-scores:

  • Have a mean of zero and a SD of one
  • To convert raw scores into z-scores: z = X – μ / σ
  • By framing the meaning of a score in terms of ‘distance from the mean’, the z score frees us from worrying about the units of the original test score
  • Can be used to compare scores across tests that are on different sized units

Converted standard scores:

  • Ideally don’t want to use negative numbers (in z scores), so you can easily convert them into alternated standard scores
  • Accomplished by rescaling the scores so that the converted scores have a different mean and standard deviation- T scores.
  • With t-scores the mean is always 50 and the standard deviation is always 10

T-scores:

  • Convert raw scores to z scores

Convert z scores using the formula:  T=z(10) +50