Chapter 3

• Test-taker


  Person who takes a test
• Test user


  Person who interprets test scores by applying knowledge and skill
• Test scores are frequently expressed as numbers, and statistical tools are used to describe, make inferences from, and draw conclusions about numbers

  
• Measurement


  The act of assigning numbers or symbols to characteristics of things (people, events, whatever) according to rules
• Scale


  A set of numbers (or other symbols) whose properties model empirical properties of the objects to which the numbers are assigned
• Scales of measurement

  • Nominal
  • Ordinal
  • Interval
  • Ratio

  
• Nominal scales


  • Simplest form of measurement
  • Involve classification or categorization based on one or more distinguishing characteristics
  • All things measured must be placed into mutually exclusive and exhaustive categories
• Ordinal scales


  • Permit classification and rank ordering on some characteristic
  • Have no absolute zero point
  • Imply nothing about how much greater one ranking is than another
• Ordinal scale examples


  • Data derived from an intelligence test
  • Triage
• Interval scales


  • Contain equal intervals between numbers
  • Contain no absolute zero point
  • It is possible to average a set of measurements and obtain a meaningful result
• Ratio scales


  • Have a true zero point
  • All mathematical operations can meaningfully be performed
• Distribution


  A set of test scores arrayed for recording or study

  
• Raw score


  A straightforward, unmodified accounting of performance that is usually numerical
• Frequency distribution


  All scores are listed alongside the number of times each score occurred

  
• Measures of central tendency


  Statistics that indicate the average or midmost score between the extreme scores in a distribution

  
• Mean


  The arithmetic average of a set of scores

  
• Median


  The middle score in a distribution
• Mode


  The most frequently occurring score in a distribution of scores

  
• Measures of central tendency

  • Mean
  • Median
  • Mode
• Mean


  • Most commonly used measure of central tendency
  • Most appropriate measure of central tendency for interval or ratio data
  • Can be used for both continuous and discrete numeric data

  
• Median


  • Appropriate measure of central tendency for ordinal, interval, and ratio data
  • Less affected by outliers and skewed data than the mean

  
• Mode


  • Can be found for both numerical and categorical (non-numerical) data
  • May not reflect the centre of the distribution very well
  • Possible to have more than one mode
• Symmetrical distributions

  Mode, median and mean are all in the middle of the distribution
• Skewed distributions

  • Mode remains the most commonly occurring value
  • Median remains the middle value
  • Mean is generally 'pulled' in the direction of the tails

  
• Measures of variability


  Statistics that describe the amount of variation in a distribution

  
• Measures of variability

  • Range
  • Interquartile and semi-interquartile ranges
  • Mean absolute deviation (MAD)
  • Standard deviation

  
• Range


  Difference between the highest and lowest scores
• Interquartile range

  Difference between Q3 and Q1
• Semi-interquartile range

  Interquartile range divided by 2
• Mean absolute deviation (MAD)


  Average of the absolute values of the deviations from the mean
• Standard deviation


  Square root of the average squared deviations about the mean

  
• Low standard deviation


  Data are clustered around the mean
• High standard deviation


  Data are more spread out
• Standard deviation


  A measure of variability equal to the square root of the average squared deviations about the mean

  
• The problem of the sum of all deviation scores around the mean equaling zero still exist

  
• Solution to the problem of deviation scores summing to zero

  Use the square of each score
• Standard deviation


  Equal to the square root of the variance—equal to the arithmetic mean of the squares of the differences between the scores in a distribution and their mean

  
• Standard deviation close to zero


  Data points are close to the mean
• High or low standard deviation


  Data points are respectively above or below the mean
• Calculating standard deviation

  1. Step 1: Calculate the differences between the scores and their mean (x-x̄)

  2. Step 2: Square each (x-x̄)2

  3. Step 3: Calculate the mean of the squared differences

  4. Step 4: Calculate the square root of the variance