Z-scores

Z-score

• Recall from the last module, standard deviation is the average amount of deviation around the mean scores have for a particular distribution.
  • So, if S = 12, then the average deviation for scores from our Planning Ability test mean is 12.
• The number of standard deviations a particular score is above or below the mean is called its Z-score.
  • If a score is below the mean, the Z-score is negative.
  • If a score is above the mean, the Z-score is positive.
• The standard deviation now becomes a kind of yardstick, a unit of measure in its own right.

Rulers Analogy

The two scales are something like a ruler with inches lined up on the one side and centimeters on the other.

• A raw score of 78 is 1 standard deviation below the mean and so, it has a Z-score of -1.
• Raw scores: and Z-scores: .

Z-scores' usefulness

• Z-scores provide a helpful way to compare scores on measures that are on completely different scales.
• If a person scored:
  • 26 on Leadership (example 1: ) and
  • 84 on Planning Ability (example 2: )
• We can say that person's scores are much higher than average on Leadership and slightly lower than average on Planning Ability.

Converting from Raw score to Z-score units.

• Formula for Z-score:
  
  
  
• Example 2:
  
  
  
• Example 3:
  
  
  
• Example 4:

Converting from Z-score to Raw score units.

• Formula for raw score:
  
  
  
• Example 2:
  
  
  
• Example 3:
  
  
  
• Example 4: