Z-scores

1
Z-scores

• Presentation 1.12

2
Z-scores

• Z-scores are a measure of relative standing
  (relative to the mean standard deviation)
• A z-score is simply how many standard deviations
  you are away from the mean.

In a population
In a sample
3
Z-scores

• So, what does a z-score do?
• Essentially, it standardizes the distribution.
• We have already seen that there are an infinite
  number of different normal distributions (with
  different means and standard deviations).
• Standardizing essentially rescales the existing
  distribution to be a normal distribution with a
  mean of 0 and a standard deviation of 1.

becomes
N(31.3,2)
N(0,1)
4
Z-scores

• Through this re-scaling, two different
  distributions may be compared.
• Z-scores are immensely useful for comparing data
  from different distributions.
• Which test score is better?
• Who really is taller (relative to gender)?
• Are speeding violations worse in urban or rural
  areas?
• All these questions (and many like them) are
  answered through z-scores.

5
Which test score is better?

• Haley takes both the SAT and the ACT.
• Her SAT score is 1320.
• The mean SAT score is 1059 and the test has a
  standard deviation of 143 (2004 data from the
  College Board)
• Her ACT score is 32.
• The mean ACT score is 26.4 and the standard
  deviation is 3.

6
Which test score is better?

• To compare the scores, we need to put them on the
  same scale.
• This is analogous to asking the question, which
  is hotter, 212 degrees Fahrenheit of 100 degrees
  Celsius?

SAT z-score
ACT z-score
7
Which test score is better?

• For the SAT, Haley scored 1.8252 standard
  deviations above (since its positive) the mean.
• For the ACT, Haley scored 1.8667 standard
  deviations above the mean.
• Therefore, relative to the test, her ACT score is
  slightly better.

8
Which test score is better?
SAT score

• Too see this visually

ACT score
9
Z-scores

• This concludes the presentation.