6|\(z\)-scores

Overview

• \(z\)-scores
• Standardized distributions
• \(z\)-scores & making inferences
• Learning checks

\(z\)-scores

\(z\)-score calculation

• \(z\)-score formula:

\(z = \dfrac{X - \mu}\sigma\)     or…     \(z = \dfrac{X - M}s\)

• Numerator: deviation score
• Denominator: standard deviation
• \(z\) expresses deviation in SD units

\(z\)-score description

• \(z\)-score describes exact location of any score in a distribution
• Two pieces of information:
  • Sign
    • Positive or negative
    • Indicates whether score is located above or below the mean
  • Magnitude
    • Indicates distance between score and mean in standard deviation units
    • \(z = 0\) is equal to the mean

Example: test scores

• How well did you do on a test
• Is your score good, bad, just ok?

Test	Score	M	SD
geniustest.com	80	70	5
mensa.lu	40	20	10

• \(z\)-score can describe location of a score in any distribution
  • Makes scores from different distributions comparable

Example: test scores

• Comparing scores from different distributions
  • How many SDs is a score above/below the mean?

\(z = \dfrac{80-70}{5} = \dfrac{10}{5} = 2\)

\(z = \dfrac{40-20}{10} = \dfrac{20}{10} = 2\)

Determining raw score from \(z\)-score

\(z = \dfrac{X - \mu}\sigma\)     so…     \(X = \mu + z\sigma\)

• Algebraically solve for \(X\)
• Raw score \(X\) equals population mean plus \(z\) multiplied by standard deviation

Determining raw score from \(z\)-score

\[\begin{align} X & = \mu + z\sigma \\ & = 70 + 2*5 \\ & = 70 + 10 \\ & = 80 \end{align}\]

\[\begin{align} X & = \mu + z\sigma \\ & = 20 + 2*10 \\ & = 20 + 20 \\ & = 40 \end{align}\]

Standardized distributions

\(z\) distribution

• Every \(X\) value can be transformed to a \(z\)-score
  • \(z\)-score distribution is called a standardized distribution
• Characteristics of \(z\)-score transformation
  • Same shape as original distribution
  • Mean of \(z\)-score distribution is always \(0\)
    • Because the mean is the balance point; \(\Sigma (X - \mu)\) always equals \(0\)
  • Standard deviation is always \(1.00\)
    • Because \(SD\) is the denominator

Example

Other standardized distributions

• Standardized: Predetermined mean & SD
  • \(z\) distribution has \(\mu = 0\) and \(\sigma = 1\)
  • SAT has \(\mu = 500\) and \(\sigma = 100\)
  • IQ has \(\mu=100\) and \(\sigma=15\) points
    
    
• Standardizing a distribution has two steps
  1. Original raw scores transformed to \(z\)-scores
  2. The \(z\)-scores are transformed to new \(X\) values so that the specific predetermined \(\mu\) and \(\sigma\) are attained
     2a. Multiply to set SD
     2b. Add or subtract a constant to set the mean

Standardizing scores

Original

1. \(z\)-scores

2a. Set SD

2b. Set \(M\)

geniustest

mensa.lu

\(z\)-scores and inferential stats

• What’s the use of \(z\)-scores?
  • Research questions are often about looking for differences
  • Does one individual seem different from others?
  • Is a sample “noticeably different” from the population?
  • \(z\) is one indication of how typical a score is
  • \(z \approx 0\) is a typical score; \(z \le -2.00\) or \(\ge 2.00\) are less typical

\(z\)-scores and inferential stats

\(z\)-scores and inferential stats

• https://humanbenchmark.com/tests/reactiontime
  • Do 5 reaction time trials
  • (click when the color changes)
  • Note your reaction time in \(ms\)
  • Then click play to resume lecture when you’re done

Peter Parker

• Peter Parker \(RT = 159ms\)
  • Impressive?
  • Depends on population characteristics

\(\begin{align} z & = \dfrac{X - \mu}{\sigma} \\ & = \dfrac{159 - 284}{50} = -2.5 \end{align}\)

CBT for OCD

• Efficacy of CBT for OCD¹
  • Pre-treatment population \(M = 30.25; SD = 14.89\)
  • Suppose a treated individual scores \(X = 15.49\)

\(\begin{align} z & = \dfrac{X - M}{s} \\ & = \dfrac{15.49 - 30.25}{14.89} \\ & = -0.99 \end{align}\)

1. Abramowitz et al. (2010)

Learning checks

1. For a population with \(\mu = 50\) and \(\sigma = 10\), what is the \(X\) value corresponding to \(z = 0.4\)?

2. In a sample distribution, \(X = 56\) corresponds to \(z = 1.00\), and \(X = 47\) corresponds to \(z = -0.50\). Find the mean and standard deviation for the sample.

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