Lecture Notes: Chapter 5 - Z-Scores and the Standardization of Statistics

Z-scores represent a fundamental topic in statistics, serving as a bridge between foundational descriptive concepts and advanced inferential statistics.
They unify concepts from previous study, specifically Chapter 3 (the mean) and Chapter 4 (standard deviations), integrated directly into the z-score formula.
Understanding z-scores is the primary tool for moving toward hypothesis testing, which constitutes the majority of statistical practice in higher-level coursework.

To understand z-scores, one must first master the concept of deviations, as introduced in Chapter 4.
Standard Deviation: Historically referred to as the "average deviation." While a standard deviation represents the group average, every individual within that group possesses their own specific deviation.
Individual Deviation: This represents how a single score compares to the group average. It is calculated by the formula:
- For a population: $X - \mu$
- For a sample: $X - M$
Logic of Calculations: If an average (mean) can be found for a group, it implies every individual contributed a score. Similarly, if a standard deviation exists for a group, it implies every individual has a specific deviation from that mean.

The deviation formula is always $Score - Mean$ .
The Sign of the Deviation:
- A positive deviation indicates the core is above the mean (e.g., a test score of $80$ in a class with a mean of $70$ results in $80 - 70 = +10$ ).
- A negative deviation indicates the score is below the mean (e.g., a test score of $65$ in a class with a mean of $70$ results in $65 - 70 = -5$ ).
The Absolute Value of the Deviation: The number part (without the sign) represents the distance of the score ( $X$ ) from the mean in the specific units of measurement being used.
Units associated with deviations:
- Test scores: points.
- Height: inches or centimeters.
- Weight: pounds ( $lbs$ ) or kilograms ( $kg$ ).
- Income: dollars ( $USD$ ).
Examples of Deviation Magnitude:
- A deviation of $+1\,inch$ for height indicates an individual is slightly taller than average.
- A deviation of $+9\,inches$ indicates an individual is significantly taller than average.

Scenario: A room of people has a mean weight ( $\mu$ ) of $190\,lbs$ . Jose weighs $178\,lbs$ .
Calculation:
- $Deviation = Score - Mean$
- $Deviation = 178 - 190$
- $Deviation = -12\,lbs$
Interpretation: A deviation of $-12\,lbs$ means Jose weighs $12\,lbs$ less than the average weight of the group.

There are two primary ways to write the z-score formula depending on whether one is dealing with a population or a sample, though the mathematical process is identical.
Population Notation (Greek Letters):
- $z = \frac{X - \mu}{\sigma}$
- Uses $\mu$ (mu) for population mean and $\sigma$ (sigma) for population standard deviation.
Sample Notation (Latin/English Letters):
- $z = \frac{X - M}{s}$
- Uses $M$ for sample mean and $s$ for sample standard deviation.
Conceptual Formula:
- $z = \frac{\text{Deviation}}{\text{Standard Deviation}}$
- The numerator of a z-score is always the individual's deviation ( $X - Mean$ ).

Interpreting what z-scores mean is considered more important than the calculation itself.
The Sign of the Z-Score: Identical to deviations. A positive z-score means the score is above the mean; a negative z-score means it is below the mean.
The Value of the Z-Score: Unlike deviations, the unit for a z-score is always standard deviations. It tells you exactly how many standard deviations ( $SDs$ ) a score is from the mean.
Global Interpretive Values:
- $z = 0$ : The score is exactly at the mean (zero standard deviations away).
- $z = 1.0$ or $-1.0$ : Moderately far from the mean.
- $z = 2.0$ or $-2.0$ : Really far from the mean; getting into outlier territory.
- $z = 3.0$ or $-3.0$ : Way out in the tails of the distribution; these are extreme outliers.

Scenario: Mean weight ( $\mu$ ) = $190\,lbs$ , Standard Deviation ( $\sigma$ ) = $8\,lbs$ , Jose's weight ( $X$ ) = $178\,lbs$ .
Calculation:
- $z = \frac{178 - 190}{8}$
- $z = \frac{-12}{8}$
- $z = -1.5$
Interpretation: Jose's weight is $1.5$ standard deviations below the average weight of the group.
Contrast with Deviation: While the deviation was $-12\,lbs$ , that specific number only translates to a z-score of $-1.5$ because the standard deviation for this specific group was $8$ . If the standard deviation were different, the z-score would change even if the deviation remained $-12$ .

Raw scores and deviations are difficult to interpret when the unit of measurement is unfamiliar.
The Schmidt Psychopathy Scale Example:
- Imagine a variable measuring psychopathy on a scale where you have no prior knowledge of the range or average.
- If an individual scores $42$ and the mean is $40$ , their deviation is $+2$ "Schmidt Psychopathy Scale points."
- Without context, it is impossible to know if being $2$ points above average makes someone a clinical psychopath (an outlier) or just a slightly above-average person.
Standardization: By dividing the deviation ( $+2$ ) by the standard deviation of the scale, we get a z-score.
- If the resulting $z = +3.0$ , the person is a significant outlier and more psychopathic than almost everyone.
- If the resulting $z = +0.2$ , the person is barely above average and essentially normal.
This allows researchers to interpret clinical psychology data, economic measures (like the Gini coefficient), and other specialized data without being a specialist in every specific unit of measurement.

Data: $X = 93$ , $\mu = 89$ , $\sigma = 5.5$ .
Calculation:
- $z = \frac{93 - 89}{5.5}$
- $z = \frac{4}{5.5}$
- $z \approx +0.73$
Interpretation: The score is $0.73$ standard deviations above the mean. This is a respectable score, almost one whole standard deviation above average, but not a crazy outlier.

Data: $M = 102$ , $s = 7$ , $X = 87$ .
Calculation:
- $z = \frac{87 - 102}{7}$
- $z = \frac{-15}{7}$
- $z \approx -2.14$
Interpretation: The score is $2.14$ standard deviations below average. This is really far below the mean, indicating a significantly low score relative to the distribution.

Data: $\mu = 55$ , $\sigma = 5$ , $z = +1$ .
Formula Setup:
- $1 = \frac{X - 55}{5}$
Algebraic Steps:
- Multiply both sides by $5$ : $5 = X - 55$
- Add $55$ to both sides: $60 = X$
Conclusion: The raw score $X$ is $60$ .

Data: $\sigma = 100$ , $X = 800$ . The problem states the individual is "two standard deviations below the mean," which translates to $z = -2.0$ .
Formula Setup:
- $-2 = \frac{800 - \mu}{100}$
Algebraic Steps:
- Multiply both sides by $100$ : $-200 = 800 - \mu$
- Subtract $800$ from both sides: $-1000 = -\mu$
- Multiply by $-1$ to solve for positive mu: $\mu = 1000$
Check of Work: If the mean is $1000$ and the standard deviation is $100$ , then one standard deviation below is $900$ and two standard deviations below is $800$ . This matches the data given in the problem, confirming the answer is correct.