Detailed Study Notes on Z-scores for PSY 360

Definition: A z-score is a statistical measurement that describes how far away a raw score is from the mean of a distribution in terms of standard deviations.
Contextual Importance: Raw deviations from a mean lack context, therefore transforming a raw score into a standardized z-score allows
- Meaningful inference of how a score compares to other scores in a distribution.
- Comparison between different distributions.
- For example: Comparing a test score of 76 against the broader class average.
- Further illustrates using a practical example: A student tells their parent they scored 27 on a quiz, which might initially alarm them, but they lack context about what that score signifies without information on averages and variability (central tendency and variability).

Positive vs. Negative Z-scores:
- A Positive z-score indicates the raw score is above the mean.
- A Negative z-score indicates the raw score is below the mean.
- The absolute value of the z-score gives the precise distance from the mean, measured in standard deviations.
Numerical Value: The z-score's numerical value directly represents the number of standard deviations the raw score is from the mean (M).

Standardization: The mean of the z-score distribution will always be 0, regardless of the original distribution's mean.
Standard Deviation: Each z-score standard deviation (SD) will always be 1.
Distribution Characteristics:
- 50% of scores will fall above the mean.
- 50% of scores will fall below the mean.
- The shape of the z-score distribution mirrors the shape of the original distribution, maintaining the same skewness and kurtosis.

Approximately 68% of scores will fall within +/- 1 standard deviation from the mean.
Approximately 95% of scores will fall within +/- 2 standard deviations from the mean, commonly considered extremes (tails).
Approximately 99.7% of scores will fall within +/- 3 standard deviations from the mean, known as the empirical rule or the 68-95-99.7 rule.

Population Z-score Formula:
$z = \frac{X - \mu}{\sigma}$
where:
- $X$ = raw score from the dataset
- $\mu$ = mean of the dataset
- $\sigma$ = standard deviation of the dataset
Sample Z-score Formula:
$z = \frac{X - M}{s}$
where:
- $M$ = mean of the sample
- $s$ = standard deviation of the sample

Which of the following z-scores fall furthest below the mean? A. +2.00 B. -0.50 C. -1.25 D. -2.00
- Answer: D: -2.00

A z-score of 0.00 will be equal to: A. 0 B. 1 C. M D. Cannot be determined
- Answer: C: M (the mean)

A z-score falls relative to the mean:
- A z-score of -0.75 falls 0.75 standard deviations below the mean.
- A z-score of 1.50 falls 1.50 standard deviations above the mean.
- A z-score of -1.25 falls 1.25 standard deviations below the mean.
- A z-score of 2.50 falls 2.50 standard deviations above the mean.

Context: Comparing weights and heights of two brothers to determine who is bigger.
- Data:
- Weight: $
  u = 95$, $ heta = 10$ (Mean and Standard Deviation)
- Younger brother's weight: 93 lb
- Older brother's height:
  - Height: $
    u = 67$, $ heta = 5$ (Mean and Standard Deviation)
  - Height: 62 inches
Calculations:
1. For the younger brother's weight, calculate:
  $z = \frac{93 - 95}{10} = -0.20$
2. For the older brother's height, calculate:
  $z = \frac{62 - 67}{5} = -1.00$
Conclusion: The older brother’s score indicates he falls further below the mean, indicating the younger brother is bigger.

Context: Comparison of Maria and Joe's exam scores:
- Mean: 57, SD: 14
- Maria's score: 64
- Joe's score: 43
Calculations:
- For Maria:
  $Z = \frac{64 - 57}{14} = +0.50$
- For Joe:
  $Z = \frac{43 - 57}{14} = -1.00$
Conclusion: Maria did slightly better than average with a z-score of +0.50, whereas Joe had a significantly poorer performance with -1.00.

Context: Comparing scores of Mary and Jason on different standardized tests:
- Mean ACT Score: 22, SD: 2
- Mean SAT Score: 1000, SD: 100
Mary's Score: 26
Jason's Score: 1150
Calculations:
- For Mary:
  $z = \frac{26 - 22}{2} = +2.00$
- For Jason:
  $z = \frac{1150 - 1000}{100} = +1.50$
Conclusion: Mary did significantly better on the ACT than Jason did on the SAT, as indicated by her higher z-score of +2.00 compared to Jason's +1.50.

Scoring:
- Exam Mean ($\mu$): 75, SD ($\sigma$): 10, Score ($X$): 85
- Calculation:
  $z = \frac{85 - 75}{10} = 1.00$
Height Evaluation:
- Height Mean ($\mu$): 160 cm, SD ($\sigma$): 7 cm, Height ($X$): 172 cm
- Calculation:
  $z = \frac{172 - 160}{7} \approx 1.71$
Test Score:
- Test Mean ($\mu$): 50, SD ($\sigma$): 5, Z-score: -1.5
- Calculation:
  $-1.5 = \frac{X - 50}{5} \ X = -1.5 * 5 + 50 = 42.5$
Exam Score for a Z-score of -2.5 with a mean of 60 and SD of 8:
- $-2.5 = \frac{X - 60}{8} \ X = -2.5 * 8 + 60 = 40$
IQ Score:
- Mean IQ ($\mu$): 100, SD ($\sigma$): 15, IQ ($X$): 130
- Calculation:
  $z = \frac{130 - 100}{15} = 2$

Z-scores can shape an individual’s self-esteem and identity, as they inform how well one perceives their abilities relative to a group.
Awareness of one's z-score can contribute positively or negatively to self-perception based on whether one feels average, above average, or below average.

The educational and medical sectors utilize z-scores, influencing concepts like fairness and meritocracy.
Z-scores may perpetuate or challenge societal norms tied to performance and expectations in various contexts, including college admissions or health assessments.

Outliers identified through z-score analysis challenge conceptions of statistical normality and authenticity of data representation.
The existence of extreme values can shape our understanding of what is defined as "normal" behavior or performance.

Z-scores enable researchers to standardize data effectively, promoting comparability across various datasets and studies in fields ranging from psychology to education.
Usage of z-scores as a quantitative research instrument prompts philosophical reflection on the implications of measuring human behavior and capability against a defined statistical norm.