Understanding Z-scores or WTF is Z score core idea Z-score

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Definition

A Z-score measures how far a value is from the mean, in units of standard deviations:

z=x−μσ z = \frac{x - \mu}{\sigma}
  • xx = observed value
  • μ\mu = mean
  • σ\sigma= standard deviation

It tells you how extreme or unusual a value is relative to the distribution.

How to visualize it (via integral / area)

Think of the normal distribution curve:

  • The x-axis = Z (distance from the mean)
  • The area under the curve to the left of a Z-score = cumulative probability

Mathematically:

P(Z≤z)=∫−∞z12πe−t2/2dt P(Z \leq z) = \int_{-\infty}^{z} \frac{1}{\sqrt{2\pi}} e^{-t^{2}/2} dt

  • This integral gives the probability of observing a value ≤ z

  • Percentiles, confidence levels, and power thresholds are just Z-scores corresponding to specific cumulative areas !641 source: https://en.wikipedia.org/wiki/Normal_distribution Example:

  • Z = 0.84 → 80% of the area under the curve lies to the left (80th percentile)

  • Z = 1.96 → 97.5% of the area lies to the left (for 95% confidence, two-sided test)

How Z-score is generally computed

  • From data: z=x−μσz = \frac{x - \mu}{\sigma} → requires observed value, mean, SD

  • From probability (hypothesis testing / power / confidence):

    • Use inverse standard normal CDF to find Z corresponding to a desired cumulative probability
    • Tools: Z-tables, Excel (=NORM.S.INV(p)), Python (scipy.stats.norm.ppf(p))

No raw data needed — you’re just looking at theoretical standard normal percentiles.

Simplification for the standard normal

  • Standard normal distribution: mean = 0, SD = 1

  • Formula: z=x−01=xz = \frac{x-0}{1} = x

  • Now the Z-score is literally the X-axis value itself, and cumulative probability = integral of the curve to the left

TL;DR (one sentence)

A Z-score tells how many SDs a value is from the mean; you can visualize it as the X-axis cutoff on the standard normal curve, where the area to the left of it equals a probability; it’s computed from data using z=(x−μ)/σz = (x - \mu) / \sigma, or from probability using the inverse normal CDF, and simplifies to z=xz = x for the standard normal.

Resources:

https://en.wikipedia.org/wiki/Normal_distribution https://latexconvert.com/en/

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