Z-Score Calculator

A z-score (also called a standard score) measures how many standard deviations a value is from the mean:

$z = \frac{x - \mu}{\sigma}$

where $x$ is the raw value, $\mu$ is the population mean, and $\sigma$ is the population standard deviation.

Interpretation:

• $z = 0$: the value equals the mean.
• $z = 1$: one standard deviation above the mean.
• $z = -2$: two standard deviations below the mean.
• $|z| > 2$ is conventionally 'unusual'; $|z| > 3$ is 'extreme'.

Why standardize?

• Comparability: z-scores let you compare values from different distributions (e.g., a $z = 1.5$ on an SAT math test vs a $z = 1.5$ on a verbal test means the same relative performance).
• Probability lookup: if the underlying distribution is approximately normal, $z$ maps directly to a probability via the standard normal CDF $\Phi(z)$.
• Outlier detection: large $|z|$ flags potential outliers.

Sample version: when working from sample data, replace $\mu$ with $\bar{x}$ and $\sigma$ with $s$:

$z = \frac{x - \bar{x}}{s}$

Step-by-Step

1. Identify the value $x$, the mean $\mu$ (or $\bar{x}$), and the standard deviation $\sigma$ (or $s$).
2. Subtract the mean: $x - \mu$.
3. Divide by the standard deviation: $z = (x - \mu)/\sigma$.

Reverse: Find $x$ from $z$

$x = \mu + z\sigma$

Useful when given a percentile and asked for the corresponding raw value.

Probability via the Standard Normal

For a normally distributed variable $X \sim N(\mu, \sigma^2)$, the standardized variable $Z = (X - \mu)/\sigma$ follows the standard normal $N(0, 1)$.

Common probabilities:

Symmetry: $P(Z < -z) = 1 - P(Z < z)$.

Empirical Rule (68-95-99.7)

For a normal distribution:

• ~68% of values fall within $\pm 1\sigma$ of the mean.
• ~95% within $\pm 2\sigma$.
• ~99.7% within $\pm 3\sigma$.

This is the foundation for confidence intervals and many quick estimates.

Critical Z-Values for Confidence Intervals

These are the values $z^*$ such that $P(-z^* < Z < z^*) =$ confidence level.

• Wrong order: $z = (x - \mu)/\sigma$, not $(\mu - x)/\sigma$. Putting the mean second flips the sign.
• Using variance instead of standard deviation: divide by $\sigma$, not $\sigma^2$. A value 'one variance away' is meaningless — you want one standard deviation.
• Sample vs population: with sample data, use $\bar{x}$ and $s$. With known parameters, use $\mu$ and $\sigma$. Conflating them inflates/deflates z-scores.
• Assuming normality without checking: z-scores can be computed for any distribution, but the probability lookup $\Phi(z)$ only applies if the underlying distribution is normal (or approximately so by the CLT).
• Forgetting the sign: $z = -2$ means 'below the mean.' Reporting $z = 2$ misrepresents direction.
• Confusing one-tailed and two-tailed probabilities: $P(|Z| > 2)$ is both tails combined ($\approx 0.0456$). $P(Z > 2)$ is one tail ($\approx 0.0228$). Read the question carefully.