Z-Scores and Percentiles

By the end of this lesson, you will be able to compute $z$-scores, use a standard normal table to find probabilities and percentiles, and interpret $z$-scores in context.

The Big Idea

A $z$-score standardizes any normally distributed value by measuring how many standard deviations it is from the mean. This converts any normal distribution to the standard normal ($\mu = 0$, $\sigma = 1$), allowing us to use one universal table for all normal distributions.

The Z-Score Formula

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

• $x$ = individual data value
• $\mu$ = population mean
• $\sigma$ = population standard deviation
• $z$ = number of standard deviations above (positive) or below (negative) the mean

Interpreting Z-Scores

• $z = 0$: value is exactly at the mean
• $z = 1$: one standard deviation above the mean
• $z = -2$: two standard deviations below the mean
• $z > 2$ or $z < -2$: unusual value (outside 95% of data)
• $z > 3$ or $z < -3$: very unusual (outside 99.7%)

Example 1 — Computing a Z-Score

Test: $\mu = 70$, $\sigma = 10$. A student scores 85.

$$z = \frac{85 - 70}{10} = 1.5$$

This student scored 1.5 standard deviations above the mean — in the top 6.7% (from the table).

Using the Z-Table

The standard normal table gives $P(Z \leq z)$ = the area to the left of $z$.

For $z = 1.5$: table gives $P(Z \leq 1.5) = 0.9332$

Interpretation: 93.32% of values fall below a $z$-score of 1.5.

Finding Probabilities from Z-Scores

$P(Z \leq z)$: look up directly in the table.

$P(Z > z) = 1 - P(Z \leq z)$: complement.

$P(z_1 \leq Z \leq z_2) = P(Z \leq z_2) - P(Z \leq z_1)$: area between two values.

Example 2 — Probability from Z-Score

$\mu = 500$, $\sigma = 100$. Find $P(X \leq 650)$.

$z = \dfrac{650 - 500}{100} = 1.5$

$P(X \leq 650) = P(Z \leq 1.5) = 0.9332$

Example 3 — Between Two Values

$P(450 \leq X \leq 650)$ for the same distribution.

$z_1 = \dfrac{450-500}{100} = -0.5$; $z_2 = 1.5$

$P(-0.5 \leq Z \leq 1.5) = P(Z \leq 1.5) - P(Z \leq -0.5)$

$= 0.9332 - 0.3085 = 0.6247$

Percentiles

The $p$th percentile is the value below which $p%$ of the data falls.