How to Calculate a Z-Score (Formula and Z-Table)

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

• $x$ is the value you are looking at.
• $\mu$ is the mean of the population.
• $\sigma$ is the standard deviation of the population.

For a sample, the same idea uses the sample mean and sample standard deviation: $z = \frac{x - \bar{x}}{s}$.

How to read the sign:

• $z > 0$: the value is above the mean.
• $z < 0$: the value is below the mean.
• $z = 0$: the value equals the mean.

A z-score of 2 means "2 standard deviations above the mean," no matter what the original units were.

Example 1: A brand of phone battery lasts a mean of 20 hours with a standard deviation of 4 hours. Find the z-score for a battery that lasts 26 hours.

1. Subtract the mean: $26 - 20 = 6$.
2. Divide by the standard deviation: $\frac{6}{4} = 1.5$.
3. Answer: $z = 1.5$. This battery lasted 1.5 standard deviations longer than average.

Example 2: Same batteries. Find the z-score for 17 hours.

1. $17 - 20 = -3$.
2. $\frac{-3}{4} = -0.75$.
3. Answer: $z = -0.75$, three quarters of a standard deviation below the mean.

Working backward. To turn a z-score into a raw value, solve the formula for $x$:

$$x = \mu + z\sigma$$

Example 3: What battery life has a z-score of $-1.5$? $x = 20 + (-1.5)(4) = 20 - 6 = 14$ hours.

Z-scores put different scales on the same footing.

Example: Maya is 70 inches tall. In her group, heights have a mean of 64 inches and a standard deviation of 2.5 inches. Leo is 74 inches tall, in a group with a mean of 69 inches and a standard deviation of 3 inches. Who is taller relative to their own group?

1. Maya: $z = \frac{70 - 64}{2.5} = \frac{6}{2.5} = 2.4$.
2. Leo: $z = \frac{74 - 69}{3} = \frac{5}{3} \approx 1.67$.
3. Maya's z-score is higher, so she is taller relative to her group, even though Leo is taller in inches.

If the data follow a normal distribution, a standard normal table (z-table) tells you the area to the left of a z-score. That area is the proportion of values below it.

Reading the table for $z = 1.25$:

1. Find the row for the first two digits: 1.2.
2. Find the column for the second decimal place: 0.05.
3. Where they meet, the table shows 0.8944.

So about 89.44% of values fall below $z = 1.25$.

Some values you will see often:

z	Area to the left
$-1.00$	0.1587
$-0.50$	0.3085
$0.00$	0.5000
$1.00$	0.8413
$1.25$	0.8944
$1.28$	0.8997
$1.50$	0.9332
$2.00$	0.9772

Use the battery example again: normally distributed with $\mu = 20$ hours and $\sigma = 4$ hours. There are three kinds of questions.

Less than: $P(X < 25)$.

1. $z = \frac{25 - 20}{4} = 1.25$.
2. Table: 0.8944. So $P(X < 25) = 0.8944$.

Greater than: $P(X > 25)$.

1. The table gives area to the left, so subtract from 1.
2. $P(X > 25) = 1 - 0.8944 = 0.1056$.

Between: $P(18 < X < 26)$.

1. Convert both ends: $z = \frac{18 - 20}{4} = -0.5$ and $z = \frac{26 - 20}{4} = 1.5$.
2. Look up both: 0.3085 and 0.9332.
3. Subtract the smaller area from the larger: $0.9332 - 0.3085 = 0.6247$.

About 62.5% of batteries last between 18 and 26 hours.

A percentile is the percent of values below a given value. Since the z-table already gives area to the left, just multiply by 100.

• $z = 1.25$: area 0.8944, so about the 89th percentile.
• $z = -1$: area 0.1587, so about the 16th percentile.
• $z = 0$: area 0.5000, exactly the 50th percentile (the median of a normal distribution).

Working backward from a percentile: What battery life marks the 90th percentile?

1. Search the inside of the table for the area closest to 0.9000. That is 0.8997, at $z = 1.28$.
2. Convert to a raw value: $x = 20 + 1.28(4) = 20 + 5.12 = 25.12$ hours.

Quick check with the empirical rule. In a normal distribution, about 68% of values have $z$ between $-1$ and $1$, about 95% are between $-2$ and $2$, and about 99.7% are between $-3$ and $3$. Values with a z-score beyond $\pm 2$ are often called unusual.

Paste a problem into the VeilGPT solver to check your z-scores and table lookups.