How to find a z-score and read a z-table

What a z-score is and the formula

A z-score tells you how many standard deviations a value sits above or below the mean. The formula is:

Here $x$ is the raw value you are standardizing, $\mu$ (mu) is the population mean, and $\sigma$ (sigma) is the population standard deviation. A positive $z$ means $x$ is above the mean, a negative $z$ means it is below, and $z = 0$ means $x$ equals the mean.

When you only have sample data, replace the population values with the sample mean $\bar{x}$ (x-bar) and the sample standard deviation $s$. The idea does not change: you are measuring distance from the center in standard-deviation units.

How to find a z-score from a raw value

Work through three steps.

1. Subtract the mean from your raw value to get $x - \mu$.
2. Divide that difference by the standard deviation $\sigma$.
3. Keep the sign. A negative result is correct whenever $x$ is below the mean.

Round the z-score to two decimal places, because standard normal tables are indexed to the hundredths. You can check any calculation with the z-score calculator.

How to read a z-score table (row vs column)

A standard normal table (see the z-table) lists the area under the curve to the left of a given z-score. That area is the cumulative proportion below the z-score. Multiply it by 100 to get the percentile for that z.

Read the table in two pieces:

• The row gives the ones and tenths digits of z, for example 1.2.
• The column gives the hundredths digit, for example 0.08.

To look up $z = 1.28$, find row 1.2, then move across to column 0.08. That cell holds 0.8997.

The most common misread is putting a digit in the wrong place. If you ignore the hundredths column and read row 1.2 with column 0.00, you get 0.8849, which is the value for $z = 1.20$, not 1.28. If you transpose the digits and look up 1.82 instead, you land on 0.9656. Match the tenths to the row and the hundredths to the column every time.

One more check: confirm your table is the area-to-the-left version. Some older tables give the area between 0 and z, which shifts every lookup.

Z-score to percentile

Once you have a z-score, the area to its left is the cumulative proportion below the value. Look up z, read that area, and multiply by 100 to get the percentile.

For $z = 1.28$, the table gives 0.8997, so the value sits at about the 90th percentile: roughly 90% of the distribution falls below it.

For the area to the right, subtract from 1. The area above $z = 1.28$ is $1 - 0.8997 = 0.1003$, about 10%. For the area between two z-scores, subtract the smaller cumulative area from the larger.

From a percentile back to a raw score

To go from a percentile to a raw value, run the steps backward.

1. Find the z-score whose cumulative area matches your target percentile. Search the body of the table for that area, then read the z from its row and column labels.
2. Solve the formula for $x$: $x = \mu + z\sigma$.
3. Substitute your mean and standard deviation.

For the 90th percentile, the closest table area to 0.9000 is 0.8997 at $z = 1.28$. That z then converts to a raw score using your own $\mu$ and $\sigma$, which the second worked example shows.

Negative z-scores and symmetry

A negative z-score means the value is below the mean. The symmetry of the normal curve makes these quick.

Because the curve is symmetric, the area to the left of a negative z equals 1 minus the area to the left of the matching positive z. For $z = -0.70$:

So a value with $z = -0.70$ sits at about the 24th percentile. Many tables print a separate negative-z page; if yours does not, use this symmetry.

A quick check with the empirical rule

The empirical rule is a shortcut for the round z-scores. About 68% of values fall between $z = -1$ and $z = 1$, about 95% between $z = -2$ and $z = 2$, and about 99.7% between $z = -3$ and $z = 3$.

The table gives the exact areas. For example, 0.8413 lies to the left of $z = 1$ and 0.1587 lies to the left of $z = -1$, so the middle 68% comes from $0.8413 - 0.1587 = 0.6826$. Use the empirical rule for fast estimates that sanity-check a table lookup, and use the normal distribution calculator when you want a precise area for any z.