How to Calculate a Z-Score Without a Standard Deviation

A z-score tells you how many standard deviations a value is from a mean. To calculate one from a raw value, you need both the mean and a standard deviation. If the standard deviation is genuinely unavailable—and cannot be calculated or inferred from other information—there is no unique numerical z-score.

This article explains what information is needed, how to get the standard deviation from a dataset, and what you can do if you have a percentile or probability rather than raw data.

Why the standard deviation is necessary

For a value $x$ from a population with mean $\mu$ and standard deviation $\sigma$, the z-score is

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

A finite z-score requires a positive standard deviation; if the standard deviation is zero, the expression is undefined. The numerator, $x-\mu$, is the value’s distance from the mean in the original units. Dividing by $\sigma$ converts that distance into standard-deviation units. A positive z-score means the value is above the mean; a negative one means it is below.

Knowing just $x$ and $\mu$ gives the distance, but not how large that distance is relative to the spread. For example, if a value is 10 and the mean is 8, the distance is 2. If the standard deviation is 2, the z-score is 1; if it is 4, the z-score is 0.5. Both spreads are compatible with the same value and mean, so those two pieces of information alone cannot determine a unique z-score.

Find or estimate the standard deviation

The standard deviation may be available directly, or it may be recoverable from other information. For example, if the variance is known, take its square root: $\sigma=\sqrt{\operatorname{Var}(X)}$. If you have the full population, calculate its population standard deviation. If you have a sample, calculate its sample standard deviation to estimate the population spread.

For observations $x_1, x_2, \ldots, x_n$, the sample mean is

$$\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i,$$

and the sample standard deviation is

$$s=\sqrt{\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}}.$$

The denominator is $n-1$ for the usual sample standard deviation. If the data are the entire population you want to describe, use the population standard deviation instead, with the population mean and denominator $N$.

Once you have the appropriate mean and standard deviation, calculate the standardized value. For sample-based standardization, this is often written

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

Worked example: calculate the spread from data

Suppose the observations are $4, 6, 8, 10, 12$, and you want the z-score of $x=10$. Treat these observations as a sample.

First calculate their mean:

$$\bar{x}=\frac{4+6+8+10+12}{5}=8.$$

Next, find the squared deviations from the mean:

$$(4-8)^2+(6-8)^2+(8-8)^2+(10-8)^2+(12-8)^2 =16+4+0+4+16=40.$$

The sample standard deviation is therefore

$$s=\sqrt{\frac{40}{5-1}}=\sqrt{10}\approx 3.162.$$

Finally, standardize 10:

$$z=\frac{10-8}{\sqrt{10}}\approx 0.632.$$

So 10 is about 0.63 sample standard deviations above the sample mean. If these five numbers were the entire population instead, the population standard deviation would be $\sqrt{40/5}=\sqrt{8}\approx 2.828$, giving a population z-score of about $0.707$. Choose the sample or population convention according to what your data represent; do not mix the sample mean with a population standard deviation without a reason.

If you have a percentile or probability

There is another meaning of “without a standard deviation”: you may want the z-score corresponding to a position in a normal distribution, without converting an original measurement. If you know the cumulative probability $p$, with $0<p<1$, find the standard normal quantile $z=\Phi^{-1}(p)$, where $\Phi$ is the standard normal cumulative distribution function. For instance, the 97.5th percentile corresponds to approximately $z=1.96$.

This gives a z-score on the standard normal scale; it does not give the z-score of a particular raw observation unless you also know how that observation relates to the distribution. To convert a raw value to a normal-distribution value, you still need the distribution’s mean and standard deviation. Using a normal table or inverse-normal function also assumes you are working with the standard normal distribution or using it as an appropriate model.

Common mistakes

• Using only the distance from the mean. The quantity $x-\mu$ is not a z-score until you divide by a standard deviation.
• Using the range as the standard deviation. The range is maximum minus minimum; it is not interchangeable with standard deviation.
• Confusing standard deviation with standard error. A standard error describes the variability of a statistic, such as a sample mean. A z-statistic for a sample mean may use a standard error, but that is not the same calculation as standardizing an individual observation.
• Assuming every z-score requires normally distributed data. The formula can standardize values from any dataset. Normality matters when you use a normal model or normal table to interpret probabilities.
• Treating an estimate as an exact population value. A sample standard deviation estimates population spread; results based on it may differ from z-scores calculated with the true population standard deviation.

If you have only a value and a mean, report the distance from the mean or obtain more information about the spread. Do not invent a standard deviation or substitute another measure of spread and call the result a z-score.