What Is a Z-Score? Meaning, Formula and How to Interpret It
By Markets & Investing DeskUpdated Nov 10, 2026
Definition
A z-score measures how far a value sits from the mean of its data set in standard deviations, so values on different scales, such as stock returns or test results, can be compared.
Key takeaways
- A z-score equals the distance between a value and the mean, divided by the standard deviation: z = (x − μ) / σ.
- The sign shows direction (above or below the mean). The size shows how unusual the value is relative to the spread of the data.
- If the data are roughly normally distributed, about 68% of values fall between z = -1 and 1, about 95% between -2 and 2, and about 99.7% between -3 and 3.
- In finance, z-scores are used to compare returns, combine factors, flag statistical extremes and test trading systems.
- The z-score is a different measure from the Altman Z-score, which estimates a company's risk of bankruptcy from its financial statements.
A z-score (also called a standard score) measures how far a single value sits from the mean of its data set, expressed in standard deviations. A z-score of 0 means the value equals the mean. A z-score of 1.0 means it is one standard deviation above the mean, and a z-score of -2.0 means it is two standard deviations below. Converting raw numbers into z-scores puts them on a common scale, so a stock return, a test result and a trading spread can each be judged against their own history or peer group.
How a Z-Score Works
The formula needs three inputs:
z = (x − μ) / σ
- x is the value being evaluated (in statistics, x simply denotes the observation).
- μ (mu) is the mean of the data set or population.
- σ (sigma) is the standard deviation.
- z is the resulting standard score.
The numerator tells you how far the value is from average in the original units: percentage points, dollars or test points. Dividing by the standard deviation removes those units. That is why z-scores are comparable across data sets that use different scales. A value that is 6 points above a mean means little on its own. If the standard deviation is 2, that is z = 3, an extreme reading. If the standard deviation is 20, it is z = 0.3, close to typical.
The letter z is the conventional symbol for a standard normal variable, a distribution with a mean of 0 and a standard deviation of 1. Standardizing any normally distributed data with the formula above maps it onto that distribution.
The formula can also be run in reverse to find the raw value that matches a given z-score: x = μ + zσ.
Calculating a z-score
Calculate the mean and standard deviation first, then apply the formula.
Hypothetical example: a stock returned 14% last year. A group of comparable stocks averaged 8%, with a standard deviation of 4%.
- z = (14 − 8) / 4
- z = 1.5
The stock’s return was 1.5 standard deviations above the group average. A second stock in the same group that returned 6% would have z = (6 − 8) / 4 = -0.5, half a standard deviation below average.
Converting a data set to z-scores
To standardize a whole set of values, compute one mean and one standard deviation for the set, then apply the formula to each value. In a spreadsheet:
- Enter the values in a column (say A2:A7).
- Find the mean with
=AVERAGE(A2:A7). - Find the standard deviation with
=STDEV.S(A2:A7)for a sample, or=STDEV.P(A2:A7)if the values are the entire population. - In the next column, enter
=(A2-$B$1)/$C$1(with the mean in B1 and standard deviation in C1) and copy it down. Excel’s=STANDARDIZE(x, mean, standard_dev)does the same calculation.
Hypothetical example: six monthly returns of 2%, 5%, -1%, 4%, 3% and -4%. The mean is 1.5%. The sample standard deviation is about 3.39%.
| Month | Return | Deviation from mean | Z-score |
|---|---|---|---|
| 1 | 2% | 0.5 | 0.15 |
| 2 | 5% | 3.5 | 1.03 |
| 3 | -1% | -2.5 | -0.74 |
| 4 | 4% | 2.5 | 0.74 |
| 5 | 3% | 1.5 | 0.44 |
| 6 | -4% | -5.5 | -1.62 |
Two properties hold for any standardized set: the z-scores average to zero, and their standard deviation is 1. The choice between sample and population standard deviation matters. Using the population figure here (about 3.10%) would push month 6 to roughly -1.78, so state which one you used when comparing results.
Interpreting a Z-Score
Sign. A positive z-score sits above the mean, a negative one below. The sign carries no other meaning; a negative z-score is not “bad” by definition. For a measure where lower is better, such as volatility or cost, a negative score may be the favorable one.
Magnitude. The further from zero, the more unusual the value relative to the rest of the data. Under a normal distribution, the empirical rule gives the rough shares:
| Z-score range | Share of values inside (normal distribution) |
|---|---|
| -1 to 1 | about 68% |
| -2 to 2 | about 95% |
| -3 to 3 | about 99.7% |
Probability. A standard normal table (or =NORM.S.DIST(z, TRUE) in Excel) converts a z-score into the share of values below it, assuming normality:
- z = 1: about 84% of values lie below, so the value is near the 84th percentile.
- z = 1.5: about 93% lie below.
- z = -0.5: about 31% lie below.
- z = 0: exactly 50%.
Significance thresholds. In hypothesis testing, a common rule of thumb treats |z| above 1.96 as significant at the 5% level (two-tailed) and above 2.58 as significant at the 1% level. These conventions describe statistical rarity under the stated assumptions, not economic importance.
What counts as a “good” z-score depends on the question. A high z-score for a fund’s return relative to peers is welcome; a high z-score for a position’s loss is not. Investors who screen with z-scores pick their own bands. Some flag anything beyond ±3 as an outlier, since only about 0.3% of normally distributed values land there, while others use a tighter band such as ±1.5 to stay close to typical behavior.
How Z-Scores Are Used in Finance
Relative performance. The basic use is measuring how far a stock’s return deviates from an average: its own historical average, the average of an index such as the S&P 500, or a hand-picked peer group. The same approach can place a mutual fund or ETF against its category.
Factor investing. Factor models rank stocks on attributes such as value, momentum or quality. Because each attribute uses different units (a price-to-book ratio and a 12-month return cannot be added directly), each is converted to a z-score first. The standardized scores can then be weighted and combined into a single composite rank.
Mean reversion and pairs trading. Quantitative traders often standardize a price spread between two related instruments, or a price against its moving average:
z = (current spread − mean spread) / standard deviation of spread
Hypothetical rule: a trader might enter when the spread’s z-score passes ±2 and exit as it returns toward 0. The threshold is the trader’s choice, and the rule assumes the relationship will revert, which it may not.
Testing trading systems. In forex and other markets, a z-score built on a runs test checks whether a system’s winning and losing trades cluster into streaks more or less than chance would predict. A strongly negative or positive result suggests dependence between consecutive trades, which some traders use to adjust position sizing.
Outside finance. Z-scores appear in standardized test scoring, medical measures such as growth charts and bone density, quality control and general business analysis.
Z-Score vs. Related Measures
Standard deviation. Standard deviation describes how dispersed an entire data set is. A set with a standard deviation of 6 is more spread out, and graphs as a flatter curve, than one with a standard deviation of 3. A z-score describes one point within that set, using the standard deviation as its unit of measure. You must calculate the standard deviation before you can calculate any z-score.
Altman Z-score. Developed by Edward Altman in 1968, the Altman Z-score combines five weighted financial ratios (including working capital, retained earnings, operating earnings, market value of equity and sales, each scaled to total assets or liabilities) to estimate the likelihood that a company will go bankrupt. In the original model for public manufacturers, scores above 2.99 indicated a “safe” zone and scores below 1.81 a “distress” zone. It is a credit model that shares only the letter with the statistical z-score.
T-score (t-statistic). When a sample is small and the population standard deviation is unknown, statisticians use a t-statistic and the t-distribution, which has fatter tails than the normal curve. With large samples the two give nearly identical results.
Risk-adjusted ratios. Measures such as the Sharpe ratio also divide an excess quantity by a volatility figure, but they evaluate return per unit of risk for a portfolio over time rather than locating one observation in a distribution.
Limitations of a Z-Score
- Normality is assumed for probabilities. The 68/95/99.7 shares and percentile conversions hold only for normally distributed data. Asset returns frequently show fat tails and skew, so a z-score of -4 can occur far more often than a normal table implies. The calculation still works on any data, but the probability reading does not.
- Outliers distort the inputs. One extreme value inflates the standard deviation and shifts the mean, which shrinks every other z-score in the set, including the outlier’s own.
- Results depend on the window and group. A return measured against a three-year history can produce a very different z-score than the same return measured against ten years, or against a different peer group.
- Past distributions can change. A z-score compares a value with historical behavior. If volatility or the underlying relationship shifts, the old mean and standard deviation may no longer describe what is normal.
- Rarity is not a signal by itself. An extreme z-score says a value is unusual, not that it will revert, continue or matter economically.
The Bottom Line
A z-score turns any value into a count of standard deviations from its mean, using z = (x − μ) / σ. That makes values on different scales directly comparable and, when the data are close to normal, links each score to a probability. Investors and traders use it to compare returns, build factor rankings and flag statistical extremes. Its readings are only as reliable as the mean, standard deviation and distributional assumptions behind them.
