What a z-score tells you

A z-score measures how far a raw value is from the mean in standard deviation units. If a value has a z-score of 2, it is two standard deviations above the mean. If it has a z-score of -1.5, it is one and a half standard deviations below the mean.

The formula is:

Here, $x$ is the raw value, $\mu$ is the mean, and $\sigma$ is the standard deviation. The standard deviation must be positive, because it represents spread.

Interpreting positive, negative, and zero z-scores

• A positive z-score means the raw value is above the mean.
• A negative z-score means the raw value is below the mean.
• A zero z-score means the raw value is exactly equal to the mean.

The magnitude matters too. A z-score near 0 is close to average, while a z-score such as 2 or -2 is farther from the center. Very large positive or negative z-scores should be interpreted in context: they might signal an unusual observation, a meaningful outlier, or simply a distribution where extreme values are expected.

From raw values to the standard normal distribution

Standardizing converts a raw value from its original scale into the standard normal scale, where the mean is 0 and the standard deviation is 1. This lets you compare values from different normally distributed measurements on the same scale.

Once you know the z-score, you can connect it to the standard normal distribution:

• Left-tail probability, written $P(X \le x)$, is the area under the curve to the left of the raw value.
• Percentile is the left-tail probability expressed as a percentage. A percentile of 84% means about 84% of normally distributed values are at or below that point.
• Right-tail probability, written $P(X > x)$, is the area under the curve to the right of the raw value.

These probabilities rely on the normal model. If the original distribution is strongly skewed, has multiple peaks, or contains unusual outliers, normal probabilities may be less reliable even though the z-score formula still standardizes the value.

Worked example

Suppose a test score is $x = 85$, the class mean is $\mu = 70$, and the standard deviation is $\sigma = 15$.

The raw score is 1 standard deviation above the mean. Under a normal model, a z-score of 1 has a left-tail probability of about 0.8413, so the score is around the 84th percentile. The right-tail probability is about 0.1587.

Common cautions

• $\sigma$ must be greater than 0. A standard deviation of 0 means there is no spread, so the z-score formula would divide by zero.
• Normal probabilities assume the distribution is reasonably normal. The z-score itself is just a standardized distance, but percentile and tail probability interpretations use the normal curve.
• Extreme z-scores need context. In some domains, values beyond 3 standard deviations are rare; in others, heavy-tailed behavior makes them less surprising.

FAQ

Is a higher z-score always better?

No. A higher z-score only means the value is farther above the mean. Whether that is good depends on what the variable measures. A higher exam score may be good, but a higher blood pressure reading may be a warning sign.

What does a negative z-score mean?

A negative z-score means the raw value is below the mean. For example, $z = -2$ means the value is two standard deviations below the mean.

Is percentile the same as probability?

They are closely related, but they are usually expressed differently. Left-tail probability is a decimal area under the curve, such as 0.8413. Percentile expresses the same location as a percent, such as about the 84th percentile.