What does a negative z-score mean?

The sign tells you which side of the mean

A z-score standardizes a value by measuring its distance from the mean in standard deviations:

The only thing that can make $z$ negative is the numerator $x - \mu$, so a negative z-score means exactly one thing: $x$ is below the mean. A positive z-score means above, and $z = 0$ means the value sits right at the mean.

The magnitude carries the rest of the message. A z-score of $-0.3$ is barely below average, close enough that it would be unremarkable in any data set. A z-score of $-3.0$ is three standard deviations below the mean, far out in the left tail and rare enough to be worth investigating. Read the sign for direction and the magnitude for how unusual.

One note on $\mu$. When the value comes from a probability model rather than a collected data set, the $\mu$ in that formula is the random variable's expected value, which is the same quantity under a different name; see is expected value the same as the mean.

What the size of a negative z means

For a roughly normal distribution, each z-score corresponds to a proportion of the data below it. These are computed values, not estimates:

The empirical rule is the quick sanity check on this table. About 95% of a normal distribution lies within 2 standard deviations of the mean, so roughly 2.5% should lie below $z = -2$, and the exact value 0.0228 sits right where that rough figure predicts. If your table lookup and the empirical rule disagree badly, you have made an error somewhere.

Notice how fast the proportions shrink. Moving from $z = -1$ to $z = -2$ cuts the proportion below from about 16% to about 2%, roughly a factor of 7.

Negative does not mean bad

This is the most common misreading, and it costs points in context questions. A z-score is arithmetic, not judgment. Whether below average is good depends entirely on the variable.

• Marathon finish time: $z = -1.8$ means you finished well ahead of the field. Excellent.
• Cholesterol level: a negative z is usually the healthier direction.
• Exam score: $z = -1.8$ means you scored well below the class. Not excellent.
• Golf score: negative is better, because low scores win.

So always translate the sign into the context before you judge it: "below the mean" first, then decide what below the mean means for this variable. A negative z-score also has nothing to do with the data value being a negative number. A newborn weighing 5.5 pounds in a population averaging 7.5 pounds has a negative z-score, and 5.5 pounds is a positive weight.

Reading a negative z on a z-table

A standard normal table gives the area to the left of $z$, which is the proportion of values below yours. Most z-tables print two pages, one for negative z and one for positive z.

Split your z-score at the hundredths place. For $z = -1.53$:

1. Find the row labeled $-1.5$ on the negative page.
2. Move across to the column labeled 0.03.
3. Read the entry: 0.0630.

That 0.0630 is the proportion below, so about 6.30% of the distribution falls under $z = -1.53$. To get the proportion above, subtract from 1: $1 - 0.0630 = 0.9370$.

Watch the row label carefully. On a negative page laid out like the AP Statistics Table A, the rows ascend from $-3.4$ at the top to $-0.0$ at the bottom, and the column still adds to the magnitude, so the row $-1.5$ with the column 0.03 gives $-1.53$, not $-1.47$. Grabbing the wrong side of the table is the single most common z-table error. The site's z-table uses that layout and covers both signs, and the z-score calculator will confirm the lookup.

Symmetry: why one side is enough

The standard normal curve is symmetric about 0, which forces a tidy relationship between a negative z and its positive twin:

In words, the area to the left of $-1.5$ equals the area to the right of $1.5$. Both are 0.0668. Equivalently, $P(Z < -1.5) = 1 - P(Z < 1.5) = 1 - 0.9332 = 0.0668$, since $P(Z < -1.5)$ and $P(Z < 1.5)$ add to exactly 1.

That is why a table printed only for positive z is still usable: look up $|z|$, then subtract from 1. It is also why the percentile of $-1.5$ (about the 7th) and the percentile of $+1.5$ (about the 93rd) add to 100.

One useful consequence for inference. Because of this symmetry, a two-sided p-value can be found as twice the area in one tail, and the critical values you use for confidence intervals come in matched pairs such as $-1.96$ and $1.96$.

Mistakes to avoid

• Do not drop the negative sign. $z = -1.5$ and $z = 1.5$ point to opposite sides of the mean and to completely different percentiles, 7th versus 93rd.
• Do not read the table area as the area above. Standard z-tables give area to the left. If a question asks what proportion scored higher, subtract from 1.
• Do not treat a negative z as an error. Roughly half of any symmetric distribution has a negative z-score. Getting one is normal.
• Do not compare raw scores across different tests. Standardizing is the whole point: $-0.60$ on one test is a higher relative standing than $-0.70$ on another, even if the raw scores look nothing alike, and only then do you ask whether higher is better for that variable.
• Do not mix up $\sigma$ and $s$. Use the population standard deviation when you have it; using a sample standard deviation makes the result an estimate. Full mechanics are in how to find a z-score.