Z-scores four ways: one value against a mean and SD, every value in a list with outlier flags, z to percentile and percentile to z, plus T-score, stanine and NCE for test reports.
Z-Score Calculator - Percentile, Outliers and Modified Z
Z-scores four ways: one value against a mean and SD, every value in a list with outlier flags, z to percentile and percentile to z, plus T-score, stanine and NCE for test reports.
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Z-score calculator for one value, a whole list, or a percentile
An IQ of 130 on a scale with mean 100 and standard deviation 15 sits exactly 2.00 standard deviations above the mean, higher than 97.7% of a normal population. That is the first of four jobs this z-score calculator does. It also scores every value in a list and flags outliers with the modified z-score that survives an extreme value, turns a z-score into a percentile and a two-tailed p-value, and turns a percentile back into a z-score, both one-sided and two-sided. For single values it adds the equivalent T-score, stanine and NCE, so one result reads on the scales that test reports use.
Four modes and the boxes each one opens
- What do you want to calculate - one value against a known mean and SD, z-scores for a list, a z-score to a percentile, or a percentile to a z-score. Only the matching boxes appear, and nothing in a hidden box is used.
- Value, mean and standard deviation - all three in the same units, such as test points, centimeters or seconds. The SD must be greater than zero.
- List of numbers and the kind of SD - values separated by spaces, commas or new lines, with a dot for decimals. Sample SD divides by n - 1 and fits data drawn from a larger group; population SD divides by n and fits a complete set.
- Z-score - any number, negative for values below the mean, such as 1.96 or -1.5.
- Percentage - strictly between 0 and 100, such as 95 or 97.5.
- Read the result: the z-score or percentile in the main tile, the tiles below it, and, for a list, the table with ordinary and modified z side by side.
What a z-score measures, and what it does not
A z-score is a distance measured in standard deviations: z = (value - mean) / SD. The unit disappears in the division, which is the whole point. A reaction time of 62 ms against a mean of 70 and an SD of 5 has z = -1.60, a grade point average of 3.1 against 2.8 and 0.45 has z = 0.67, and the two can now be compared on one ruler even though one is in milliseconds and the other in grade points.
The z-score itself needs no assumption about the shape of the data. The percentile printed next to it does: it is read from the normal curve, so for strongly skewed data such as incomes or waiting times it can be several points off. When the data are in front of you, the list mode prints the actual rank of each value among the others as well, the way Excel's PERCENTRANK.INC computes it, so the two can be compared.
One more property matters for outliers. In a list, the extreme value inflates the very SD it is divided by. With n values and a sample SD no z can exceed (n - 1) / √n, which is 2.85 for ten values, so the popular rule "above 3 is an outlier" cannot fire in a short list. The modified z-score of Iglewicz and Hoaglin, 0.6745 × (value - median) / MAD, uses the median and the median absolute deviation, which an outlier cannot move, and flags values above 3.5.
The z-table in twelve rows
Each row is what the calculator returns in the z-to-percentile mode. The rows at 1.645, 1.96 and 2.576 are the critical values behind 90%, 95% and 99% confidence intervals.
| z | Below | Above | Between -z and z | Two-tailed p |
|---|---|---|---|---|
| 0 | 50.0% | 50.0% | 0% | 100% |
| 0.5 | 69.1% | 30.9% | 38.3% | 61.7% |
| 1 | 84.1% | 15.9% | 68.3% | 31.7% |
| 1.282 | 90.0% | 10.0% | 80.0% | 20.0% |
| 1.5 | 93.3% | 6.7% | 86.6% | 13.4% |
| 1.645 | 95.0% | 5.0% | 90.0% | 10.0% |
| 1.96 | 97.5% | 2.5% | 95.0% | 5.0% |
| 2 | 97.7% | 2.3% | 95.4% | 4.6% |
| 2.326 | 99.0% | 1.0% | 98.0% | 2.0% |
| 2.576 | 99.50% | 0.50% | 99.00% | 1.00% |
| 3 | 99.87% | 0.13% | 99.73% | 0.27% |
| 3.291 | 99.950% | 0.050% | 99.900% | 0.100% |
For a negative z swap the Below and Above columns: z = -1 has 15.9% below it.
Three kinds of printed z-table, one answer
Textbooks print the normal table in three layouts, and reading the wrong one is a classic exam mistake. For z = 1.2 the three tables show three different numbers that all describe the same curve:
| Table type | Area it gives | Entry at z = 1.2 |
|---|---|---|
| Cumulative (left tail) | everything below z | 0.8849 |
| Upper tail (right tail) | everything above z | 0.1151 |
| Mean to z (center) | from 0 up to z | 0.3849 |
The quick check: the first two always add up to 1, and the center entry is the cumulative one minus 0.5. The calculator's "below" tile is the cumulative table, "above" is the upper-tail table, and half of "between -z and z" is the center table. A negative z needs no separate table, because the curve is symmetric: the area below -1.2 equals the area above 1.2.
One z-score on five reporting scales
Standardized tests rarely print z itself. They rescale it: a T-score is 50 + 10z, the IQ scale 100 + 15z, a normal curve equivalent 50 + 21.06z, and a stanine is 2z + 5 rounded and held between 1 and 9. The single-value mode prints all of them.
| z | Percentile | T-score | IQ scale | Stanine | NCE | Band |
|---|---|---|---|---|---|---|
| -2.00 | 2.3% | 30.0 | 70 | 1 | 7.9 | unusual |
| -1.00 | 15.9% | 40.0 | 85 | 3 | 28.9 | clearly away from the mean |
| 0.67 | 74.8% | 56.7 | 110 | 6 | 64.0 | typical |
| 1.50 | 93.3% | 65.0 | 122.5 | 8 | 81.6 | clearly away from the mean |
| 2.00 | 97.7% | 70.0 | 130 | 9 | 92.1 | unusual |
Standardizing five values by hand
Take the scores 12, 15, 18, 21 and 24. Their mean is 18 and the squared distances from it are 36, 9, 0, 9 and 36, which sum to 90. Divided by n - 1 = 4 that gives a variance of 22.5 and a sample SD of 4.7434. Each z is then the distance divided by that SD:
| Value | Distance from 18 | z, sample SD | z, population SD | Rank in the list |
|---|---|---|---|---|
| 12 | -6 | -1.26 | -1.41 | 0.0% |
| 15 | -3 | -0.63 | -0.71 | 25.0% |
| 18 | 0 | 0.00 | 0.00 | 50.0% |
| 21 | 3 | 0.63 | 0.71 | 75.0% |
| 24 | 6 | 1.26 | 1.41 | 100.0% |
Two checks worth knowing. The z-scores of any list add up to exactly 0, and with the population SD their squares add up to n, here 5. If a spreadsheet column of z-scores does not sum to zero, a wrong mean went into it. The population column, with its SD of 4.2426 (the square root of 90 / 5), reaches 1.41 at the ends. The largest any z can be in a list of five, with the population SD, is the square root of n - 1, which is 2, and it is reached only when four values are equal and the fifth differs: in 10, 10, 10, 10, 20 the value 20 has z = 2.00.
Less obvious facts about z-scores
What each band of z means for a single value
| |z|, rounded to two places | Label in the result | Share of a normal population this far out or farther |
|---|---|---|
| below 1 | typical | more than 31.7% |
| 1 to 2 | clearly away from the mean | 31.7% down to 4.6% |
| 2 to 3 | unusual | 4.6% down to 0.27% |
| 3 or more | extreme | 0.27% or less |
These bands are conventions for describing a distance, not clinical or academic cut-offs. A growth chart, a lab reference range or an admissions policy has its own thresholds, and those win over any generic label.
Where z-scores turn up outside a statistics class
Growth charts. Pediatric charts from the WHO report a child's height or weight as a z-score against children of the same age and sex, and the WHO growth standards use -2 and +2 as the lines for stunting, wasting and overweight. The chart supplies its own mean and SD for each month of age, so a generic mean and SD should not be typed in for a child; the chart's own tables apply.
Bone density. A DEXA scan prints a T-score and a Z-score, and these are not the scale called T-score above. The densitometry T-score compares bone density with young healthy adults, the Z-score with people of the same age, and the WHO definition of osteoporosis sits at a densitometry T-score of -2.5 or lower.
Test scores. IQ tests, the scale scores of many aptitude tests and school reports built on stanines are all a z-score dressed in a different mean and SD, which is why the scales table above converts freely between them.
Quality control. Laboratory proficiency testing scores each lab's result as a z-score against the assigned value, and the usual reading is |z| up to 2 satisfactory, between 2 and 3 questionable and 3 or more unsatisfactory. Control charts put their action lines at 3 SD for the same reason: 0.27% of a stable process falls outside by chance.
Hypothesis tests. A z test statistic is a z-score of the sample mean, measured in standard errors instead of standard deviations, and its two-tailed p-value is exactly the "outside" column of the z-table.
Z-score questions from the search box
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