Standard Score Conversion Calculator

StatisticsLast updated: August 22, 2026

Standard scores translate raw scores from different tests into a common language. The z-score is what they all share: a unitless measure of how many standard deviations an observation sits from the mean. The T score (50 + 10z), sten (5.5 + 2z), stanine (2z + 5), Wechsler-type IQ (100 + 15z), and SAT section score (500 + 100z) are the same z information written on different scales; none of them adds new information, they simply shift the scale to avoid negative and fractional values.

This calculator takes a single value on any of these scales, converts it to a z-score first, and then produces its equivalent on all the other scales at once. The percentile rank comes from the cumulative probability of the standard normal distribution (Φ); in the other direction, entering a percentile rank gives the z-score through Φ⁻¹. The output also shows the cross-scale conversion table, the theoretical shares of the stanine bands, and the position of the score on the normal curve as a chart.

Select the scale your score is already on; every other scale is then computed for you.
You can use either a decimal point or a decimal comma (e.g. 1.5 or 1,5).
Enter the target mean if you also want the score on a scale of your own.
For a custom scale it must be filled in together with the mean, and it must be greater than zero.

Standard Score Conversion Formulas

Common ground:    z = (x − μ) / σ
T score:          T = 50 + 10 · z
Sten score:       Sten = 5.5 + 2 · z
Stanine:          S = round(2 · z + 5),  1 ≤ S ≤ 9
IQ (Wechsler):    IQ = 100 + 15 · z
SAT section:      SAT = 500 + 100 · z
Percentile rank:  P = Φ(z) × 100
Reverse:          z = Φ⁻¹(P / 100)
General scale:    x = μ_target + σ_target · z

Φ is the standard normal cumulative distribution function. The linear conversions (T, sten, IQ, SAT) hold for any distribution; the percentile rank and stanine equivalents rest on the normality assumption.

How to Calculate

  1. Select the scale your score is on: z, T, percentile rank, stanine, sten, IQ, or SAT.
  2. Enter the score value; a decimal comma is accepted as well as a decimal point (e.g. 1.5).
  3. If you like, enter the mean and standard deviation of your own scale to convert the score to that scale too.
  4. Read the equivalents on all the scales together with the percentile rank in the results card; the z-score is the common ground behind all of them.
  5. Look at the neighboring z steps in the conversion table to see how the score shifts from one scale to another.
  6. On the normal curve chart the size of the shaded area shows the proportion below the score, while the stanine chart shows the theoretical share of the band.

Worked Examples

From a z-score to every scale

A z-score of 1.50 corresponds to 65 on the T scale, 8.50 on the sten scale, 122.50 on the IQ scale, and 650 on the SAT scale. The percentile rank is 93.32% and the proportion above is 6.68%; the stanine band is 8, which lies in the upper band.

z-score (standard score): 1.5000 · T score: 65.00 · Percentile rank (proportion below): 93.32%

Working back from a percentile rank

The 90th percentile corresponds to a standard score of z = 1.2816. That means a T score of 62.82, a sten score of 8.06, an IQ of 119.22, and an SAT score of 628.16; the stanine band is 8. In a group of 10,000 people this score sits at about rank 1,001.

z-score (standard score): 1.2816 · T score: 62.82 · Percentile rank (proportion below): 90.00%

Converting an IQ score to a custom scale

An IQ of 130 equals a standard score of z = 2: the T score is 70, the sten score is 9.50, the SAT score is 700, and the percentile rank is 97.72%. On a custom scale with a mean of 300 and a standard deviation of 50, the same score becomes 400.

z-score (standard score): 2.0000 · T score: 70.00 · Percentile rank (proportion below): 97.72%

Frequently Asked Questions

What is the difference between a T score and a z-score?
The T score is the z-score shifted by the formula T = 50 + 10z; its mean is 50 and its standard deviation is 10. It carries exactly the same information as z, the only difference being that it never produces negative or fractional-looking values. That is why T scores are preferred in counseling reports, personality inventories, and educational measurement.
How many bands does a stanine have and what are their shares?
The stanine (standard nine) consists of nine bands from 1 to 9, and under a normal distribution their theoretical shares are 4%, 7%, 12%, 17%, 20%, 17%, 12%, 7%, and 4% respectively. The fifth band is centered on the mean. Each band is half a standard deviation wide, which makes the stanine a coarse but easily communicated scale.
Is a percentile rank the same as a percentage score?
No. A percentile rank is the percentage of people who scored below a given score; it has nothing to do with a raw score out of 100. A student who scored 65 out of 100 may well be in the 90th percentile if the exam was hard. This calculator computes the percentile rank from the cumulative probability of the normal distribution.
Why can I not enter a percentile rank of 0% or 100%?
In a normal distribution no finite z-score corresponds exactly to the 0th or the 100th percentile; those values correspond to minus and plus infinity. In practice, values such as 99.9% or 0.1% are used to express the highest or lowest score. With empirical percentiles, the highest observation is usually placed in the 99.x percentile.
Should an IQ score use a standard deviation of 15 or 16?
The Wechsler scales (WISC, WAIS) and most intelligence tests in use today have a mean of 100 and a standard deviation of 15, and this calculator follows that convention. The older editions of the Stanford-Binet and some publications use a standard deviation of 16, which gives a slightly different IQ for the same z-score. Always check the scale definition of the test before interpreting an IQ value.
How do I run the same conversions in Excel, SPSS, and R?
In Excel, use NORM.S.DIST(z, TRUE) for the percentile rank and NORM.S.INV(probability) for the reverse. In R, pnorm(z) and qnorm(p) give the same results, as do norm.cdf(z) and norm.ppf(p) in Python/SciPy. In SPSS, the "Save standardized values as variables" option in the Descriptives menu produces z-scores; for T scores, COMPUTE T = 50 + 10*Zvariable is enough.
Does standardizing make a distribution normal?
No. The z transformation is linear: it sets the mean to 0 and the standard deviation to 1, but it does not change the shape, skewness, or kurtosis of the distribution. Skewed data stays skewed after standardization. To bring a distribution closer to normal you need a nonlinear transformation such as a logarithm, a square root, or Box-Cox.