Z-Score Calculator

Q: How do I convert a z-score to a percentile?

Find Φ(z), the area to the left of z, and multiply by 100. For z = 1.5, Φ(1.5) = 0.933193, so the score sits at about the 93rd percentile (93.32% below). On this page, pick z-score to probability, type the z and leave Area on Below.

Q: What z-score is the 90th percentile?

z = +1.2816. On a test with mean 70 and standard deviation 8, that puts the 90th-percentile score, which is also the top-10% cutoff, at 70 + 1.2816 × 8 = 80.2524. The 1.282 row of the Common critical values table shows the same cut rounded to three decimals.

Q: What does a z-score of 0 mean?

The value equals the mean. Half of a normal distribution lies below it, so z = 0 is the 50th percentile, and on the T-score scale it is 50.

Q: Can a z-score be negative, and how do I look one up in a table that lists positive z?

Yes: a negative z means the value is below the mean. The curve is symmetric, so Φ(−z) = 1 − Φ(z). For z = -2, row 2.0 and column 0.00 of the z-table give Φ(2) = 0.9772, so the area below -2 is 1 − 0.9772 = 0.0228. Typed into the calculator, -2 returns 0.0227501 (2.28%) below without the table step.

Q: What z-score do I use for a 95% confidence level?

±1.96. It leaves 2.5% in each tail and 95% in the center. For a one-sided 95% cut the value is 1.645, and for 99% it is ±2.576; the Common critical values table on this page lists these with their below, above and central percentages.

Q: How do I find the area between two z-scores?

Subtract the smaller cumulative area from the larger: P(a < Z < b) = Φ(b) − Φ(a). Between -1 and +1 that is Φ(1) − Φ(-1) = 0.682689, or 68.27%. In Between two scores the order of the bounds does not matter, and a second card gives the share outside the interval, 31.73%.

Q: Is a z-score of 2 unusual?

It sits at the edge. In a normal distribution 4.55% of values lie more than 2 standard deviations from the mean in either direction, and 2.28% lie above +2. Beyond ±3 the share drops to 0.27%. The line for "unusual" is a convention set by your course or field; the Two-tailed area gives the rarity to compare against it.

Q: What is the difference between a z-score and a T-score?

They give the same position on two scales. A z-score has mean 0 and standard deviation 1; a T-score rescales it as T = 50 + 10z, so the mean is 50 and each standard deviation is 10 points: z = +1.5 is T = 65 and z = -1.25 is T = 37.5. This capital-T score is unrelated to Student's t statistic from small-sample tests.

Q: Should I use a z-score or a t-score for a small sample?

To locate one value against a group whose mean and standard deviation you know, use z, which is what this page computes. To test a sample mean when the standard deviation is estimated from that same sample, the statistic follows Student's t-distribution, whose tails are heavier than the normal curve's when n is small. This calculator does not compute t-distribution probabilities.

Q: How do I get the same numbers in Excel or on a TI-84?

In Excel, =STANDARDIZE(82, 70, 8) returns the z-score 1.5, =NORM.S.DIST(1.5, TRUE) returns the area below it and =NORM.S.INV(0.9) goes from a probability back to z. On a TI-84, normalcdf(-99, 1.5, 0, 1) gives the area below z = 1.5, with -99 standing in for minus infinity, and invNorm(0.9, 0, 1) returns the z for the 90th percentile. Rounded to 6 significant figures, Excel's area below 1.5 matches the page's 0.933193.

Turn a value, the mean and the standard deviation into a z-score and a T-score, read any area under the normal curve, or go back from a percentile to the z that cuts it off.

Value to z-score z-score to probability Between two scores Percentile to z-score

Area Below Above Two-tailed Central

Value (x)

Mean (μ)

Standard deviation (σ)

The population σ. For a sample, enter s: the formula is the same.

z-score

+1.5

= (82 − 70) / 8

Below 82

93.32%

0.933193

T-score

T-score = 50 + 10 × z

Where it falls on the normal curve

The areas assume a normal distribution. For a real class or test, the shares and ranks are estimates.

Every area for z = +1.5

Area	Probability	Percent
Below z P(Z < 1.5)	0.933193	93.32%
Above z P(Z > 1.5)	0.0668072	6.68%
Between −z and +z P(\|Z\| < 1.5)	0.866386	86.64%
Both tails P(\|Z\| > 1.5)	0.133614	13.36%
From the mean to z P(0 < Z < 1.5)	0.433193	43.32%

How it's calculated

Standardize: z = (82 − 70) / 8 = +1.5
T-score to z: z = (— − 50) / 10 = +1.5
Solve Φ(z) = —: z = +1.5
T-score: 50 + 10 × 1.5 = 65
Area below: P(Z < 1.5) = Φ(1.5) = 0.933193 (93.32%)
T-scores to z: (— − 50) / 10 = — and (— − 50) / 10 = —
The two tails: P(Z < —) = — and P(Z > —) = —
Between: 1 − — − — = — (—)
Between, from the upper tail: P(Z > —) − P(Z > —) = — − — = — (—)
Between, from the lower tail: P(Z < —) − P(Z < —) = — − — = — (—)
Outside: — + — = — (—)
Raw value: x = 70 + 1.5 × 8 = —
Rank: — × — = —, rounded up: —

T-score table: top share and rank out of 1,000

T-score	z	Top	Bottom	Rank of 1,000

The shares assume normally distributed scores. The rank counts from the top out of 1,000 people, rounded up.

Standard normal table (z-table)

z	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
0.0	0.5000	0.5040	0.5080	0.5120	0.5160	0.5199	0.5239	0.5279	0.5319	0.5359
0.1	0.5398	0.5438	0.5478	0.5517	0.5557	0.5596	0.5636	0.5675	0.5714	0.5753
0.2	0.5793	0.5832	0.5871	0.5910	0.5948	0.5987	0.6026	0.6064	0.6103	0.6141
0.3	0.6179	0.6217	0.6255	0.6293	0.6331	0.6368	0.6406	0.6443	0.6480	0.6517
0.4	0.6554	0.6591	0.6628	0.6664	0.6700	0.6736	0.6772	0.6808	0.6844	0.6879
0.5	0.6915	0.6950	0.6985	0.7019	0.7054	0.7088	0.7123	0.7157	0.7190	0.7224
0.6	0.7257	0.7291	0.7324	0.7357	0.7389	0.7422	0.7454	0.7486	0.7517	0.7549
0.7	0.7580	0.7611	0.7642	0.7673	0.7704	0.7734	0.7764	0.7794	0.7823	0.7852
0.8	0.7881	0.7910	0.7939	0.7967	0.7995	0.8023	0.8051	0.8078	0.8106	0.8133
0.9	0.8159	0.8186	0.8212	0.8238	0.8264	0.8289	0.8315	0.8340	0.8365	0.8389
1.0	0.8413	0.8438	0.8461	0.8485	0.8508	0.8531	0.8554	0.8577	0.8599	0.8621
1.1	0.8643	0.8665	0.8686	0.8708	0.8729	0.8749	0.8770	0.8790	0.8810	0.8830
1.2	0.8849	0.8869	0.8888	0.8907	0.8925	0.8944	0.8962	0.8980	0.8997	0.9015
1.3	0.9032	0.9049	0.9066	0.9082	0.9099	0.9115	0.9131	0.9147	0.9162	0.9177
1.4	0.9192	0.9207	0.9222	0.9236	0.9251	0.9265	0.9279	0.9292	0.9306	0.9319
1.5	0.9332	0.9345	0.9357	0.9370	0.9382	0.9394	0.9406	0.9418	0.9429	0.9441
1.6	0.9452	0.9463	0.9474	0.9484	0.9495	0.9505	0.9515	0.9525	0.9535	0.9545
1.7	0.9554	0.9564	0.9573	0.9582	0.9591	0.9599	0.9608	0.9616	0.9625	0.9633
1.8	0.9641	0.9649	0.9656	0.9664	0.9671	0.9678	0.9686	0.9693	0.9699	0.9706
1.9	0.9713	0.9719	0.9726	0.9732	0.9738	0.9744	0.9750	0.9756	0.9761	0.9767
2.0	0.9772	0.9778	0.9783	0.9788	0.9793	0.9798	0.9803	0.9808	0.9812	0.9817
2.1	0.9821	0.9826	0.9830	0.9834	0.9838	0.9842	0.9846	0.9850	0.9854	0.9857
2.2	0.9861	0.9864	0.9868	0.9871	0.9875	0.9878	0.9881	0.9884	0.9887	0.9890
2.3	0.9893	0.9896	0.9898	0.9901	0.9904	0.9906	0.9909	0.9911	0.9913	0.9916
2.4	0.9918	0.9920	0.9922	0.9925	0.9927	0.9929	0.9931	0.9932	0.9934	0.9936
2.5	0.9938	0.9940	0.9941	0.9943	0.9945	0.9946	0.9948	0.9949	0.9951	0.9952
2.6	0.9953	0.9955	0.9956	0.9957	0.9959	0.9960	0.9961	0.9962	0.9963	0.9964
2.7	0.9965	0.9966	0.9967	0.9968	0.9969	0.9970	0.9971	0.9972	0.9973	0.9974
2.8	0.9974	0.9975	0.9976	0.9977	0.9977	0.9978	0.9979	0.9979	0.9980	0.9981
2.9	0.9981	0.9982	0.9982	0.9983	0.9984	0.9984	0.9985	0.9985	0.9986	0.9986
3.0	0.9987	0.9987	0.9987	0.9988	0.9988	0.9989	0.9989	0.9989	0.9990	0.9990
3.1	0.9990	0.9991	0.9991	0.9991	0.9992	0.9992	0.9992	0.9992	0.9993	0.9993
3.2	0.9993	0.9993	0.9994	0.9994	0.9994	0.9994	0.9994	0.9995	0.9995	0.9995
3.3	0.9995	0.9995	0.9995	0.9996	0.9996	0.9996	0.9996	0.9996	0.9996	0.9997
3.4	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9998

Each cell is Φ(z), the area to the left of z. The row gives z to one decimal and the column adds the second: z = 1.96 is row 1.9, column 0.06. For a negative z, Φ(−z) = 1 − Φ(z); the calculator above gives any z exactly.

Common critical values

z	Below	Above	Central
1.000	84.13%	15.87%	68.27%
1.282	90%	10%	80%
1.645	95%	5%	90%
1.960	97.5%	2.5%	95%
2.000	97.72%	2.28%	95.45%
2.326	99%	1%	98%
2.576	99.5%	0.5%	99%
3.000	99.87%	0.13%	99.73%
3.291	99.95%	0.05%	99.9%

Each z is the exact cut-off for its percentage, rounded to three decimals: 1.960 leaves 2.5% in each tail and 95% in the center.

Have every score but not the mean or σ? Work them out from the list with the standard deviation calculator.

Updated: September 26, 2026

Z-score calculator with z-table. Percentile and normal-curve area from a score, the mean and the standard deviation.

This z-score calculator turns a score, the mean and the standard deviation into a z-score and the percentage of a normal distribution below or above it. It also works backwards from a percentile to the z and the raw score that cut it off, and its folded z-table highlights the cell for your z.

What a z-score tells you, and how it becomes a percentile

A z-score is the number of standard deviations a value sits above or below the mean, found with z = (x − μ) / σ, where x is the value, μ the mean and σ the standard deviation. A score of 82 on a test with a mean of 70 and a standard deviation of 8 has z = (82 − 70) / 8 = +1.5: the 12 points above average are one and a half standard deviations of that test.

The sign gives the side of the mean and the size gives the distance, which puts results from different scales on one ruler. An 82 on the test above (z = +1.5) is a stronger result than a 90 on a test with mean 84 and standard deviation 6 (z = +1.0), although 90 is the bigger number. A z of 0 is the mean itself, and a negative z is a value below it.

If scores follow a normal distribution, the z-score also fixes the share of the group on each side of the value. That share is Φ(z), the area to the left of z under the standard normal curve, the bell curve with mean 0 and standard deviation 1. For z = +1.5, Φ(1.5) = 0.933193: about 93.32% of a normally distributed class scores below 82 and 6.68% above it, so 82 sits near the 93rd percentile. The NIST/SEMATECH e-Handbook points out that Φ has no closed-form formula, so it is computed numerically or read from a printed z-table.

This z-score calculator covers both directions with four modes. Value to z-score takes x, μ and σ and returns z, the area picked in the Area control (Below, Above, Two-tailed or Central) and the same position as a T-score, T = 50 + 10z, which makes +1.5 a 65. The z-score to probability mode starts from a z or a T-score. Between two scores gives the share between two cuts and the share outside them. Percentile to z-score runs backwards: type a percentage and get the z that cuts it off, plus the raw cutoff score when the mean and σ are filled in.

Under the results, a table lists every area for the current z, followed by three folded panels: the worked steps, a standard normal table (z-table) for z from 0.00 to 3.49 that highlights the cell for your z, and the common critical values such as 1.645 and 1.960. If you have the list of scores rather than their mean and standard deviation, get those two numbers first from the standard deviation calculator.

How to use it: one exam score, from z-score to the A cutoff

Jordan scored 82 on a statistics midterm, and the instructor posted a class mean of 70 with a standard deviation of 8.

1. The page opens on Value to z-score with 82, 70 and 8 already in Value (x), Mean (μ) and Standard deviation (σ), so Jordan's case is the default. The main card reads z-score +1.5 with the working (82 − 70) / 8 under it, the card labeled Below 82 reads 93.32%, and the T-score card reads 65.

2. To see how many classmates scored higher, Jordan switches Area from Below to Above. The area card drops to 6.68%, and the shading on the bell curve moves to the right of the mark at +1.5.

3. Back on Below, Jordan opens How it's calculated, which lists two steps: z = (82 − 70) / 8 = +1.5, then P(Z < 1.5) = Φ(1.5) = 0.933193. Opening Standard normal table (z-table) shows where the four-decimal version comes from: row 1.5 and the 0.9332 cell under column 0.00 are highlighted.

4. The syllabus gives an A to the top 10%. Jordan picks Percentile to z-score. With Area on Below the field is labeled Percentile (% below), and 90 there is the same cut as 10 in Top % (% above) with Area on Above. The main card gives z-score +1.2816; because the mean and σ are still 70 and 8, the Raw value at the cut-off card adds 70 + 1.2816 × 8 = 80.2524. Jordan's 82 clears the A line by 1.75 points.

5. For the middle of the class, Jordan picks Between two scores, where the Area control disappears. This mode reads z-scores or T-scores, so the raw scores 62 and 78 go in as (62 − 70) / 8 = -1 and (78 − 70) / 8 = +1, which are also the default Lower bound and Upper bound. The main card reads P(-1 < Z < 1) = 68.27%, and Outside the interval reads 31.73%.

Z-score examples: below the mean, T-scores, confidence levels and tails

A score below the mean: 60 on the same test (z = -1.25)

On the test with mean 70 and σ 8, a 60 gives z = (60 − 70) / 8 = -1.25. In z-score to probability, with -1.25 typed and Area on Above, the card reads P(Z > -1.25) = 89.44% (0.89435): almost nine in ten of a normal class scored higher. The raw-value card confirms 70 − 1.25 × 8 = 60, and the T-score card reads 37.5.

The page's z-table lists positive z only, and the symmetry of the curve covers the rest. Row 1.2, column 0.05 holds Φ(1.25) = 0.8944, which is also the area above -1.25. The area below 60 is 1 − 0.8944 = 0.1056, and the calculator on Below gives 0.10565 to six significant figures, or 10.56%.

Reading a T-score of 70

A T-score puts the mean at 50 and each standard deviation at 10 points, so a report showing T = 70 describes a result 2 standard deviations above average. In z-score to probability, set Entered as to T-score and type 70. The main card shows P(T < 70) = 97.72% and the other-scale card shows z-score +2; Area on Above gives the 2.28% that scored higher. With the mean 70 and σ 8 still filled in, the raw-value card translates T = 70 into a test score of 70 + 2 × 8 = 86.

Switching Entered as back to z-score converts the typed number as well, so the 70 becomes 2 instead of being read as z = 70. In Japan the same scale is called hensachi: Benesse, publisher of the Shinken Moshi mock exams, defines it as (score − mean) ÷ standard deviation × 10 + 50, and Study Sapuri reads a hensachi of 70 as the top 2.28% of test takers, about 1 in 44.

The z for a 95% confidence level, and the one-sided 1.645

In Percentile to z-score, set Area to Central, and the field becomes Central area (confidence level). Typing 95 returns z-score ±1.96, and the steps show the equation it solved, 2 × Φ(z) − 1 = 0.95. With the mean 70 and σ 8 left in, the raw-value card gives both cutoffs, 54.3203 to 85.6797: the middle 95% of a normal class on that test.

A one-sided 95% cut is a different number. With Area on Below and 95 typed, the card gives +1.6449, which the Common critical values table rounds to 1.645.

How rare is z = 1.5 in both directions?

With 82, 70 and 8 in Value to z-score and Area on Two-tailed, the card reads Further from the mean than 82 = 13.36% (0.133614), about 1 in 7. That is the share of a normal class at least 12 points from 70 on either side, below 58 or above 82. For a z statistic in a hypothesis test, this two-tailed area is the two-sided p-value; the page gives the number and leaves the comparison with your α to you.

One z-score, five areas

Each area answers its own question about z = +1.5 on the mean-70, σ-8 test, and the page lists all five under the curve:

Area	Question it answers	Probability	Percent
Below z	What share scored below 82?	0.933193	93.32%
Above z	What share scored above 82?	0.0668072	6.68%
Between −z and +z	What share scored between 58 and 82?	0.866386	86.64%
Both tails	What share scored below 58 or above 82?	0.133614	13.36%
From the mean to z	What share scored between 70 and 82?	0.433193	43.32%

Adding 0.5 to the last row gives the first: 0.5 + 0.433193 = 0.933193.

Z-score to percentile chart, -3 to +3

z-score	Percentile (% below)	% above	T-score
-3.0	0.13%	99.87%	20
-2.5	0.62%	99.38%	25
-2.0	2.28%	97.72%	30
-1.5	6.68%	93.32%	35
-1.0	15.87%	84.13%	40
-0.5	30.85%	69.15%	45
0.0	50%	50%	50
+0.5	69.15%	30.85%	55
+1.0	84.13%	15.87%	60
+1.5	93.32%	6.68%	65
+2.0	97.72%	2.28%	70
+2.5	99.38%	0.62%	75
+3.0	99.87%	0.13%	80

Z-score formula, the T-score and the areas built on them

z = \frac{x - \mu}{\sigma}

$z$ = the z-score, or standard score: how many standard deviations x lies from the mean, positive above it and negative below
$x$ = the value you are locating, such as one exam score
$\mu$ = the mean of the group or distribution
$\sigma$ = the standard deviation; a sample standard deviation s goes in the same place

Solved for x, the same equation gives the raw value at any z: x = μ + zσ. The raw-value card carries it in its label, (x = μ + zσ), and it is how a percentile turns into a cutoff score: on the test with mean 70 and σ 8, z = +1.2816 lands at 70 + 1.2816 × 8 = 80.2524. The T-score is one more rescaling, T = 50 + 10z, so the mean maps to 50 and each standard deviation is worth 10 points; the way back is z = (T − 50) / 10.

Every area on the page comes from Φ, the cumulative standard normal distribution, and from its upper tail Q(z) = 1 − Φ(z):

P(Z<z)=\Phi(z)\qquad P(Z>z)=Q(z)\qquad P(a<Z<b)=\Phi(b)-\Phi(a)

P(|Z|>|z|)=2\,Q(|z|)\qquad P(|Z|<|z|)=1-2\,Q(|z|)\qquad P(0<Z<|z|)=\Phi(|z|)-0.5

On a computer, the subtraction 1 − Φ(z) wipes out a small tail: once Φ(z) rounds to 1 in floating point, the upper tail comes out as 0. The calculator computes each tail on its own side with W. J. Cody's rational approximations to erfc, which the netlib code says theoretically reach at least 18 significant digits, and turns a percentile back into z with M. J. Wichura's algorithm AS 241, accurate to about 16. Probabilities print with 6 significant figures, z with up to 4 decimals and T with 1. At the extremes the large side prints as > 0.999999 and a tail below 10⁻³⁰⁰ prints as < 1 × 10⁻³⁰⁰, so neither side ever reads as a flat 1 or 0.

Common z-score mistakes

Reading a 0-to-z table as if it were cumulative. Printed z-tables come in different styles: cumulative from the left, from the mean to z, and sometimes the upper tail. calculator.net's table is titled "Z Table from Mean (0 to Z)", so its cell for z = 1.50 is the area between the mean and z, 0.43319 at five decimals; the table on this page is cumulative from the left and gives 0.9332 for the same z. Check a table's header before copying a cell, and for a positive z add 0.5 to a 0-to-z value to get the percentile.
Typing a probability where a percentage goes. The Percentile to z-score field takes a percentage. Typed as 0.975, it asks for the 0.975th percentile and returns z = -2.3358; the page catches this with a line under the results that says to type 97.5 for a probability of 0.975.
Subtracting in the wrong order. The formula is value minus mean. (70 − 82) / 8 gives -1.5, and the percentile flips with the sign: 6.68% below instead of 93.32%.
Dividing by the variance. σ is the standard deviation, the square root of the variance. A test reported with a variance of 64 has σ = 8; dividing 12 points by 64 gives z = 0.1875 where the right answer is +1.5.
Picking one tail when the question has two. "More extreme than" and "at least this far from the mean" are two-tailed: for z = 1.5 the answer is 13.36%, twice the 6.68% in the upper tail. A confidence level is a central area, so a 95% level splits the remaining 5% between the two tails.
Using the single-score formula for a sample mean. z = (x − μ) / σ locates one value. The average of n values varies less, so its z divides by the standard error σ / √n, the method CalculatorSoup offers as its sample-mean option; this page works with single values.

Where a z-score percentile is only an estimate

The z-score itself holds for any data: (82 − 70) / 8 is +1.5 whatever shape the class's scores have. The percentages are areas under a normal curve, and the page prints that assumption under every result: "The areas assume a normal distribution. For a real class or test, the shares and ranks are estimates."

Three cases pull the real share away from the model. Skewed scores, such as an easy test where most of the class bunches near the top, put more or fewer people below a given z than the bell curve says. Small groups move in steps: in a class of 20 each student is 5% of the class, so the real share below a score can land on 90% or 95% but not on the modeled 93.32%. A hard ceiling, such as several perfect scores on a 100-point test, cuts off the upper tail the model assumes. With the full list of scores, counting the ones below yours gives the real percentile rank.

The page also stops short of inference. It has no t-distribution for small samples, no hypothesis-test decision against α and no confidence interval for a mean. Medical z-scores, such as bone-density T- and Z-scores and children's growth-chart z-scores, are out of scope as well.

Z-score and z-table questions

How do I convert a z-score to a percentile?

Find Φ(z), the area to the left of z, and multiply by 100. For z = 1.5, Φ(1.5) = 0.933193, so the score sits at about the 93rd percentile (93.32% below). On this page, pick z-score to probability, type the z and leave Area on Below.

What z-score is the 90th percentile?

z = +1.2816. On a test with mean 70 and standard deviation 8, that puts the 90th-percentile score, which is also the top-10% cutoff, at 70 + 1.2816 × 8 = 80.2524. The 1.282 row of the Common critical values table shows the same cut rounded to three decimals.

What does a z-score of 0 mean?

The value equals the mean. Half of a normal distribution lies below it, so z = 0 is the 50th percentile, and on the T-score scale it is 50.

Can a z-score be negative, and how do I look one up in a table that lists positive z?

Yes: a negative z means the value is below the mean. The curve is symmetric, so Φ(−z) = 1 − Φ(z). For z = -2, row 2.0 and column 0.00 of the z-table give Φ(2) = 0.9772, so the area below -2 is 1 − 0.9772 = 0.0228. Typed into the calculator, -2 returns 0.0227501 (2.28%) below without the table step.

What z-score do I use for a 95% confidence level?

±1.96. It leaves 2.5% in each tail and 95% in the center. For a one-sided 95% cut the value is 1.645, and for 99% it is ±2.576; the Common critical values table on this page lists these with their below, above and central percentages.

How do I find the area between two z-scores?

Subtract the smaller cumulative area from the larger: P(a < Z < b) = Φ(b) − Φ(a). Between -1 and +1 that is Φ(1) − Φ(-1) = 0.682689, or 68.27%. In Between two scores the order of the bounds does not matter, and a second card gives the share outside the interval, 31.73%.

Is a z-score of 2 unusual?

It sits at the edge. In a normal distribution 4.55% of values lie more than 2 standard deviations from the mean in either direction, and 2.28% lie above +2. Beyond ±3 the share drops to 0.27%. The line for "unusual" is a convention set by your course or field; the Two-tailed area gives the rarity to compare against it.

What is the difference between a z-score and a T-score?

They give the same position on two scales. A z-score has mean 0 and standard deviation 1; a T-score rescales it as T = 50 + 10z, so the mean is 50 and each standard deviation is 10 points: z = +1.5 is T = 65 and z = -1.25 is T = 37.5. This capital-T score is unrelated to Student's t statistic from small-sample tests.

Should I use a z-score or a t-score for a small sample?

To locate one value against a group whose mean and standard deviation you know, use z, which is what this page computes. To test a sample mean when the standard deviation is estimated from that same sample, the statistic follows Student's t-distribution, whose tails are heavier than the normal curve's when n is small. This calculator does not compute t-distribution probabilities.

How do I get the same numbers in Excel or on a TI-84?

In Excel, =STANDARDIZE(82, 70, 8) returns the z-score 1.5, =NORM.S.DIST(1.5, TRUE) returns the area below it and =NORM.S.INV(0.9) goes from a probability back to z. On a TI-84, normalcdf(-99, 1.5, 0, 1) gives the area below z = 1.5, with -99 standing in for minus infinity, and invNorm(0.9, 0, 1) returns the z for the 90th percentile. Rounded to 6 significant figures, Excel's area below 1.5 matches the page's 0.933193.

Does the percentile box take 0.975 or 97.5?

A percentage, so type 97.5. A number between 0 and 1 is still read as a percentage (0.975 gives z = -2.3358), and the page adds a line under the results: "This field takes a percentage: for a probability of 0.975, type 97.5."

Does a z-score of +1.5 mean I got an A?

That depends on how your instructor grades. The z says you are 1.5 standard deviations above the mean, near the 93rd percentile if scores are close to normal. If the syllabus gives the A to the top 10%, Percentile to z-score turns that rule into a cutoff score (80.2524 on a test with mean 70 and σ 8). If grades use fixed point bands, the z does not change the letter.

Can I use this for a bone-density T-score or a child's growth z-score?

No. According to MedlinePlus, the T-score on a bone-density scan compares your bone density with a healthy young adult of the same sex, and the Z-score compares it with people of your age, weight, sex and ethnic origin. Neither is the 50 + 10z T-score this page prints, so a -2.5 from a DXA report cannot be typed here as a T-score. Growth-chart z-scores for children are out of scope too.

How accurate is this z-score calculator?

The areas come from W. J. Cody's erfc approximations, which the netlib source says theoretically reach 18 or more significant digits, and the percentile-to-z direction from algorithm AS 241, at about 16 digits. The page computes in double precision, which holds about 16 significant digits, so that is the practical limit. Each tail is computed on its own side, so a far tail stays a real number instead of collapsing to 0. The page prints 6 significant figures, and the z-table shows 4 decimals.

Is this z-score calculator free, and can I share a result?

Yes, it is free and needs no account. The scenario you set up travels in the page address, so a copied link reopens the same mode, area and numbers for a classmate, and the same calculator can be embedded on a course page.

Sources & References

Content verified by the Smart Calculators Team