There are three ways to use this calculator. ① Find the z-score from a value, the mean and the standard deviation. ② Convert between a z-score and a probability (fill in just one field). ③ Find the probability between two z-scores. Fill in only the fields of the group you want to use.
Fill in the fields for only one way of use, and leave the other fields blank. Enter probabilities as numbers from 0 to 1 (for example, 0.975 for 97.5%).
Result and graph
Enter numbers in the fields on the left and press "Calculate". The result and a graph will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formula and graph
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter a value \(x\), the mean \(\mu\) and the standard deviation \(\sigma\), and you get the z-score (standard score) and the probability of a value below it in one go
- Instead of looking up a z table (standard normal table), convert between z-scores and probabilities in either direction (z-score to probability, or probability back to z-score)
- Also find the probability between two z-scores (for example, between −1.96 and 1.96) on the spot
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
This calculator assumes the data follows a normal distribution (a symmetric bell shape around the mean). It does not work for data whose distribution is strongly skewed.
What is this calculation used for?
Knowing where you stand - test scores and percentiles (school and exams)
On a test with a mean of 65 and a standard deviation of 5, a score of 75 is \(z = 2\). If the scores follow a normal distribution, that puts you in about the top 2.3% (\(1 - \Phi(2) \approx 0.0228\)), which is roughly the 97.7th percentile. IQ scores are built directly from z-scores (IQ \(= 100 + 15 \times z\)), so an IQ of 130 is the same position.
This is the most familiar use of z-scores: turning a raw score into a percentile, and reading how big a gap of a few points really is, even between tests with different maximum scores and averages.
Comparing scores from different subjects or tests on the same ruler
"80 in English (mean 70, standard deviation 10)" and "65 in math (mean 50, standard deviation 15)" look like English is higher on raw scores, but as z-scores both are \(z = 1\), the same position.
Even when units, maximum scores or grading strictness differ, z-scores let you compare fairly by "how far above the group" a score is. The same idea is used to scale scores across different test forms and to normalize ratings in performance reviews.
Reading a child's growth chart (parenting and health)
Pediatric growth charts, such as the WHO charts used for young children, rate height and weight with z-scores (also called SD scores), which tell how many standard deviations (SD) a child is from the average.
About 2.3% of children fall outside each of the ±2 SD lines (\(z = \pm 2\)). These lines, which match the 2.3rd and 97.7th percentiles, are one of the guides doctors use to decide whether to keep an eye on growth (children grow at different rates, and the doctor makes the call after following the trend). Knowing what the "−2 SD" line stands for makes checkup explanations easier to follow.
Quality control in a factory - how many products miss the spec (manufacturing)
The sizes and weights of products often vary close to a normal distribution, so turning the upper and lower spec limits into z-scores lets you estimate the defect rate. If the limits sit at the mean \(\pm 3\sigma\), about 0.3% fall outside (\(2 \times (1 - \Phi(3)) \approx 0.0027\)).
The name of the quality improvement method "Six Sigma" also comes from this z-score idea (how many σ away).
Risk management in finance - "a 99% chance the loss stays within this" (banking and investing)
Banks and asset managers measure the risk of losses on their holdings with a measure called value at risk (VaR). If returns are assumed to follow a normal distribution, the boundary where "the loss stays within this range with 99% probability" is the z-score \(\Phi^{-1}(0.99) \approx 2.33\), that is, 2.33 standard deviations from the mean.
This "probability to z-score" conversion is at the base of risk calculations at financial institutions around the world.
Formula and graph
Definition of the z-score (how many standard deviations from the mean)
Graph
Standard notation
(the usual math form)
\(z\)
\(=\)
\((\)
\(x\)
\(-\)
\(\mu\)
\()\)
\(\div\)
\(\sigma\)
In words
(symbols replaced with words)
④
\(z\): z-score
\(=\)
\((\)
①
\(x\): value
\(-\)
②
\(\mu\): mean
\()\)
\(\div\)
③
\(\sigma\): standard deviation
The formula in words
①
Take the
\(x\): value
②
subtract the
\(\mu\): mean
③
divide by the
\(\sigma\): standard deviation
④
and you get the
\(z\): z-score
Quick example
For someone who scored 75 on a test with a mean of 65 and a standard deviation of 5, the z-score is
\(z\): z-score
\(=\)
\((\)
score (75)
\(-\)
mean (65)
\()\)
\(\div\)
standard deviation (5)
\((75 - 65) \div 5 = 10 \div 5 = 2\)
Key idea
The z-score (standard score) tells how many standard deviations a value is from the mean. A positive z-score is above the mean, a negative one is below it, and 0 is exactly the mean.
Even data with different units or different maximum scores can be compared on the same ruler once turned into z-scores. IQ scores use this idea: IQ \(= 100 + 15 \times z\) (the \(z = 2\) in the example matches an IQ of 130).
From a z-score to "the probability of a value below \(Z\)"
Graph
Standard notation
(the usual math form)
\(P(x<Z)\)
\(=\)
\(\Phi(Z)\)
In words
(symbols replaced with words)
②
probability below \(Z\)
\(=\)
①
area to the left, \(\Phi(Z)\)
The formula in words
①
Put the z-score \(Z\) into the cumulative distribution function \(\Phi\) (phi) to get the
value \(\Phi(Z)\)
(\(\Phi(Z)\) is the area under the standard normal curve to the left of \(Z\))
②
and that is the
probability of a value below \(Z\), \(P(x<Z)\)
Quick example
When the z-score is 2, the probability of a value below 2 is
probability of a value below 2
\(=\)
value of the cumulative distribution function, \(\Phi(2)\)
\(\Phi(2) \approx 0.97725\ \ (97.7\%)\)
Key idea
Values of \(\Phi\) used to be looked up in the "standard normal table (z table)" at the back of a textbook. This calculator works as a replacement for that z table.
The opposite side, "the probability of a value above \(Z\)", is the total probability 1 minus it: \(1 - \Phi(Z)\). To go back from a probability to a z-score, use the inverse function \(\Phi^{-1}\) (example: \(\Phi^{-1}(0.975) \approx 1.96\)). This calculator converts in both directions.
Probability between \(-Z\) and \(Z\) (both sides together)
Graph
Standard notation
(the usual math form)
\(P(-Z<x<Z)\)
\(=\)
\(2\)
\(\times\)
\(\Phi(Z)\)
\(-\)
\(1\)
In words
(symbols replaced with words)
④
probability between \(-Z\) and \(Z\)
\(=\)
②
times 2 (both sides)
\(\times\)
①
area to the left, \(\Phi(Z)\)
\(-\)
③
total probability \(1\)
The formula in words
①
Take the
value \(\Phi(Z)\) (the area to the left of \(Z\))
②
multiply it
by 2 (the curve is symmetric, so two of them)
③
subtract the
total probability \(1\)
④
and you get the
probability between \(-Z\) and \(Z\)
Quick example
When the z-score is 1.96 (\(\Phi(1.96) \approx 0.975\)), the probability between −1.96 and 1.96 is
probability between −1.96 and 1.96
\(=\)
times 2
\(\times\)
value of the cumulative distribution function (0.975)
\(-\)
total probability (1)
\(2 \times 0.975 - 1 = 1.95 - 1 = 0.95\ \ (95\%)\)
Key idea
This formula works when \(Z\) is 0 or more. The standard normal distribution is symmetric around 0, so the area "from \(-Z\) to \(Z\)" is the area to the left of \(Z\), \(\Phi(Z)\), doubled, minus the one whole (1) that was counted twice.
The probability of landing outside (in the two tails) is the complement, \(1 - (2\Phi(Z) - 1)\). With \(Z = 1.96\) as in the example, 95% is inside and 5% is outside. This z-score is the cutoff for the "5% significance level" that comes up so often in statistical tests.
Probability between two z-scores \(Z_1\) and \(Z_2\)
Graph
Standard notation
(the usual math form)
\(P(Z_1<x<Z_2)\)
\(=\)
\(\Phi(Z_2)\)
\(-\)
\(\Phi(Z_1)\)
In words
(symbols replaced with words)
③
probability between \(Z_1\) and \(Z_2\)
\(=\)
①
area to the left, \(\Phi(Z_2)\)
\(-\)
②
area to the left, \(\Phi(Z_1)\)
The formula in words
①
Take the
area to the left of the upper bound \(Z_2\), \(\Phi(Z_2)\)
②
subtract the
area to the left of the lower bound \(Z_1\), \(\Phi(Z_1)\)
③
and you get the
probability between \(Z_1\) and \(Z_2\)
Quick example
When \(Z_1 = -1\) and \(Z_2 = 0\) (\(\Phi(0) = 0.5\), \(\Phi(-1) \approx 0.15866\)), the probability between −1 and 0 is
probability between −1 and 0
\(=\)
area to the left of the upper bound, \(\Phi(0)\)
\(-\)
area to the left of the lower bound, \(\Phi(-1)\)
\(0.5 - 0.15866 = 0.34134\ \ (34.1\%)\)
Key idea
All it does is "the area up to the larger one minus the area up to the smaller one", so it works for any \(Z_1\) and \(Z_2\) (even when both are negative).
Enter \(Z_1 = -1\) and \(Z_2 = 1\) and you get about 0.683 (68.3%). This lets you check the well-known rule of thumb for the normal distribution, "about 68% falls within one standard deviation of the mean" (the 68–95–99.7 rule).
The z-score is a common ruler for where a value sits in the data: how many standard deviations it is from the mean. Once values are z-scores, you can compare data with different units, and the cumulative distribution function \(\Phi\) turns a z-score into "the probability (area) up to that value".
Symbols and terms
Symbols
| \(z\) | zee (z-score) | A number that tells how many standard deviations a value is from the mean, found with \(z = (x - \mu) \div \sigma\). A positive value is above the mean, and a negative value is below it. On this page, a z-score that you specify, as in ② and ③, is written with a capital \(Z\). |
| \(x\) | ex | The value you measured, such as a test score or a height. It is the original value you want to turn into a z-score. |
| \(\mu\) | mu | The mean. It is where the center of the peak of the distribution sits. A Greek letter that corresponds to the English m. |
| \(\sigma\) | sigma | The standard deviation, the size of the spread of the data. The larger it is, the lower and wider the peak. A Greek letter that corresponds to the English s. |
| \(\Phi(Z)\) | phi of Z | The cumulative distribution function of the standard normal distribution (mean 0, standard deviation 1). It returns the probability of a value below \(Z\), which is the area under the curve to the left of \(Z\). |
| \(\Phi^{-1}\) | phi inverse (inverse of Φ) | The function that works the other way from \(\Phi\). It takes a probability (area) and returns the z-score that gives it. (Example: \(\Phi^{-1}(0.975) \approx 1.96\)) |
| \(Z_1,\ Z_2\) | Z sub 1, Z sub 2 | The two z-scores you enter in ③. \(Z_1\) is the left (smaller) bound and \(Z_2\) is the right (larger) bound. |
Terms
| z-score (standard score) | A number with no units that tells how many standard deviations a value is from the mean. It is also called a standard score or z-value. Its biggest strength is that it puts data with different units or different maximum scores on the same ruler. |
| standardization | Turning an original value into a z-score. Once standardized, data from any normal distribution can be handled with the single standard normal distribution with mean 0 and standard deviation 1. |
| standard normal distribution | The normal distribution with mean 0 and standard deviation 1. It is the distribution of z-scores and serves as the common ruler for probability calculations with any normal distribution. |
| normal distribution | A distribution spread symmetrically around the mean in a bell shape. Many real-world data, such as test scores, heights and the weights of products, come close to this shape. |
| cumulative distribution function | A function that returns the probability of getting that value or less. For the standard normal distribution it is written \(\Phi\) (phi). It gets its name because it adds up (accumulates) the area on the left side of the curve. |
| z table (standard normal table) | A table that lists z-scores and the matching values (areas) of \(\Phi\). It is printed at the back of statistics textbooks. Way ② on this page can be used instead of a z table. |
| T-score | A z-score rescaled so that the mean is 50 and one standard deviation is 10 (T-score \(= 50 + 10 \times z\)). It is used in some psychological and educational tests. IQ scores work the same way with a mean of 100 and 15 per standard deviation. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
| The mean (Grade 6) |
|
| Operations with negative numbers (Grade 7) |
|
| Data analysis and standard deviation (high school statistics) |
|
| The normal distribution and reading areas (high school statistics / AP Statistics) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the z-score
| Value x | 75 |
| Mean μ | 65 |
| Standard dev. σ | 5 |
| z-score | =(B1-B2)/B3 |
Table to find the probability of a value below the z-score, P(x<Z)
| z-score Z | 2 |
| P(x<Z) | =NORM.S.DIST(B1,TRUE) |
Table to find the probability between −Z and Z
| z-score Z | 1.96 |
| Φ(Z) | =NORM.S.DIST(B1,TRUE) |
| Probability between −Z and Z | =2*B2-1 |
Table to find the probability between two z-scores Z1 and Z2
| Left Z1 | -1 |
| Right Z2 | 0 |
| Φ(Z1) | =NORM.S.DIST(B1,TRUE) |
| Φ(Z2) | =NORM.S.DIST(B2,TRUE) |
| Probability between Z1 and Z2 | =B4-B3 |
After pasting the table, the numbers in column B work as input fields. "NORM.S.DIST(value, TRUE)" is the function for the cumulative distribution function \(\Phi\), and TRUE tells it to "return the cumulative value (the area to the left)".
In the first table, change the value, mean and standard deviation (B1 to B3) and B4 shows the z-score (with the example values, B4 is 2). The second table finds the probability (B2) from the z-score (B1); with the example value, B2 is about 0.97725 (97.7%).
In the third table, enter the z-score 1.96 and you get the probability between −1.96 and 1.96 (B3) of about 0.95 (95%). The fourth table is the probability between two z-scores; with the example values (−1 and 0), B5 is about 0.34134 (34.1%).
To go the other way, "from a probability to a z-score", use "=NORM.S.INV(probability)" (example: `=NORM.S.INV(0.975)` is about 1.95996).
In the first table, change the value, mean and standard deviation (B1 to B3) and B4 shows the z-score (with the example values, B4 is 2). The second table finds the probability (B2) from the z-score (B1); with the example value, B2 is about 0.97725 (97.7%).
In the third table, enter the z-score 1.96 and you get the probability between −1.96 and 1.96 (B3) of about 0.95 (95%). The fourth table is the probability between two z-scores; with the example values (−1 and 0), B5 is about 0.34134 (34.1%).
To go the other way, "from a probability to a z-score", use "=NORM.S.INV(probability)" (example: `=NORM.S.INV(0.975)` is about 1.95996).
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the z-score
| Value x | 75 |
| Mean μ | 65 |
| Standard dev. σ | 5 |
| z-score | =(B1-B2)/B3 |
Table to find the probability of a value below the z-score, P(x<Z)
| z-score Z | 2 |
| P(x<Z) | =NORMSDIST(B1) |
Table to find the probability between −Z and Z
| z-score Z | 1.96 |
| Φ(Z) | =NORMSDIST(B1) |
| Probability between −Z and Z | =2*B2-1 |
Table to find the probability between two z-scores Z1 and Z2
| Left Z1 | -1 |
| Right Z2 | 0 |
| Φ(Z1) | =NORMSDIST(B1) |
| Φ(Z2) | =NORMSDIST(B2) |
| Probability between Z1 and Z2 | =B4-B3 |
In Google Sheets, calculate the cumulative distribution function \(\Phi\) with "NORMSDIST(value)" (it works the same as "NORM.S.DIST(value, TRUE)" in Excel, but TRUE is not needed). Copy the whole table, paste it into cell A1, and change the numbers in column B.
The results are the same as in the Excel version: the first table gives a z-score of 2 from 75, 65 and 5, and the second gives a probability of about 0.97725 from a z-score of 2.
To find a z-score from a probability, use "=NORMSINV(probability)" (example: `=NORMSINV(0.975)` is about 1.95996).
The results are the same as in the Excel version: the first table gives a z-score of 2 from 75, 65 and 5, and the second gives a probability of about 0.97725 from a z-score of 2.
To find a z-score from a probability, use "=NORMSINV(probability)" (example: `=NORMSINV(0.975)` is about 1.95996).
How to calculate it in Python
import math
def phi(z):
# Cumulative distribution function of the standard normal distribution, Φ(z) (calculated with the error function erf)
return 0.5 * (1 + math.erf(z / math.sqrt(2)))
x = 75 # value
mean = 65 # mean
sd = 5 # standard deviation
z_score = (x - mean) / sd # z-score
p_less_than_z = phi(z_score) # probability of a value below the z-score, P(x<Z)
z = 1.96
p_between_minus_z_and_z = 2 * phi(z) - 1 # probability between -z and z
z1, z2 = -1, 0
p_between_z1_z2 = phi(z2) - phi(z1) # probability between z1 and z2
print(f"z-score: {z_score}")
print(f"P(x<Z): {p_less_than_z}")
print(f"Probability between -z and z: {p_between_minus_z_and_z}")
print(f"Probability between z1 and z2: {p_between_z1_z2}")
The standard library's math.erf (the error function) is all you need to calculate the cumulative distribution function \(\Phi\). If you need the other direction (probability to z-score), use NormalDist().inv_cdf() from the standard library's statistics module (example: `statistics.NormalDist().inv_cdf(0.975)` is about 1.95996).
How to write it in LaTeX and other math languages (copy and paste)
Definition of the z-score (how many standard deviations from the mean)
z = (x − μ) ÷ σ
z = \dfrac{x - \mu}{\sigma}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>z</mi>
<mo>=</mo>
<mfrac>
<mrow><mi>x</mi><mo>−</mo><mi>μ</mi></mrow>
<mi>σ</mi>
</mfrac>
</mrow>
</math>
z = (x - mu)/sigma
(x - mu)/sigma
z := (x - mu)/sigma;
z = (x - mu)/sigma;
z = (x-μ)/σ
From a z-score to "the probability of a value below \(Z\)"
P(x<Z) = Φ(Z)
P(x<Z) = \Phi(Z)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>P</mi><mo>(</mo><mi>x</mi><mo><</mo><mi>Z</mi><mo>)</mo>
<mo>=</mo>
<mi>Φ</mi><mo>(</mo><mi>Z</mi><mo>)</mo>
</mrow>
</math>
P(x<Z) = Phi(Z)
CDF[NormalDistribution[0, 1], Z]
with(Statistics): CDF(RandomVariable(Normal(0, 1)), Z);
p = normcdf(Z);
P(x<Z) = Φ(Z)
Probability between \(-Z\) and \(Z\) (both sides together)
P(−Z<x<Z) = 2Φ(Z) − 1
P(-Z<x<Z) = 2\Phi(Z) - 1
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>P</mi><mo>(</mo><mo>−</mo><mi>Z</mi><mo><</mo><mi>x</mi><mo><</mo><mi>Z</mi><mo>)</mo>
<mo>=</mo>
<mn>2</mn><mi>Φ</mi><mo>(</mo><mi>Z</mi><mo>)</mo><mo>−</mo><mn>1</mn>
</mrow>
</math>
P(-Z<x<Z) = 2 Phi(Z) - 1
2 CDF[NormalDistribution[0, 1], Z] - 1
with(Statistics): 2*CDF(RandomVariable(Normal(0, 1)), Z) - 1;
p = 2*normcdf(Z) - 1;
P(-Z<x<Z) = 2Φ(Z) - 1
Probability between two z-scores \(Z_1\) and \(Z_2\)
P(Z₁<x<Z₂) = Φ(Z₂) − Φ(Z₁)
P(Z_1<x<Z_2) = \Phi(Z_2) - \Phi(Z_1)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>P</mi><mo>(</mo><msub><mi>Z</mi><mn>1</mn></msub><mo><</mo><mi>x</mi><mo><</mo><msub><mi>Z</mi><mn>2</mn></msub><mo>)</mo>
<mo>=</mo>
<mi>Φ</mi><mo>(</mo><msub><mi>Z</mi><mn>2</mn></msub><mo>)</mo>
<mo>−</mo>
<mi>Φ</mi><mo>(</mo><msub><mi>Z</mi><mn>1</mn></msub><mo>)</mo>
</mrow>
</math>
P(Z_1<x<Z_2) = Phi(Z_2) - Phi(Z_1)
CDF[NormalDistribution[0, 1], Z2] - CDF[NormalDistribution[0, 1], Z1]
with(Statistics): CDF(RandomVariable(Normal(0, 1)), Z2) - CDF(RandomVariable(Normal(0, 1)), Z1);
p = normcdf(Z2) - normcdf(Z1);
P(Z1<x<Z2) = Φ(Z2) - Φ(Z1)
How to have ChatGPT do the calculation
You are a statistics calculation assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). Someone scored 75 points on a test with a mean of 65 points and a standard deviation of 5 points. 1. Find this person's z-score. 2. Find the probability of scoring below that value, P(x<Z) (use the cumulative distribution function Φ of the standard normal distribution; you may use Python's standard library, such as math.erf or statistics.NormalDist). 3. Also find the probability that a z-score falls between −1.96 and 1.96. Show the formulas you used and the numbers from the execution result.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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