Probability
Z-Score Probability Calculator
Find the standard-normal probability to the left of z, to the right of z, or between z and a second z score.
Worksheet
Press Calculate to see the result.
Find the standard-normal probability to the left of z, to the right of z, or between z and a second z score.
What this calculator answers
Once a measurement has already been standardized, the only remaining question is how much of the standard normal curve sits in a region. This page answers that question for a left tail, a right tail, or the band between two z values. It does not ask for a mean or a raw score, and it does not convert a raw score into z. That conversion belongs to the z-score calculator. Bring z here only after the standardization is finished.
Left of z is the standard normal cdf. Right of z is one minus that cdf. Between z and z2 is the absolute difference of the two cdf values, so the order of the endpoints does not matter. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
Standard normal tail and central probability uses this relationship.
Left: Φ(z). Right: 1 − Φ(z). Between: |Φ(z2) − Φ(z)|.
- z. A standard normal score, already standardized.
- z2. The second endpoint, used only for a between region.
- Φ. The standard normal cumulative distribution.
- region. Left of z, right of z, or between the two scores.
The standard normal cdf is defined as the probability to the left of its argument. The right tail is the complement because the total probability is 1. A band between two scores is the cdf at the right endpoint minus the cdf at the left endpoint, which is why the page sorts the two inputs before subtracting. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The form is prefilled with a small example so every arithmetic step can be checked by hand. Change one field after you finish the walkthrough and confirm the ticket moves the way the formula predicts.
The region is Left of 1.96. This page starts from z. It does not convert a raw score. That conversion is the z-score calculator. The standard normal cdf of 1.96 is 0.975.
The left-tail probability is the cdf itself: 0.975. The right tail would be 1 − 0.975 = 0.025, but that is not the selected region. z2 is ignored for a left-tail question. The loaded second z was not used.
No mean was requested, because z is already on the standard scale. The headline probability for this region is 0.975.
| Probability | 0.975 |
|---|---|
| Region | Left of 1.96 |
| Φ(z) | 0.975 |
| z | 1.96 |
A second situation
The central region between −1.96 and 1.96
The region is Between −1.96 and 1.96. This page starts from z. It does not convert a raw score. That conversion is the z-score calculator. The standard normal cdf of −1.96 is 0.025.
The second z is 1.96 and its cdf is 0.975. Ordered from low to high, the endpoints are −1.96 and 1.96. The between probability is 0.975 − 0.025 = 0.95.
The order you typed does not matter because the page sorts the endpoints before subtracting. The headline probability for this region is 0.95.
| Probability | 0.95 |
|---|---|
| Region | Between −1.96 and 1.96 |
| Φ(z) | 0.025 |
| z | −1.96 |
Switching from a left tail at 1.96 to the band between −1.96 and 1.96 changes the question from one tail to the central body. The same cdf routine answers both, but the subtraction uses two endpoints instead of one. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
z moved from 1.96 to 1 while the region stayed to the left of z. The headline result went from 0.975 to 0.8413.
The nudged inputs
The region is Left of 1. This page starts from z. It does not convert a raw score. That conversion is the z-score calculator. The standard normal cdf of 1 is 0.8413.
The left-tail probability is the cdf itself: 0.8413. The right tail would be 1 − 0.8413 = 0.1587, but that is not the selected region. z2 is ignored for a left-tail question. The loaded second z was not used.
No mean was requested, because z is already on the standard scale. The headline probability for this region is 0.8413.
| Probability | 0.8413 |
|---|---|
| Region | Left of 1 |
| Φ(z) | 0.8413 |
| z | 1 |
How to read the result
A result near 0.025 for the right tail of 1.96 is the familiar 2.5 percent tail. A result near 0.95 for the matching central band is that same curve with both tails removed. Neither number is a raw score. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Write the result with its unit, the inputs that produced it, and the name of the formula. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- z is already a standard score with mean 0 and standard deviation 1.
- The probability model is the continuous standard normal.
- For a between region, both endpoints refer to the same standard curve.
- No continuity correction is applied, because the input is already continuous z.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Typing a raw score such as 115 into the z box. This page will treat 115 as an enormous z.
- Forgetting the second z when the region is between two values.
- Subtracting the cdf values in the wrong order and reporting a negative probability. The page uses the absolute difference.
- Confusing this page with the z-score calculator, which is the one that converts a raw score.
Where a different calculator is the honest choice
Use the normal distribution calculator when you still have a raw score, a mean, and a standard deviation. Use the inverse normal calculator when the probability is known and the score is not. The z-score calculator, not this page, is where a raw score becomes z. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: Normal Distribution Calculator, Inverse Normal Distribution Calculator, Binomial Distribution Calculator.
Check yourself
What is the standard-normal probability to the left of z = 1.96?
The left-tail probability is the cdf of 1.96, which equals 0.975.
Using the result in a sentence
Homework on the standard normal table, checking a published z statistic's tail, and shading a central band are the jobs this page is built for. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
Tails this far out are sensitive to the normal assumption. A t distribution with small degrees of freedom puts more probability beyond 1.96 than the standard normal does. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the Z-Score Probability Calculator commit to?
This page uses Standard normal tail and central probability: Left: Φ(z). Right: 1 − Φ(z). Between: |Φ(z2) − Φ(z)|.. The standard normal cdf is defined as the probability to the left of its argument. The right tail is the complement because the total probability is 1. A band between two scores is the cdf at the right endpoint minus the cdf at the left endpoint, which is why the page sorts the two inputs before subtracting. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for Region, z, Second z, and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the probability is 0.975. A result near 0.025 for the right tail of 1.96 is the familiar 2.5 percent tail. A result near 0.95 for the matching central band is that same curve with both tails removed. Neither number is a raw score. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Typing a raw score such as 115 into the z box. This page will treat 115 as an enormous z. Tails this far out are sensitive to the normal assumption. A t distribution with small degrees of freedom puts more probability beyond 1.96 than the standard normal does. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
Use the normal distribution calculator when you still have a raw score, a mean, and a standard deviation. Use the inverse normal calculator when the probability is known and the score is not. The z-score calculator, not this page, is where a raw score becomes z. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
The left-tail probability is the cdf of 1.96, which equals 0.975. Name the formula, the sample size or the inputs, and the unit before you compare the figure with someone else's. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.