Probability
Inverse Normal Distribution Calculator
Convert a tail probability into a z-score and then into a raw score on the mean and standard deviation you supply.
Worksheet
Press Calculate to see the result.
Convert a tail probability into a z-score and then into a raw score on the mean and standard deviation you supply.
What this calculator answers
Sometimes the probability is the input and the score is the unknown. A cutoff, a growth chart percentile, or a specification limit often arrives as a share of the curve, and the decision needs the matching measurement. This page converts that share into a lower-tail probability, finds the standard normal z, and shifts it onto your mean and standard deviation. The headline is the raw score, not the z. The page keeps the definition fixed while the ticket updates, so a hand check can match every intermediate term. The worksheet asks for Probability, Which tail, Mean, Standard deviation, and it refuses a blank that the formula cannot interpret. Treat the loaded example as a rehearsal for the live numbers you will type next, not as a claim about your own data.
The selected tail is converted to a lower-tail probability. The inverse standard normal supplies z, and the raw score is mean plus z times the standard deviation. A percentile-to-z tool would stop at z. This page does not. 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
Inverse normal, returned on the raw scale uses this relationship.
p_lower = p (lower tail) or 1 − p (upper tail), z = Φ⁻¹(p_lower), x = mean + z × sd
- p. The tail probability you typed, between 0 and 1.
- p_lower. The probability to the left of the cutoff.
- z. The standard normal quantile of p_lower.
- x. The cutoff on the original measurement scale.
The standard normal inverse answers which z leaves probability p_lower to its left. Multiplying by the standard deviation stretches that distance, and adding the mean slides it to the center of your scale. Upper-tail input is flipped to a lower tail first so one inverse routine covers both questions. 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 entered probability is 0.95 and the tail choice is lower. The selected tail is lower, so the probability 0.95 is already a lower-tail probability. The inverse standard normal of 0.95 is z = 1.6449.
The mean is 100 and the standard deviation is 15. x = 100 + (1.6449) × 15 = 124.6728. The headline is 124.6728 on the original scale, not the z-score 1.6449.
A percentile-to-z calculation would have stopped at 1.6449. This page multiplies by the standard deviation and adds the mean. The probability was kept as a proportion. A percent such as 95 would have been rejected because it is not between 0 and 1.
| Raw score x | 124.6728 |
|---|---|
| z-score | 1.6449 |
| Lower-tail probability | 0.95 |
| Tail entered | Lower |
A second situation
The lower 5 percent point on the same scale
The entered probability is 0.05 and the tail choice is lower. The selected tail is lower, so the probability 0.05 is already a lower-tail probability. The inverse standard normal of 0.05 is z = −1.6449.
The mean is 100 and the standard deviation is 15. x = 100 + (−1.6449) × 15 = 75.3272. The headline is 75.3272 on the original scale, not the z-score −1.6449.
A percentile-to-z calculation would have stopped at −1.6449. This page multiplies by the standard deviation and adds the mean. The probability was kept as a proportion. A percent such as 95 would have been rejected because it is not between 0 and 1.
| Raw score x | 75.3272 |
|---|---|
| z-score | −1.6449 |
| Lower-tail probability | 0.05 |
| Tail entered | Lower |
A lower-tail probability of 0.05 sits below the mean, so z is negative and the raw score drops below 100. The arithmetic is the same formula with a smaller lower-tail probability, which is why the headline moves while the mean and standard deviation stay put. 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
The lower-tail probability moved from 0.95 to 0.975. The headline result went from 124.6728 to 129.3995.
The nudged inputs
The entered probability is 0.975 and the tail choice is lower. The selected tail is lower, so the probability 0.975 is already a lower-tail probability. The inverse standard normal of 0.975 is z = 1.96.
The mean is 100 and the standard deviation is 15. x = 100 + (1.96) × 15 = 129.3995. The headline is 129.3995 on the original scale, not the z-score 1.96.
A percentile-to-z calculation would have stopped at 1.96. This page multiplies by the standard deviation and adds the mean. The probability was kept as a proportion. A percent such as 95 would have been rejected because it is not between 0 and 1.
| Raw score x | 129.3995 |
|---|---|
| z-score | 1.96 |
| Lower-tail probability | 0.975 |
| Tail entered | Lower |
How to read the result
Read the headline as a measurement in the units of the mean, such as points or centimeters. Read z only as the intermediate distance. A probability of 0.95 in the lower tail is the 95th percentile of this normal model. 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
- The variable is continuous and normal with the mean and standard deviation you typed.
- The probability is a tail area, not a density height.
- A lower-tail choice means the area to the left, and an upper-tail choice means the area to the right.
- The standard deviation is the scale of a single observation, not a standard error, unless that is truly the model.
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
- Stopping at z and treating it as the raw score. This page exists to finish the conversion.
- Entering 95 instead of 0.95. The box takes a proportion.
- Choosing the upper tail when the probability was already a lower-tail percentile, which flips the score to the other side of the mean.
- Using 0 or 1. The inverse normal goes to infinity there and is not returned.
Where a different calculator is the honest choice
Use the normal distribution calculator when you already have the raw score and want the probability. Use the z-score probability calculator when the work should stay on the z scale. A percentile-to-z page is the wrong stop if the decision needs a number in the original units. 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, Z-Score Probability Calculator, Mean Confidence Interval Calculator.
Check yourself
What raw score leaves lower-tail probability 0.95 on a normal curve with mean 100 and standard deviation 15?
The lower-tail probability is already 0.95, z is 1.6449, and the raw score is 100 + 1.6449 × 15 = 124.6728.
Using the result in a sentence
Cut scores, growth-chart percentiles, and specification limits are the usual reasons to start from a probability and finish on the raw scale. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
The score is only as trustworthy as the normal model. A skewed measurement can put the true 95th percentile somewhere else even when the mean and standard deviation are right. 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 Inverse Normal Distribution Calculator commit to?
This page uses Inverse normal, returned on the raw scale: p_lower = p (lower tail) or 1 − p (upper tail), z = Φ⁻¹(p_lower), x = mean + z × sd. The standard normal inverse answers which z leaves probability p_lower to its left. Multiplying by the standard deviation stretches that distance, and adding the mean slides it to the center of your scale. Upper-tail input is flipped to a lower tail first so one inverse routine covers both questions. 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 Probability, Which tail, Mean, Standard deviation, 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 raw score x is 124.6728. Read the headline as a measurement in the units of the mean, such as points or centimeters. Read z only as the intermediate distance. A probability of 0.95 in the lower tail is the 95th percentile of this normal model. 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?
Stopping at z and treating it as the raw score. This page exists to finish the conversion. The score is only as trustworthy as the normal model. A skewed measurement can put the true 95th percentile somewhere else even when the mean and standard deviation are right. 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 already have the raw score and want the probability. Use the z-score probability calculator when the work should stay on the z scale. A percentile-to-z page is the wrong stop if the decision needs a number in the original units. 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 lower-tail probability is already 0.95, z is 1.6449, and the raw score is 100 + 1.6449 × 15 = 124.6728. 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.