Keep the value, mean, and positive standard deviation from one reference group, then choose the population, sample, inverse, comparison, or normal-model relation that matches the task.
The population form is z = (x minus mu) divided by sigma, and the sample-referenced form uses x bar and s. Reverse the population relation with x = mu + z sigma. Compare unlike scales only after standardizing each value with its own reference group, and convert z to cumulative probability only when a normal model is justified.
Express a raw value as a signed number of standard deviations from its reference mean.
z=σx−μ
Population z-scoreSigma is the population standard deviation and is greater than zero. x and mu use the same units.
Use a population z-score to measure how many population standard deviations a value lies above or below the mean.
z=sx−xˉ
Sample-standardized scores is greater than zero. x, x bar, and s refer to the same sample and use matched units.
Use this descriptive standardization to locate a value relative to the mean and standard deviation of the observed sample.
Reverse and compare
Recover a raw value or compare measurements recorded on different scales.
x=μ+zσ
Recover a raw valueMu and sigma are the same reference values used to define z. Sigma is greater than zero.
Use this rearrangement to recover a raw value from a population z-score and its original reference distribution.
z1>z2⟺σ1x1−μ1>σ2x2−μ2
Compare values from different scalesEach standard deviation is greater than zero. Each value is matched to its own relevant reference mean and standard deviation.
Use matched z-scores to compare relative positions from different populations or measurement scales.
Normal-model probability
Connect a standardized score to cumulative probability only under a justified normal model.
P(X≤x)=Φ(σx−μ)=Φ(z)
Normal cumulative probability from a z-scoreX follows a normal distribution with mean mu and positive standard deviation sigma. Phi is the cumulative distribution function of the standard normal distribution.
Use this relation to convert a z-score to a lower-tail probability when a normal distribution model is justified.
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Notation
Symbols used on this sheet
Interpret each symbol with the conditions attached to its formula.
Symbol
Meaning
z
Signed distance from the reference mean in reference-standard-deviation units.
x
The raw observed value in its original measurement units.
μ
The population mean used as the reference center.
σ
The positive population standard deviation used as the reference scale.
xˉ
A sample mean when standardization is explicitly relative to a sample.
s
A positive sample standard deviation when the reference is a sample.
Use the reference
Short applications
These examples show when to choose a formula; detailed instruction belongs in the linked guide or calculator.
Standardize above the mean
Subtract the matching mean and express the eight-unit deviation in four-unit SDs.
x=78,μ=70,σ=4
z=(78−70)/4=2
Condition check: The inverse relation 70 + 2(4) recovers 78.
Recover a raw score
Convert the negative standardized distance back to raw units before adding the mean.
Raw differences are not comparable until each uses its own center and spread.
A:(82,70,6),B:(640,600,25)
zA=2,zB=1.6
Condition check: Reverse-standardizing both z-scores returns their original raw values.
Use conditions, not memory alone
Common formula confusions
Reversing the subtraction
Using mean minus x and flipping the interpretation sign.
Fix: Use x minus mean, then predict the sign from whether x is above or below the mean.
Mixing reference groups
Using a value from one group with the mean or SD of another.
Fix: Keep x, center, and spread tied to one meaningful reference group.
Treating z as a percentile
Assigning a normal-table percentile without a justified normal model.
Fix: Interpret z as standardized distance first; state any distribution model separately.
Allowing zero or negative SD
Dividing by zero or reporting a negative standard deviation while rearranging.
Fix: Standard deviation must be positive for z to be defined; reject inconsistent sign information.
Common questions
Z-score formulas in both directions FAQ
What is the z-score formula?
For a population reference, z equals x minus the population mean, divided by the positive population standard deviation.
How do I find x from a z-score?
Multiply z by the same reference standard deviation and add the matching mean: x = mean + z times standard deviation.
What does z = 0 mean?
The raw value equals its reference mean. It does not mean the raw value itself is zero.
Can I use a z-score to compare different scales?
Yes, when each raw value is standardized with the mean and positive standard deviation of its own relevant reference group. The resulting unitless z-scores can then be compared.
Sources and curriculum alignment
This page follows standard introductory mathematics notation and learning sequences. Use these references to continue with a complete course treatment.
