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Z-score calculator

Find how far a value is from its mean

Enter a raw value, reference mean, and positive standard deviation to express the value in standard-deviation units.

Choose what to find, then enter the values

The standard deviation must be greater than zero.

Try an example

Result

The raw value is 2 standard deviations above the mean of its reference distribution.

z=78−704=2z=\frac{78-70}{4}=2
Conditions
  • The raw value and mean must use the same measurement units.
  • The standard deviation must be positive; division by zero is undefined.
  • Use the mean and standard deviation from the same reference group as the raw value.
  • A z-score states relative position, not the probability of observing the value unless an appropriate distribution model is also supplied.

Steps

  1. Find the signed deviation Subtract the reference mean from the raw value.x−μ=78−70=8x-\mu=78-70=8
  2. Standardize the deviation Divide by the positive standard deviation of 4.z=84=2z=\frac{8}{4}=2
  3. Interpret the sign A positive result places the value above the mean.z=2>0z=2>0
  4. Recover the raw score Substitute the z-score back into x equals mean plus z times standard deviation.x=70+(2)(4)=78x=70+(2)(4)=78
Independent check

The inverse calculation x = mean + z times standard deviation returns the original raw value 78 exactly.

Scope

What this raw-score-to-z-score covers

For a raw score of 78, mean 70, and standard deviation 4, the z-score is 2. The value is two standard deviations above its reference mean.

Raw value to z-score

Subtract the reference mean and divide by its positive standard deviation.

Examples: x=78,\ \mu=70,\ \sigma=4, z=2

Z-score to raw value

Rearrange the standardization formula to recover a value on the original scale.

Examples: z=1.5,\ \mu=50,\ \sigma=10, x=65

Signed position

Identify whether the value is below, at, or above the mean from the sign of z.

Examples: z = -2 is below, z = 0 is at the mean, z = 2 is above

Cross-scale comparison

Compare relative positions only when each raw score is standardized with the mean and standard deviation of its own relevant group.

Examples: z_1=(x_1-\mu_1)/\sigma_1, z_2=(x_2-\mu_2)/\sigma_2

How to use it

Enter enough information for one clear task

  1. 1
    Enter the reference values

    Use one raw value together with the mean and positive standard deviation of its reference distribution.

  2. 2
    Choose the direction

    Calculate z from a raw value, or calculate a raw value when z, the mean, and standard deviation are known.

  3. 3
    Read the sign and magnitude

    The sign gives direction from the mean. The absolute value gives the distance in standard-deviation units.

  4. 4
    Keep probability separate

    Do not turn a z-score into a percentile unless the problem also specifies a suitable probability distribution model.

Worked inputs

Examples to try

Use these examples to recognize the method, compare equivalent forms, and check your own work.

Two standard deviations above

Subtract 70, then divide by 4.

x=78, μ=70, σ=4x=78,\ \mu=70,\ \sigma=4

Expected result

z=2z=2

Two standard deviations below

The signed deviation is negative 6; divide by 3.

x=64, μ=70, σ=3x=64,\ \mu=70,\ \sigma=3

Expected result

z=−2z=-2

Recover a raw score

Use x equals mean plus z times standard deviation.

z=1.5, μ=50, σ=10z=1.5,\ \mu=50,\ \sigma=10

Expected result

x=50+(1.5)(10)=65x=50+(1.5)(10)=65

Fractional z-score

Divide the signed deviation 20 by 15 and reduce the fraction.

x=120, μ=100, σ=15x=120,\ \mu=100,\ \sigma=15

Expected result

z=2015=43≈1.333z=\frac{20}{15}=\frac{4}{3}\approx1.333

Value at the mean

The numerator is zero.

x=8, μ=8, σ=2x=8,\ \mu=8,\ \sigma=2

Expected result

z=0z=0

Negative raw values

The signed deviation is positive 6 even though both raw values are negative.

x=−4, μ=−10, σ=3x=-4,\ \mu=-10,\ \sigma=3

Expected result

z=−4−(−10)3=2z=\frac{-4-(-10)}{3}=2

Complete example

Standardize a score of 78

The raw value and mean are on the same scale, and the standard deviation is positive, so direct standardization is defined.

x=78,μ=70,σ=4x=78,\qquad \mu=70,\qquad \sigma=4
  1. 1
    Subtract the mean

    This preserves the direction of the value relative to the center.

    78−70=878-70=8
  2. 2
    Divide by standard deviation

    One standard-deviation unit is 4 raw-score units.

    8/4=28/4=2
  3. 3
    Interpret the standardized value

    The positive sign means above the mean, and the magnitude 2 means two standard deviations away.

    z=2z=2
  4. 4
    Reverse the transformation

    The inverse calculation should reproduce the original score.

    70+(2)(4)=7870+(2)(4)=78
z=2z=2

Verification: Substitution into x = mean + z times standard deviation gives 78. Units cancel in the ratio, so the z-score is dimensionless.

Avoidable errors

Common mistakes and how to fix them

Reversing the subtraction

Problem: z=(\mu-x)/\sigma

Why it matters: This reverses the sign and places above-mean values below the mean.

Better approach: Use raw value minus mean in the numerator.

Using a variance in the denominator

Problem: z=(x-\mu)/\sigma^2

Why it matters: A z-score measures distance in standard-deviation units, not squared units.

Better approach: Divide by the standard deviation, which is the square root of variance.

Mixing reference groups

Problem: Using a mean from one class and a standard deviation from another class.

Why it matters: The resulting scale does not describe either reference distribution coherently.

Better approach: Use a matched mean and standard deviation for the group that defines the comparison.

Calling every z-score a percentile

Problem: Interpreting z = 1 as a fixed percentile without a distribution model.

Why it matters: The standardization formula alone gives relative distance, not an area under a specified curve.

Better approach: Use a percentile conversion only when an appropriate distribution, often a normal model, is explicitly justified.

Trust the result for the right reasons

Checks, assumptions, and limits

How results are checked

  • The standard deviation is checked to be greater than zero.
  • The sign of z is checked against whether the raw value is below or above the mean.
  • The inverse formula x = mean + z times standard deviation is used to recover the input value.
  • Units are checked to cancel so the standardized result is dimensionless.

When to stop and revise the input

  • A z-score is undefined when the standard deviation is zero.
  • A z-score describes relative position and does not by itself establish normality.
  • Percentiles and tail probabilities require a stated distribution model and are outside this calculation's basic scope.
  • Comparisons are meaningful only when each value uses an appropriate and internally consistent reference group.

Common questions

Find how far a value is from its mean FAQ

What does a negative z-score mean?

It means the raw value is below the reference mean. A z-score of -1.5 is one and a half standard deviations below that mean.

What z-score is exactly at the mean?

Zero, because the raw value minus the mean is zero.

Can I compare z-scores from different tests?

You can compare relative positions when each score was standardized using the correct mean and standard deviation for its own relevant group. That comparison does not make the tests identical in content or meaning.

Does a z-score require a normal distribution?

The arithmetic standardization does not. A normal model becomes important when you use the z-score to calculate a normal percentile or tail probability.

Continue learning

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Created by Mathos AI. Methods, conditions, and checks are shown so you can review the mathematical reasoning.