How this calculator works
Standardisation measures distance from a mean in standard-deviation units. Reversing it adds that distance back to the mean. The algebra does not require a normal distribution. Normal probabilities require the additional normal model. All quantities on the original scale must use the same unit. Recovering a standard deviation requires a non-zero z-score and a positive result.
Formula
z = (x - μ)/σ; x = μ + zσ; μ = x - zσ; σ = (x - μ)/z.
Worked example
With mean 100, standard deviation 15 and z = 2, the raw value is 130. With mean 50, standard deviation 4 and z = -1.5, it is 44.
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Frequently asked questions
Does a z-score need normally distributed data?
The standardisation formula does not. Using a normal table to assign a percentile does require a suitable normal model.
Can the raw result be negative?
Yes, if the original measurement scale permits negative values.
Why reject a recovered negative standard deviation?
Standard deviation is a non-negative scale measure, and standardisation requires it to be strictly positive.