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A normal distribution is a symmetric, bell-shaped probability distribution defined by two numbers, a mean μ and a standard deviation σ. The standard normal distribution is the single normal distribution with μ = 0 and σ = 1. Every normal distribution is the standard one stretched by σ and shifted by μ, which is why one table or one function answers questions about all of them.
Normal distribution vs standard normal distribution
NIST’s e-Handbook of Statistical Methods states it directly: “The case where mu = 0 and sigma = 1 is called the standard normal distribution,” and notes that because the general form of the probability functions can be expressed in terms of the standard distribution, its formulas are given for the standard form. In notation, X ~ N(μ, σ) for a general normal variable and Z ~ N(0, 1) for the standard one. Conversion between them is the z-score, covered in depth in the z-score guide.
| Property | Normal, N(μ, σ) | Standard normal, N(0, 1) |
|---|---|---|
| Mean, median and mode | All equal μ | All equal 0 |
| Spread | Standard deviation σ | Standard deviation 1 |
| Skewness | 0 | 0 |
| Kurtosis (NIST convention, not excess) | 3 | 3 |
| Units | Units of the measurement | Standard deviations from the mean |
Source: NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.6.1, Normal Distribution (checked 2026-10-29).
Standard normal distribution and normal distribution: how they relate
The standard normal is a member of the normal family, not a separate distribution. Standardizing a normal variable with z = (x − μ) / σ produces a variable that follows N(0, 1); the reverse, x = μ + zσ, rebuilds the original scale. OpenStax describes the standard normal distribution as “a normal distribution of standardized values called z-scores.”
Normal distribution and standard normal distribution formulas
The probability density function of the normal distribution, as given by NIST, is:
f(x) = exp( −(x − μ)² / (2σ²) ) / ( σ √(2π) )
- x: the value at which the density is evaluated.
- μ (mu): the location parameter, the mean, median and mode.
- σ (sigma): the scale parameter, the standard deviation; σ² is the variance.
- π and exp: the constant 3.14159… and the exponential function.
Setting μ = 0 and σ = 1 gives the standard normal density, φ(z) = exp(−z² / 2) / √(2π). Its peak, at z = 0, is 0.3989. The area under any normal curve is 1; the area to the left of a value is the cumulative probability, which for the standard normal is usually written Φ(z).
Worked example: converting between the two
Illustrative setup (hypothetical scores, not real data): a test score X is normally distributed with μ = 100 and σ = 15. What proportion of scores fall below 120?
- Standardize: z = (120 − 100) / 15 = 20 / 15 = 1.33 (1.3333 before rounding).
- Look up the cumulative probability: Φ(1.3333) = 0.9088.
- Interpret: about 90.9% of scores fall below 120; the remaining 1 − 0.9088 = 0.0912, about 9.1%, fall above it.
Going the other way, what score marks the 95th percentile? The standard normal value with Φ(z) = 0.95 is z = 1.6449. Then x = μ + zσ = 100 + 1.6449 × 15 = 100 + 24.67 = 124.67.
For table-reading technique, see the z-score guide and how to find a critical value. NIST’s percent point table lists p = 0.975 at Z = 1.960, the familiar two-sided 95% critical value.
Interpretation: reading the standard normal
A z value says how many standard deviations a value sits above or below the mean. The cumulative probability turns that into a percentile.
| z | Cumulative probability Φ(z) | Reading |
|---|---|---|
| −1.5 | 0.0668 | 1.5 SD below the mean; about 6.7% of values lie below |
| 0 | 0.5000 | At the mean; half of values lie below |
| 1.0 | 0.8413 | 1 SD above; about 84.1% of values lie below |
| 1.960 | 0.9750 | NIST critical value for p = 0.975 |
| 2.0 | 0.9772 | 2 SD above; about 97.7% of values lie below |
The empirical rule, also called the 68-95-99.7 rule, gives the shortcut: for normal data, about 68% of values lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3 (OpenStax, Introductory Statistics, via Statistics LibreTexts, checked 2026-10-29). Exactly, the three areas are 0.6827, 0.9545 and 0.9973.
Assumptions and common mistakes
- Standardizing does not make data normal. A z-score only rescales; a skewed variable stays skewed. Check shape separately, as in normality of distribution and the Shapiro-Wilk test in R.
- Using the standard normal table for raw values. Convert to z first, with the correct μ and σ.
- Confusing population and sample values. The formula uses the parameters μ and σ; see standard deviation for how a sample estimate differs.
- Assuming raw data must be normal for every test. Many tests rely on the sampling distribution of a mean, which the central limit theorem addresses.
- Applying the empirical rule to skewed data. Right-skewed positive quantities may fit a lognormal distribution better; see skewness.
How to report it (APA example)
Example sentence: “Scores were approximately normally distributed (M = 100.00, SD = 15.00). A score of 120 corresponds to z = 1.33, about the 91st percentile.” State the mean, standard deviation and the z value, and say how normality was assessed if you tested it.
How to calculate it in Excel, R or Python
Each snippet reproduces the worked example (μ = 100, σ = 15, x = 120).
Excel
=STANDARDIZE(120, 100, 15) ' z = 1.3333
=NORM.S.DIST(1.3333, TRUE) ' 0.9088
=NORM.DIST(120, 100, 15, TRUE) ' 0.9088, no standardizing needed
=NORM.INV(0.95, 100, 15) ' 124.67R
z <- (120 - 100) / 15 # 1.3333
pnorm(z) # 0.9088
pnorm(120, 100, 15) # same
qnorm(0.95, 100, 15) # 124.67Python
from statistics import NormalDist
z = (120 - 100) / 15
NormalDist().cdf(z) # 0.9088
NormalDist(100, 15).cdf(120) # same
NormalDist(100, 15).inv_cdf(0.95) # 124.67FAQ
Is the standard normal distribution a normal distribution?
Yes. It is the normal distribution with mean 0 and standard deviation 1.
Why convert a normal distribution to the standard normal?
One set of probabilities, such as a printed table or a software function, then serves every normal distribution regardless of its units.
What is the difference between a normal distribution and a standard normal distribution?
A normal distribution can have any mean and standard deviation; the standard normal fixes them at 0 and 1 and measures values in standard deviations.
Does a z-score require normally distributed data?
A z-score can be computed for any data. Reading it as a percentile through the standard normal table assumes the data are approximately normal.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.1 Normal Distribution: https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm (checked 2026-10-29).
- NIST/SEMATECH e-Handbook, 1.3.6.7.1 Cumulative distribution function of the standard normal distribution: https://www.itl.nist.gov/div898/handbook/eda/section3/eda3671.htm (checked 2026-10-29).
- OpenStax, Introductory Statistics, The Standard Normal Distribution (CC BY 4.0), as hosted by Statistics LibreTexts: https://stats.libretexts.org/Courses/Marian_University/Applied_Statistics_for_Social_Science_(19-20)/03%3A_The_Normal_Distribution/3.2%3A_The_Standard_Normal_Distribution (checked 2026-10-29).
