Z-score and the standard normal distribution

A z-score is a standardized value that says how many standard deviations a value lies from the mean, z = (x − μ) / σ: positive above the mean, negative below it.

Last updated: 07 Oct, 2026 · SciPy 1.18

Empirical rule (68-95-99.7) talks in whole standard deviations. Most values fall between those steps, and the z-score says exactly where.

Measuring a value in standard deviations

The video's dataset has mean 4 and standard deviation 1, so the axis runs 3, 2, 1 to the left of the mean and 5, 6, 7 to the right. Where does 4.5 fall? 5 is one standard deviation to the right, so 4.5 is half of one: +0.5 standard deviations.

For 4.75 the steps are harder to count by eye, so the video uses the z-score: the distance from the mean divided by the standard deviation.

For 4.75, z = (4.75 − 4)/1 = 0.75: 4.75 lies 0.75 standard deviations to the right of the mean. The sign gives the side; positive means right. For 3.75, z = (3.75 − 4)/1 = −0.25: a quarter of a standard deviation to the left.

A z-score has no units. Whether x is in runs, rupees or kilograms, the units cancel in (x − μ)/σ, which is what lets you compare values from different scales.

With data instead of a known population, use the sample mean x̄ and standard deviation s: z = (x − x̄)/s. Tools differ in which s they use. scipy.stats.zscore divides by n by default (ddof=0), the population formula from Variance and standard deviation.

Converting a normal distribution to the standard normal

Apply the z-score to every point on the axis of the video's N(4, 1): 1 becomes (1 − 4)/1 = −3, 2 becomes −2, 3 becomes −1, 4 becomes 0, and 5, 6, 7 become 1, 2, 3. The curve stays the same; only the axis is relabelled. The result is the standard normal distribution, the normal distribution with mean 0 and standard deviation 1, written Z ~ N(0, 1).

Every normal distribution turns into this one curve, so one table of Φ answers questions about all of them. That table is the subject of Z-table and normal probabilities.

Z-scoring a dataset keeps its shape

Z-scoring data, rather than the axis of a known distribution, uses the data's own mean and standard deviation. The seven values 1, 2, 3, 4, 5, 6 and 7 have mean 4 and a population standard deviation of 2, so their z-scores are (x − 4)/2: −1.5, −1, −0.5, 0, 0.5, 1 and 1.5.

Z-scored values always have mean 0 and standard deviation 1. Their shape does not change: z = (x − μ)/σ only shifts and rescales, so evenly spaced values stay evenly spaced, and skewed data stays skewed by the same amount. Z-scored data follows the standard normal distribution only when the original data was normal.

Computing z-scores in Python

One value by hand

mu, sigma = 4, 1
z = (4.75 - mu) / sigma        # 0.75: three quarters of an SD above the mean

A whole array with scipy.stats.zscore

import numpy as np
from scipy.stats import zscore

x = np.arange(1, 8)            # 1, 2, ..., 7
zscore(x)                      # (x - x.mean()) / x.std(), ddof=0 by default

from scipy.stats import skew

for value in [4.5, 4.75, 3.75]:
    print(f"x = {value}: z = {(value - 4) / 1:+.2f}")         # mean 4, SD 1

print("mean and SD of 1..7:", x.mean(), x.std())
print("z-scores:", zscore(x))
print("mean and SD after:", zscore(x).mean(), zscore(x).std())

data = np.array([1, 2, 2, 3, 4, 5])                        # a right-skewed set
print("skewness before:", round(skew(data), 4), " after:", round(skew(zscore(data)), 4))

x = 4.5: z = +0.50
x = 4.75: z = +0.75
x = 3.75: z = -0.25
mean and SD of 1..7: 4.0 2.0
z-scores: [-1.5 -1.  -0.5  0.   0.5  1.   1.5]
mean and SD after: 0.0 1.0
skewness before: 0.3053  after: 0.3053

What the z-scores show

• 4.5, 4.75 and 3.75 give +0.50, +0.75 and −0.25, the distances the video reads off the curve.
• The values 1 to 7 have mean 4.0 and SD 2.0, so zscore returns −1.5 to 1.5 in steps of 0.5.
• After z-scoring, the mean is 0.0 and the SD is 1.0, whatever the data.
• The skewness of 1, 2, 2, 3, 4, 5 is 0.3053 before and after: z-scoring moved and squeezed the data but kept its lopsided shape.

Population z-score vs sample z-score

Where you use z-scores

• Comparing across scales. Two scores from different series or exams become comparable once each is measured in standard deviations of its own group; Z-table and normal probabilities does this for two cricket series.
• Outliers. |z| > 3 flags a value, as in Outlier detection with IQR and z-score.
• Feature scaling. Machine learning standardizes whole columns this way, in Standardization and normalization.
• Test statistics. The One-sample z-test applies the same idea to a sample mean.

Related

• Previous: Empirical rule (68-95-99.7)
• Next: Standardization and normalization
• See also: Variance and standard deviation
• Reference: scipy.stats.zscore