Standardizing a Normal Distribution

Statistics • Continuous Random Variables and the Normal Distribution

Written by STEM Calculators Team Published December 20, 2025 Updated February 24, 2026

Standardizing a Normal Distribution (z-scores)

Convert values from a normal distribution (μ, σ) into standard units (z), then compute probabilities as areas under the curve. This calculator also supports bulk standardization by pasting values or CSV data.

Single / Interval Probability

Mean (μ)

Standard deviation (σ)

Task

Left-tail probability p (0 < p < 1)

Example: μ=50, σ=10, left of x=55

Example: μ=25, σ=4, between x=18 and x=34

Example: Standard normal, right of z=2.32

Example: μ=40, σ=5, right of x=55

Continuous-variable reminders: The probability for a single exact value is 0, so P(X = x) = 0 and P(X ≤ x) = P(X < x). Probabilities are interpreted as areas under the curve and are always nonnegative.

Step-by-step

Move marker (x)

This updates the shaded area based on the current task.

Animation

Empirical rule (standard normal idea): Approximately 68.26% of the area lies within z ∈ [-1, 1], 95.44% within z ∈ [-2, 2], and 99.74% within z ∈ [-3, 3].

Bulk: Paste values / Paste CSV

Paste numbers (one per line) or paste CSV text (first numeric column will be used). You can also load a CSV file. Then generate a table of x, z, and Φ(z) (area to the left).

Paste data

Use the μ and σ from above Estimate μ and σ from pasted data

Sample σ (divide by n−1) Population σ (divide by n)

#	x	z	Φ(z) (left area)	Percentile

Note: Φ(z) is the cumulative probability to the left. Right-tail area is 1 − Φ(z).

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Standardizing a Normal Distribution

Normal distribution (key properties)

The total area under the curve is 1.
The curve is symmetric about the mean μ.
The tails extend indefinitely in both directions (they approach 0 but do not “end”).
A family of normal curves is determined by its parameters μ (center) and σ (spread).

Standard normal distribution

The standard normal distribution has μ = 0 and σ = 1.
Values on the horizontal axis are called z-values (z-scores, standard units).
If z is positive, the point lies to the right of the mean; if negative, to the left.
Areas under the curve are always positive and represent probabilities.

Standardizing (converting x to z)
To convert a value x from a normal distribution with mean μ and standard deviation σ into a z-score:

\[ z = \frac{x - \mu}{\sigma} \]

Interpretation: z tells how many standard deviations x is from the mean.

Probabilities as areas (continuous variable)

Probabilities correspond to areas under the curve.
For a continuous distribution, the probability of a single exact value is zero: P(X = a) = 0, so P(X ≤ a) = P(X < a).
If Φ(z) denotes the standard normal cumulative area to the left of z:
- Left tail: P(X ≤ x) = Φ(z)
- Right tail: P(X ≥ x) = 1 − Φ(z)
- Between a and b: P(a ≤ X ≤ b) = Φ(z_b) − Φ(z_a)
- Outside [a, b]: P(X ≤ a or X ≥ b) = 1 − (Φ(z_b) − Φ(z_a))

Empirical rule (useful reference)

Within 1 standard deviation: about 68.26% of the total area
Within 2 standard deviations: about 95.44%
Within 3 standard deviations: about 99.74%

Practical workflow

Compute z for each endpoint using:
\[ z = \frac{x - \mu}{\sigma} \]
Use Φ(z) as the “area to the left.”
Use subtraction for “between” areas and 1 − for right/outside areas.

Frequently Asked Questions

What does it mean to standardize a normal distribution?

Standardizing converts a value x from X ~ N(mu, sigma) into a z-score measured in standard deviations from the mean. After standardization, you interpret the value on the standard normal scale.

How do you calculate a z-score from x, mu, and sigma?

Use z = (x - mu) / sigma with sigma > 0. The result tells you how many standard deviations x is above (positive) or below (negative) the mean.

How do I standardize an interval from x1 to x2?

Compute z1 = (x1 - mu) / sigma and z2 = (x2 - mu) / sigma. The interval probability on X corresponds to the same bounds on the Z scale.

Why must sigma be greater than 0 when standardizing?

Sigma is a standard deviation and cannot be 0 or negative. If sigma were 0, the z-score formula would involve division by 0 and the distribution would not be a valid normal spread.

What does a negative z-score mean?

A negative z-score means the x value is below the mean mu. Its magnitude tells how many standard deviations below the mean the value lies.

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