Parameters of the Normal Distribution

Learning Objectives

Understand the notation and interpret the parameters of a normal distribution
Standardize a normally distributed random variable
Calculate the mean and standard deviation of a normal distribution

Why This Matters

Your SAT score is just a number until you know two things: the average (μ ≈ 1050) and the spread (σ ≈ 210). Those two parameters decide whether 1280 is impressive or ordinary. Medical licensing boards, actuarial exams, and every data-driven hiring assessment convert raw scores into z-scores for exactly this reason -- two parameters turn any scale into a universal one.

How to Use This Simulation

Select a preset distribution or drag the μ and σ sliders to shape the curve. Notice how μ controls position and σ controls spread.
Enter a raw value in the Z-Score Calculator to see the standardization formula with substituted values. Annotation lines appear on both charts.
Click Standardize to animate the left curve transforming into the standard normal. Watch it become identical to the right panel.
Read the Explanation Panel — it updates in real time as you interact and directly addresses a key misconception about what standardization does to data.

Notation note: This simulation writes N(μ, σ) — the second argument is the standard deviation. Some textbooks write N(μ, σ²) where the second argument is the variance. Mixing the two conventions corrupts every calculation downstream. The Quick Check below tests this distinction directly.

Preset:

SAT Preset loaded: μ = 1050, σ = 210. Sliders are disabled for this preset -- values are outside the generic slider range. The z-score calculator and Standardize button remain fully functional.

μ (mean):

0.0

Controls location. Drag left or right to shift the entire curve without changing its shape.

σ (std. dev.):

1.0

Controls spread. σ must be > 0. Narrowing the curve raises the peak density but area under the curve always equals 1.

Your Distribution X ~ N(0, 1)

Standard Normal Reference Z ~ N(0, 1)

Always shows N(0, 1) — the universal reference

Z-Score Calculator

Enter a value from the current distribution to see the standardization formula with substituted values. Annotation lines appear on both charts.

Value (x):

Notation

X ~ N(0, 1)

Distribution is fully described by μ and σ

Peak Density f(μ)

0.3989

Z-Score

What's Happening

You're looking at the standard normal distribution, written N(0, 1). The mean μ = 0 is the center of the curve — it tells you where the distribution is located. The standard deviation σ = 1 controls how spread out the values are. Drag the μ slider to move the entire curve left or right without changing its shape. Drag the σ slider to widen or narrow it. The standard normal is the reference distribution that makes z-scores possible. You'll use it constantly in the probability and inference simulations that follow.

Quick Check

A statistics textbook writes X ~ N(100, 225). Which of the following correctly identifies the parameters of this distribution?

Try This

Load the Wide Spread (σ = 3) preset. Before touching any slider, look at the curve and predict: if σ changes from 3 to 1, will the peak of the curve go up or down? Write your prediction and your reasoning in one sentence.

Now drag the σ slider from 3 to 1 and check your prediction. Write one sentence explaining why the peak moved in the direction it did, using the word "area."

Load the Adult Male Heights preset (μ = 70 inches, σ = 3 inches). A friend is 76 inches tall. Enter x = 76 in the Z-Score Calculator and note the z-score. Now calculate it by hand: z = (76 − 70) / 3. Do your answers match?

Next, switch to the IQ Scores preset (μ = 100, σ = 15). Enter x = 118 and compare the z-score to the height result. Both values are "above average," but which one represents a more unusual position in its population? Write one sentence explaining what the z-scores tell you that the raw numbers cannot.

You're comparing two scholarship applicants. Applicant A scored 1280 on the SAT (μ = 1050, σ = 210). Applicant B scored 28 on the ACT (μ = 21, σ = 5.2). Load the SAT preset and use the z-score calculator for x = 1280. Then mentally calculate the ACT z-score by hand: z = (28 − 21) / 5.2.

Compare the two z-scores. Determine which applicant performed better relative to their test's population. Write a two-sentence recommendation that names the stronger relative performance AND acknowledges one thing a z-score comparison cannot tell you about these two applicants.

Instructor Notes

Teaching Notes

This simulation's most powerful moment is the Standardize animation — especially with the SAT preset loaded. Students who watch N(1050, 210) animate into the same curve as N(0, 1) consistently report the "aha" that makes z-tables work. Do not narrate before they click; let the animation speak, then debrief.

The notation callout (N(μ, σ) vs N(μ, σ²)) surfaces a confusion that causes silent calculation errors throughout the inference unit. The Quick Check directly tests this. Consider running it in class before any computation involving normal distributions.

Common Student Errors

Reading N(100, 225) as μ = 100, σ = 225. The second argument in this notation is the variance; σ = √225 = 15.
Believing standardization "changes" the data. The z-score changes the scale, not the relative position. A value at the 84th percentile stays at the 84th percentile.
Confusing μ and σ roles: "μ controls shape, σ controls position." The opposite is true — μ is position, σ is shape/spread.
Assuming a taller curve has more total probability. Area is always 1; height is the density, not the probability.

Discussion Questions

If SAT scores and ACT scores are both normally distributed, why do colleges convert them to a common scale for comparison? What's the limitation of that approach?
Two students are told they scored "above average" on an exam. One has z = 0.2, the other has z = 1.8. What's the difference in their relative standing? Why doesn't "above average" tell you that?
A company standardizes employee performance scores across departments with different rating scales. What assumption does this standardization require about the shape of performance in each department?

Exam Connection

Exam items frequently give a normal distribution in N(μ, σ²) notation and ask students to compute a z-score. Students who don't recognize the variance convention apply σ = 225 instead of σ = 15, producing a wildly incorrect z-score. This simulation's Quick Check is a direct warmup for that item type. The standardization practice in the Stretch tier mirrors the step-by-step format of z-score computation items on most introductory stats exams.