Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for algebra 2 normal distribution and z-scores? Then this Algebra 2 lesson on Normal Curves, z-Scores and Study Design explains each idea step by step, with worked examples.
Quick answer: Standard deviation measures how far data typically fall from the mean. In a normal distribution, about 68% of values lie within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. A z-score, z = (x − μ)/σ, tells how many standard deviations a value is from the mean, so values on different scales can be compared.
Standards: S-ID.A.4, S-IC.A.1, S-IC.B.3, S-IC.B.4 · Level: Grade 11 Algebra 2
What do the mean and standard deviation tell you?
The mean μ is the balance point of the data. The standard deviation σ measures the typical distance from the mean. To compute it, square each deviation, average the squares, then take the square root.
Data: 6, 8, 10, 12, 14. Mean = 10. Deviations: −4, −2, 0, 2, 4, with squares 16, 4, 0, 4, 16. Average = 40 ÷ 5 = 8, so σ = √8 ≈ 2.83.
What is the 68-95-99.7 rule?
For a normal distribution (a symmetric bell curve), almost all values sit near the mean.
- About 68% of values lie within 1σ of the mean.
- About 95% lie within 2σ.
- About 99.7% lie within 3σ.
How do z-scores compare values?
z = (x − μ) ÷ σ
It counts how many standard deviations a value is above (+) or below (−) the mean.
- Compute the z-score of each value with its own mean and σ.
- Compare the z-scores, not the raw scores.
- Flag |z| > 2 as unusual.
How do surveys, experiments and observational studies differ?
A survey samples a population to estimate a parameter, and random selection reduces bias. An observational study records what happens without assigning anything. An experiment randomly assigns treatments, which is the only design that can support a cause-and-effect claim.
What is a margin of error?
Repeating a random sample many times by simulation shows how much a sample proportion varies. The middle 95% of the simulated results spans about ±2 standard deviations, and half that width is the margin of error.
Comparing raw scores from tests with different means and spreads. A higher score is not always the better performance.
Draw the bell curve and mark μ ± σ, 2σ and 3σ. The picture shows the percent you need.
- σ is the square root of the average squared deviation from the mean.
- Empirical Rule: 68%, 95% and 99.7% within 1, 2 and 3 standard deviations.
- z = (x − μ)/σ; |z| greater than 2 marks an unusual value.
- Only a randomized experiment can support a cause-and-effect conclusion.
- Simulating many random samples shows the margin of error: about twice the standard deviation of the sample proportions.
Normal Curves, z-Scores and Study Design: frequently asked questions
What does a z-score of −1.5 mean?
The value lies 1.5 standard deviations below the mean. In a normal distribution it is somewhat below average but not unusual, because only values beyond z = ±2 are rare.
Why divide by σ in a z-score?
Dividing by σ removes the units and the spread of the data. A z-score of 2 means the same position, two standard deviations above the mean, whether the data are test scores or heights.
When can I use the 68-95-99.7 rule?
Use it when the data distribution is roughly symmetric and bell-shaped. For skewed data the percentages can be far off, so check a histogram first.
What is the difference between a survey and an experiment?
A survey asks questions of a sample to learn about a population. An experiment assigns different treatments to subjects at random and compares results, which lets it show cause and effect.
How does simulation give a margin of error?
Simulate many random samples of the same size, record each sample proportion, and look at the spread. The interval containing the middle 95% of results has half-width equal to the margin of error.
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