The normal distribution is a symmetric, bell-shaped model used to describe many measurements such as test scores, heights, measurement errors, and averages from repeated samples. Normal distribution calculations help you find the probability that a value falls in a certain range. These calculations are useful in science, engineering, medicine, business, and standardized testing.
The main idea is to turn any normal value into a standard score that can be compared using one common table or calculator.
Key Facts
- A normal distribution is written as X ~ N(μ, σ), where μ is the mean and σ is the standard deviation.
- The z-score formula is z = (x - μ) / σ.
- For the standard normal distribution, Z ~ N(0, 1).
- A left-tail probability is P(X < x) = P(Z < (x - μ) / σ).
- A between-values probability is P(a < X < b) = P(Z < (b - μ) / σ) - P(Z < (a - μ) / σ).
- A cutoff value can be found from x = μ + zσ after finding the needed z-score.
Vocabulary
- Normal distribution
- A symmetric bell-shaped probability distribution described by its mean and standard deviation.
- Mean
- The center value μ of a normal distribution, located at the peak of the bell curve.
- Standard deviation
- A measure σ of how spread out the values are around the mean.
- Z-score
- The number of standard deviations a data value is above or below the mean.
- Percentile
- A value below which a given percentage of the distribution lies.
Common Mistakes to Avoid
- Using x directly in the z-table, which is wrong because most z-tables use standard normal values only. Convert first with z = (x - μ) / σ.
- Forgetting to subtract when finding a middle area, which is wrong because P(a < X < b) is the area left of b minus the area left of a.
- Mixing up left-tail and right-tail probabilities, which gives the complement of the desired answer. For a right-tail area, use P(Z > z) = 1 - P(Z < z).
- Using the wrong sign for a z-score, which changes which side of the mean the value is on. Values below the mean must have negative z-scores.
Practice Questions
- 1 Scores on a test are normally distributed with μ = 75 and σ = 8. Find the z-score for a score of 87, then find P(X < 87) using a standard normal table or calculator.
- 2 A machine fills bottles with amounts normally distributed with μ = 500 mL and σ = 12 mL. Find P(488 < X < 518).
- 3 Two students have scores from different normal distributions. Student A scored 84 on a test with μ = 78 and σ = 6. Student B scored 90 on a test with μ = 82 and σ = 10. Explain which student performed better relative to their group.