The Normal Distribution and Z-Scores
Understand why the bell curve appears everywhere in statistics, how to use the 68-95-99.7 rule, and how Z-scores let you compare observations on any scale.
You'll learn
How to read z-scores and probability areas under the normal curve.
Use this when
You need to interpret standardized scores or understand probability regions.
What Is a Normal Distribution?
The normal distribution is a symmetric, bell-shaped probability curve fully described by two parameters: the mean (μ) and standard deviation (σ). It appears throughout nature and statistics — heights, blood pressure, measurement errors, and standardized test scores all approximate it.
📖 The 68-95-99.7 Rule
In any normal distribution: 68% of data falls within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. This lets you quickly judge whether an observation is "unusual."
Z-Scores: Standardized Distance
A Z-score measures how many standard deviations an observation is from the mean. This standardization lets you compare observations from completely different scales.
Z = (X − μ) / σ
- ●Z = 0 → exactly at the mean
- ●Z = 1 → one standard deviation above the mean
- ●Z = −2 → two standard deviations below the mean
- ●|Z| > 2 → unusual (outside 95% of data)
- ●|Z| > 3 → very unusual (outside 99.7% of data)
Z-scores are the foundation of hypothesis testing. A Z-score of 1.96 corresponds to the boundary of the outer 5% of the standard normal — the p = 0.05 threshold for a two-tailed test.
Why Normal? The Central Limit Theorem
The Central Limit Theorem (CLT) states that the sampling distribution of the mean becomes approximately normal as sample size increases — regardless of the original distribution. This is why so many statistical methods assume normality: they apply to means, not individual observations.
| Sample Size (n) | Sampling Distribution Shape |
|---|---|
| n = 5 | Follows the original distribution shape closely |
| n = 15 | Becomes more symmetric and bell-shaped |
| n = 30 | Approximately normal for most distributions |
| n ≥ 30 | Normal approximation is reliable (CLT holds) |
⚠️ Normality assumption applies to the mean, not raw data
Many tests assume the sampling distribution of the mean is normal — not that your raw data must be normal. With n ≥ 30, the CLT gives you normality of the mean even if individual observations are skewed.
Practice with your own dataset
Check if your continuous variable is approximately normally distributed.
- 1.Upload a dataset with a continuous variable (e.g., age, BMI)
- 2.Select the variable and open the Distribution panel
- 3.View the histogram overlaid with a normal curve
- 4.Check the Shapiro-Wilk p-value — p > 0.05 supports normality
Trusted sources behind this lesson
Further reading
Read next
Statistical Power and Sample Size
Understand the relationship between α, β, power, and effect size through interactive visualizations. Learn why underpowered studies fail to find real effects.