Math · Introductory statistics · Concept
z-scores and the normal distribution
A z-score measures how many standard deviations a value lies above or below the mean. Standardizing puts different scales on one footing, and for data that follow a normal distribution it turns any value into a probability: the share of the distribution below or above it.
Standardizing
Subtract the mean and divide by the standard deviation. A positive z lies above the mean, a negative z below it, and z = 1.5 means one and a half standard deviations away. The units cancel.
The normal distribution
A normal distribution is a symmetric, bell-shaped curve set by its mean μ and standard deviation σ. The standard normal distribution has μ = 0 and σ = 1, and any normal variable becomes standard normal when standardized.
The 68–95–99.7 rule
For normal data, about 68% of values lie within one standard deviation of the mean, about 95% within two and about 99.7% within three.
| Within | Share |
|---|---|
| 1 standard deviation | 68.27% |
| 2 standard deviations | 95.45% |
| 3 standard deviations | 99.73% |
From z to a probability
The cumulative probability Φ(z) is the area under the standard normal curve to the left of z. The area to the right is 1 − Φ(z), and the area between two values is the difference of their Φ values.
From a percentile to a value
Run the table backward: find the z whose cumulative probability is the percentile you want, then convert back to the original scale.
When the normal model applies
The probabilities are only as good as the normal model. Check that the data are roughly symmetric and single-peaked, without heavy tails, before using them; standardizing alone does not make data normal.
Common mistakes
- Dividing by the variance instead of the standard deviation.
- Reading Φ(z) as the area to the right of z: the table gives the area to the left.
- Using normal probabilities for data that are clearly skewed.
- Dropping the sign of z: a value below the mean has a negative z-score.
Key terms
- z-score
- How many standard deviations a value is from the mean: z = (x − μ)/σ. A z-score of 2 is 2 standard deviations above the mean; standardizing doesn’t make data normal.
- Normal distribution
- The symmetric, bell-shaped distribution set by its mean and standard deviation. Real data can look roughly symmetric without actually being normal.
- Standard normal distribution
- The normal distribution with mean 0 and standard deviation 1, used with z-scores to find probabilities when a normal model fits.
- Empirical rule
- For a normal distribution, about 68% of values lie within one standard deviation of the mean, about 95% within two and about 99.7% within three. It is a quick check, not a substitute for exact normal probabilities.
- Percentile
- The value below which a stated percentage of the data or of a distribution falls. The 25th, 50th and 75th percentiles are the first quartile, the median and the third quartile.
- Standard deviation
- How spread out the values are around the mean, in the data’s own units: the square root of the variance. It describes individual values, not how precise the mean is.
Work through an example
Adult heights in a population are approximately normal with mean 170 cm and standard deviation 8 cm. What fraction are taller than 186 cm?
Find a normal probability with a z-score →Find a percentile of a normal distribution →
Sources and scope
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Open the example inputs, change a value and keep a useful result on your board.
Find the tail area in Statistics Check the arithmetic in Math Open worked example on a board z-score in Math ReferenceYour existing work stays on this device. Examples open as editable copies.