Quick Answer

A value of 85 with a mean of 100 and standard deviation of 15 produces a z-score of -1.0, placing it at the 15.87th percentile with a cumulative probability of 0.1587.

Observed Value (x)

Population Mean (μ)

Standard Deviation (σ)

Common Examples

Input	Result
x = 85, mean = 100, SD = 15	Z = -1.0, Percentile: 15.87%
x = 115, mean = 100, SD = 15	Z = 1.0, Percentile: 84.13%
x = 130, mean = 100, SD = 15	Z = 2.0, Percentile: 97.72%
x = 70, mean = 100, SD = 15	Z = -2.0, Percentile: 2.28%
x = 100, mean = 100, SD = 15	Z = 0.0, Percentile: 50.00%

How It Works

The z-score (also called the standard score) measures how many standard deviations a particular value lies from the mean of a distribution. The formula is:

z = (x - μ) / σ

Where:

• x = the observed value
• μ = the population mean
• σ = the population standard deviation

A z-score of 0 means the value equals the mean. A z-score of +1 means the value is one standard deviation above the mean. A z-score of -2 means the value is two standard deviations below the mean.

Cumulative Probability

The cumulative probability (also called the p-value or CDF value) represents the proportion of values in a standard normal distribution that fall at or below a given z-score. This calculator uses the error function approximation (Abramowitz and Stegun method) to compute the cumulative distribution function:

P(Z ≤ z) = 0.5 × (1 + erf(z / √2))

Percentile

The percentile is the cumulative probability expressed as a percentage. A percentile of 84.13% means that approximately 84.13% of all values in the distribution fall at or below the observed value.

Common Z-Score Reference Points

In a standard normal distribution, approximately 68% of values fall within one standard deviation of the mean (z between -1 and +1), approximately 95% fall within two standard deviations (z between -2 and +2), and approximately 99.7% fall within three standard deviations (z between -3 and +3). This pattern is known as the 68-95-99.7 rule.

Worked Example

Suppose a student scores 85 on a test where the class mean is 100 and the standard deviation is 15. The z-score is (85 - 100) / 15 = -15 / 15 = -1.0. Using the cumulative distribution function, the probability P(Z ≤ -1.0) = 0.1587. This means the student scored at the 15.87th percentile, meaning approximately 15.87% of the class scored at or below 85. Conversely, a score of 115 produces z = (115 - 100) / 15 = +1.0 with a cumulative probability of 0.8413, placing it at the 84.13th percentile.

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Frequently Asked Questions

What does a z-score tell you?

A z-score tells you how far a value is from the mean in terms of standard deviations. A z-score of +2 means the value is 2 standard deviations above the mean, while a z-score of -1.5 means it is 1.5 standard deviations below the mean. Z-scores allow comparison across different scales and distributions.

What is a good or bad z-score?

There is no universally good or bad z-score. The interpretation depends on context. In many statistical tests, z-scores beyond +1.96 or below -1.96 are considered statistically significant at the 95% confidence level. In standardized testing, z-scores near 0 indicate average performance, while scores above +1 or below -1 indicate above- or below-average performance.

Can z-scores be used for non-normal distributions?

The z-score formula works for any data set, but the cumulative probability and percentile interpretations assume a normal (bell-curve) distribution. For non-normal distributions, the z-score still measures distance from the mean in standard deviation units, but the percentile values may not be accurate.

What is the relationship between z-scores and percentiles?

The percentile is the cumulative probability of a z-score expressed as a percentage. For a standard normal distribution: z = 0 corresponds to the 50th percentile, z = +1 to the 84.13th percentile, z = +2 to the 97.72nd percentile, z = -1 to the 15.87th percentile, and z = -2 to the 2.28th percentile.

Why does the calculator show 'standard deviation cannot be zero'?

A standard deviation of zero means every value in the data set is identical, so there is no spread to measure. Dividing by zero is undefined in the z-score formula. If all values are the same, the concept of how far a value deviates from the mean does not apply.