Z-Score Calculator

What is a Z-Score?

A z-score (also called a standard score) measures how many standard deviations a data point is from the mean of a distribution. A positive z-score indicates the value is above the mean, while a negative z-score means it is below the mean. Z-scores are fundamental in statistics for comparing data across different scales, performing hypothesis testing, and determining statistical significance.

For example, a z-score of 1.96 means the data point is 1.96 standard deviations above the mean. In a standard normal distribution, approximately 95% of values fall between z = -1.96 and z = 1.96, making this a commonly used critical value for 95% confidence intervals.

The Z-Score Formula

The z-score formula converts a raw score into its standardized form:

• x is the raw value (data point)
• μ (mu) is the population mean
• σ (sigma) is the population standard deviation

Once you know the z-score, you can use the standard normal distribution table (or this calculator) to find the probability associated with that score. The cumulative distribution function (CDF) gives the probability that a standard normal random variable Z is less than or equal to a given z-score.

Worked Example

Problem: A class has a mean test score of 80 with a standard deviation of 5. A student scored 90. What percentage of students scored below this student?

Step 1: Calculate the z-score: z = (90 - 80) / 5 = 2.0

Step 2: Look up z = 2.0 in the standard normal table (or use this calculator). The left-tail probability P(Z ≤ 2.0) = 0.9772.

Answer: Approximately 97.72% of students scored below 90. The student is in the 97.72nd percentile.

Understanding Tail Types

• Left Tail: P(Z ≤ z) — the probability that Z is less than or equal to your z-score. Used when testing if a value is significantly low.
• Right Tail: P(Z ≥ z) — the probability that Z is greater than or equal to your z-score. Used when testing if a value is significantly high.
• Two-Tailed: P(|Z| ≥ |z|) — the probability in both tails combined. Used in two-sided hypothesis tests where you care about deviation in either direction. For z = 1.96, the two-tailed p-value is approximately 0.05.
• Between: P(z1 ≤ Z ≤ z2) — the probability of Z falling between two z-scores. Useful for confidence intervals and range probabilities.

Frequently Asked Questions