Z‑Score and Probability Calculator

Z‑Score and Probability Calculator

Quickly calculate Z‑scores, convert them to probabilities, and find probability between two Z‑scores with this easy‑to‑use online tool

What is a Z-Score?

The z-score, also called standard score, z-value, or normal score, is a dimensionless quantity that measures how many standard deviations a data point is from the mean.

A positive z-score means the value is above the mean, while a negative z-score means it is below the mean.

Formula for Z-Score

The z-score is calculated by subtracting the population mean from the raw score (data point), then dividing the result by the population standard deviation:

z = ^{x − μ}⁄_σ

• x = Raw score (data point)
• μ = Population mean
• σ = Population standard deviation

For a sample, the formula is similar but uses the sample mean and sample standard deviation.

Applications of Z-Score

Z-scores are widely used in:

• Performing z-tests
• Calculating prediction intervals
• Process control in manufacturing
• Comparing scores on different scales

Z-Table

A z-table (standard normal table or unit normal table) lists standardized values used to find the probability that a statistic lies below, above, or between points on the standard normal distribution.

A z-score of 0 means the data point is exactly equal to the mean.
On the standard normal distribution curve:

• z = 0 → Center of the curve
• Positive z → Value is to the right of the mean
• Negative z → Value is to the left of the mean

The values in a z-table represent the area between z = 0 and a given z-score.

How to read the z-table

In the table above,

• the column headings define the z-score to the hundredth’s place.
• the row headings define the z-score to the tenth’s place.
• each value in the table is the area between z = 0 and the z-score of the given value, which represents the probability that a data point will lie within the referenced region in the standard normal distribution.

For example, referencing the right-tail z-table above, a data point with a z-score of 1.12 corresponds to an area of 0.36864 (row 13, column 4). This means that for a normally distributed population, there is a 36.864% chance, a data point will have a z-score between 0 and 1.12.

Because there are various z-tables, it is important to pay attention to the given z-table to know what area is being referenced.

Related Calculator:
Permutation and Combination Calculator, Mean Median Mode Range Calculator, 
Standard Deviation Calculator

External Resources:
Z-Score Calculator on Calculator.net