Z-score Calculator

Use this calculator to compute the z-score of a normal distribution.

Raw Score, x
Population Mean, μ
Standard Deviation, σ

Z-score and Probability Converter

Enter one of the fields—either a z-score or a probability—to perform the conversion. This works just like consulting a z-table.

Z-score, Z
Probability, P(x<Z)
Probability, P(x>Z)
Probability, P(0 to Z or Z to 0)
Probability, P(-Z<x<Z)
Probability, P(x<-Z or x>Z)

Probability between Two Z-scores

Use this calculator to find the probability (area P in the diagram) between two z-scores.

Left Bound, Z1
Right Bound, Z2

Related

Standard Deviation Calculator

What is z-score?

The z-score (also called standard score, z-value or normal score) is a unit‑free measure that tells how many standard deviations an observation lies from the population mean, including direction. Scores above the mean yield positive z-scores; those below produce negative ones.

The z-score can be calculated by subtracting the population mean from the raw score, or data point in question (a test score, height, age, etc.), then dividing the difference by the population standard deviation:

z =

x - μ σ

Using x for the raw observation, μ for the population mean and σ for the population standard deviation, the calculation is x‑μ divided by σ. For sample data the same structure applies, but the sample mean replaces μ while σ remains the population standard deviation.

The z-score has numerous applications and can be used to perform a z-test, calculate prediction intervals, process control applications, comparison of scores on different scales, and more.

Z-table

A z-table—sometimes referred to as a standard normal or unit normal table—lists standardized values that let you find the probability that a statistic falls below, above, or between certain points of the normal distribution. A z-score of zero corresponds to the mean, placing it at the centre of the bell curve. Positive z-scores lie to the right of the mean, while negatives sit to the left. Various formats of z-tables exist.

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

How to read the z-table

In the table above,

• column headings give the hundredths digit of the z-score.
• row headings give the tenths digit of the z-score.
• each entry represents the area under the curve from 0 up to the indicated z-score, i.e., the probability that a datum falls within that segment of 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.

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