Z-Score Calculator

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

Calculate Z-Score from Raw Value

Value (x)

Mean (μ)

Standard Deviation (σ)

Z-Score and Probability Converter

Please provide any one value to convert between z-score and probability. This is the equivalent of referencing a z-table.

Z-score (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

What is a Z-Score?

The z-score, also referred to as standard score, z-value, and normal score, among other things, is a dimensionless quantity that is used to indicate the signed, fractional, number of standard deviations by which an event is above the mean value being measured. Values above the mean have positive z-scores, while values below the mean have negative z-scores. A z-score (also called a standard score) measures how many standard deviations a data point is from the mean. The formula for calculating a z-score is: \[z = \frac{x - \mu}{\sigma}\] where: \[x = \text{the individual value}\] \[\mu = \text{the mean of the population}\] \[\sigma = \text{the standard deviation of the population}\] Z-scores allow us to compare values from different datasets by standardizing them to a common scale.

The Z-Score Formula

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: \[z = \frac{x - \mu}{\sigma}\] where x is the raw score, μ is the population mean, and σ is the population standard deviation.

How to Interpret Z-Scores

Z-scores have these important properties: - A z-score of 0 means the data point's value is exactly equal to the mean - A positive z-score indicates the data point is above the mean - A negative z-score indicates the data point is below the mean - The absolute value of the z-score tells you how many standard deviations away from the mean the data point is In a normal distribution: - About 68% of values have z-scores between -1 and 1 - About 95% of values have z-scores between -2 and 2 - About 99.7% of values have z-scores between -3 and 3 (the 'three-sigma rule')

The Normal Distribution and the Empirical Rule

[Normal Distribution Curve image showing standard deviations]

-1σ to +1σ

68.27%

-2σ to +2σ

95.45%

-3σ to +3σ

99.73%

Z-Table

A z-table, also known as a standard normal table or unit normal table, is a table that consists of standardized values that are used to determine the probability that a given statistic is below, above, or between the standard normal distribution. A z-score of 0 indicates that the given point is identical to the mean. On the graph of the standard normal distribution, z = 0 is therefore the center of the curve. A positive z-value indicates that the point lies to the right of the mean, and a negative z-value indicates that the point lies left of the mean.

Z	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
0.0	0.5000	0.5000	0.5000	0.5000	0.5000	0.5000	0.5000	0.5000	0.5000	0.5000
0.1.0	0.5000	0.5050	0.5100	0.5150	0.5200	0.5250	0.5300	0.5350	0.5400	0.5450
0.2.0	0.5000	0.5100	0.5200	0.5300	0.5400	0.5500	0.5600	0.5700	0.5800	0.5900
0.3.0	0.5000	0.5150	0.5300	0.5450	0.5600	0.5750	0.5900	0.6050	0.6200	0.6350
0.4.0	0.5000	0.5200	0.5400	0.5600	0.5800	0.6000	0.6200	0.6400	0.6600	0.6800
0.5.0	0.5000	0.5250	0.5500	0.5750	0.6000	0.6250	0.6500	0.6750	0.7000	0.7250
0.6.0	0.5000	0.5300	0.5600	0.5900	0.6200	0.6500	0.6800	0.7100	0.7400	0.7700
0.7.0	0.5000	0.5350	0.5700	0.6050	0.6400	0.6750	0.7100	0.7450	0.7800	0.8150
0.8.0	0.5000	0.5400	0.5800	0.6200	0.6600	0.7000	0.7400	0.7800	0.8200	0.8600
0.9.0	0.5000	0.5450	0.5900	0.6350	0.6800	0.7250	0.7700	0.8150	0.8600	0.9050

How to read the z-table

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.

Applications of Z-Scores

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-scores are used in many statistical applications: 1. Identifying outliers in data 2. Converting scores from different tests to a standard scale for comparison 3. Creating standardized test scores (like SAT or IQ tests) 4. Quality control in manufacturing 5. Financial analysis and risk assessment 6. Medical testing and diagnostics 7. Educational assessment and grading on a curve