Z-Score Calculator
Calculate standard scores, convert between z-scores and probabilities, and find probabilities between two z-scores. Complete with z-table reference.
Z-Score Calculator
Use this calculator to compute the z-score of a normal distribution.
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.
Probability between Two Z-Scores
Use this calculator to find the probability (area P in the diagram) between two z-scores.
Z-Table Reference
A z-table, also known as a standard normal table or unit normal table, provides the probability that a statistic is below, above, or between the standard normal distribution.
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.
For example, a data point with a z-score of 1.12 corresponds to an area of 0.36864 (row 1.1, column 0.02). 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.
Understanding Z-Scores: A Complete Guide
What is a Z-Score?
The z-score, also referred to as standard score, z-value, and normal score, is a dimensionless quantity that indicates 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.
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
where x is the raw score, μ is the population mean, and σ is the population standard deviation. For a sample, the formula is similar, except that the sample mean and population standard deviation are used instead of the population mean and 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 Reference
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. There are a few different types of z-tables.
The values in the table above represent the area between z = 0 and the given z-score.
Applications of Z-Scores
Z-scores are widely used in various fields:
- Education: Comparing test scores from different distributions
- Finance: Assessing investment performance relative to market
- Quality Control: Identifying outliers in manufacturing processes
- Research: Standardizing measurements for comparison
- Medicine: Evaluating growth charts and medical test results
Interpreting Z-Scores
Understanding what z-scores mean:
- Z = 0: The value is exactly at the mean
- Z = ±1: The value is one standard deviation from the mean
- Z = ±2: The value is two standard deviations from the mean
- Z = ±3: The value is three standard deviations from the mean
In a normal distribution, approximately:
- 68% of values fall between z = -1 and z = 1
- 95% of values fall between z = -2 and z = 2
- 99.7% of values fall between z = -3 and z = 3
Frequently Asked Questions
Z-scores are based on the standard normal distribution and are used when the population standard deviation is known. T-scores are used when the population standard deviation is unknown and the sample size is small. T-scores follow the t-distribution, which has heavier tails than the normal distribution, accounting for the additional uncertainty when using sample standard deviation.
Yes, z-scores can be negative. A negative z-score indicates that the data point is below the mean, while a positive z-score indicates it's above the mean. For example, a z-score of -1.5 means the data point is 1.5 standard deviations below the mean.
A z-score of 2.5 means the data point is 2.5 standard deviations above the mean. In a normal distribution, this is quite extreme - only about 0.62% of values would be expected to have a z-score greater than 2.5. This indicates that the value is unusually high compared to the rest of the distribution.
Z-scores can be converted to percentiles using the standard normal distribution. For example, a z-score of 0 corresponds to the 50th percentile (the median). A z-score of 1 corresponds to approximately the 84th percentile, meaning about 84% of values fall below this point. Our calculator automatically provides these probability conversions.
Z-scores are most useful when you need to:
- Compare values from different normal distributions
- Determine how unusual a value is within its distribution
- Standardize variables for statistical analysis
- Calculate probabilities for normally distributed variables
- Identify outliers in your data
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