Calculator
Example Data Table
| Observed Value | Mean | Standard Deviation | Z Score | Left Probability | Percentile |
|---|---|---|---|---|---|
| 78 | 70 | 10 | 0.8000 | 0.7881 | 78.81% |
| 62 | 70 | 10 | -0.8000 | 0.2119 | 21.19% |
| 95 | 80 | 12 | 1.2500 | 0.8944 | 89.44% |
Formula Used
The standard normal variable is calculated with this formula:
z = (x - μ) / σ
Here, x is the observed value. μ is the mean. σ is the standard deviation. The calculator then estimates normal curve probabilities from the z score.
Density formula:
φ(z) = e-z²/2 / √(2π)
Left area: P(Z ≤ z)
Right area: 1 - P(Z ≤ z)
Two tail area: 2 × smaller tail probability
How to Use This Calculator
- Enter the observed value you want to standardize.
- Enter the population or process mean.
- Enter the standard deviation.
- Add a second value when you need range probability.
- Enter a confidence level for critical z output.
- Choose decimal places for the displayed result.
- Press Calculate to view the result above the form.
- Use CSV or PDF buttons to save the output.
Understanding the Standard Normal Variable
What This Calculator Does
A standard normal variable calculator converts a raw value into a z score. The z score shows distance from the mean. It measures that distance in standard deviation units. This makes different scales easier to compare. A test score, height, error value, or process result can be standardized. After conversion, the value can be read against the normal curve. The calculator also gives tail areas, percentile, density, and central area.
Why Z Scores Matter
Z scores help compare values from separate data sets. A value above the mean has a positive z score. A value below the mean has a negative z score. A score near zero is close to average. Scores beyond two standard deviations are less common. Scores beyond three standard deviations are often rare. This pattern helps in quality checks, reports, surveys, and class results.
Probability and Percentile Meaning
The left probability gives the area below the z score. This is also the percentile position. For example, a left probability of 0.8413 means about 84.13 percent. The right probability shows the area above the value. The two tail probability checks how extreme the value is. It combines both outer ends of the curve. Range probability is useful when two raw values are supplied.
Advanced Result Options
This calculator includes a critical z result. It uses the selected confidence level. A 95 percent setting gives the common two sided cutoff. The raw value interval shows the matching lower and upper limits. Density shows curve height at the z score. It is not a probability by itself. Instead, it describes how high the normal curve is there. Use the export buttons when results must be stored. The CSV file supports spreadsheets. The PDF file supports simple sharing.
FAQs
What is a standard normal variable?
It is a raw value converted into a z score. The result shows how many standard deviations the value is from the mean.
What does a positive z score mean?
A positive z score means the observed value is above the mean. Larger positive values are farther above average.
What does a negative z score mean?
A negative z score means the observed value is below the mean. Larger negative distance means the value is farther below average.
Can I calculate percentile from a z score?
Yes. The left probability is the percentile expressed as a decimal. Multiply it by 100 to get a percentage.
Why must standard deviation be greater than zero?
Standard deviation measures spread. A zero or negative value cannot divide the distance from the mean correctly.
What is two tail probability?
Two tail probability measures extremeness on both sides of the normal curve. It is often used in hypothesis testing.
What is the optional second value for?
It calculates the probability between two raw values. The calculator converts both values into z scores first.
Can I download the result?
Yes. Use the CSV button for spreadsheet use. Use the PDF button for a simple printable report.