Solve z score, test, interval, and probability tasks. Switch modes and inspect every key statistic. Export results fast with examples, formulas, guidance, and FAQs.
| Scenario | Input Example | Output Example |
|---|---|---|
| Z Score | x = 72, μ = 68, σ = 4 | z = 1 |
| Probability | Lower z = -1.25, Upper z = 1.80 | P = 0.8584 |
| Raw Value | z = 1.5, μ = 68, σ = 4 | x = 74 |
| Z Test | x̄ = 104.2, μ₀ = 100, σ = 12, n = 36 | z = 2.1 |
| Confidence Interval | x̄ = 104.2, σ = 12, n = 36, 95% | 100.28 to 108.12 |
Z score: z = (x - μ) / σ
Raw value from z: x = μ + zσ
Probability between two z values: P(a ≤ Z ≤ b) = Φ(b) - Φ(a)
One sample z test: z = (x̄ - μ₀) / (σ / √n)
Confidence interval: x̄ ± zcritical(σ / √n)
Here, Φ(z) is the cumulative normal distribution function.
This Z MN Z calculator helps with several common normal distribution tasks. It computes z scores quickly. It also handles z test work. You can estimate interval limits. You can convert a z value into a raw value. You can also find probability between two z points.
Z based statistics are useful when the population standard deviation is known. They are also useful in large sample settings. The method standardizes values. That makes distant observations easier to compare. A z score shows how far a value is from the mean. The distance is measured in standard deviation units.
Probability mode is helpful for classroom work and applied analysis. Enter a lower z and an upper z. The calculator returns the probability within that range. This is useful for cutoff analysis. It is also useful for quality control review. It supports fast interpretation of tail and middle area behavior.
The z test mode compares a sample mean with a hypothesized population mean. It uses the known standard deviation and sample size. The result includes the standard error. It also gives the z statistic and p value. This is useful when testing a claim about a population mean.
The confidence interval mode estimates a likely range for the population mean. It uses the sample mean, known standard deviation, sample size, and confidence level. The result shows the margin of error and both interval limits. This gives a practical summary for reporting and decision work.
This page keeps the workflow simple. The form is compact. The output appears above the form. The result table is easy to scan. Export buttons help with reporting. The example data table shows common use cases. The formula section supports learning. The guidance section helps first time users work accurately.
A z score shows how many standard deviations a value sits above or below the mean. Positive values are above the mean. Negative values are below it.
Use a z test when the population standard deviation is known, or when large sample conditions justify the normal approach. It is mainly used for testing a population mean.
Z methods use a known population standard deviation. T methods are preferred when population standard deviation is unknown and estimated from the sample.
Yes. Choose the probability mode, enter lower and upper z values, and the tool returns the area between them plus cumulative probabilities.
A negative z score means the observed value is below the mean. The sign only shows direction. The magnitude shows distance from the mean.
It gives a plausible range for the population mean under the model assumptions. Wider intervals reflect more uncertainty. Larger samples usually narrow the interval.
Yes. After calculation, use the CSV button for spreadsheet style output or the PDF button for a print friendly report of the result block.
Yes. Use the raw value mode. Enter the z value, mean, and standard deviation. The calculator returns the original scale value.