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Z-Score Calculator

Calculate the Z-score (standard score) for a given value in a normal distribution. Enter your value, mean, and standard deviation to see the Z-score and visualize it on both the original and standard normal distribution graphs.

Calculate Your Z-Score

Enter values and click 'Calculate Z-Score'.

Calculation Explanation

Enter values above to see the step-by-step Z-score calculation.

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Comprehensive Normal Distribution Calculator

Access advanced features like calculating probabilities, inverse CDFs, and areas between values with detailed graphing and step-by-step solutions.

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Raw Score Calculator

Calculate the raw score for a given Z-score in a normal distribution. Enter your Z-score, mean, and standard deviation to find the original value and visualize it on both the standard and original normal distribution graphs.

Go to Raw Score Calculator
Why use these tools?
This Raw Score Calculator focuses on calculating and visualizing raw scores from Z-scores. The Comprehensive Normal Distribution Calculator offers advanced features like probabilities, inverse CDFs, and areas between values for a complete statistical analysis.

What is a Z-Score?

The Z-score, or standard score, measures how many standard deviations a value is from the mean of a normal distribution. It standardizes data, allowing comparison across different distributions. A positive Z-score indicates the value is above the mean, while a negative Z-score indicates it’s below.

Key Formulas:

1. Z-Score (Standard Score)

\( Z = \frac{x - \mu}{\sigma} \)

Where:

  • \(x\) is the value of the variable.
  • \(\mu\) is the mean of the distribution.
  • \(\sigma\) is the standard deviation.

2. Probability Density Function (PDF)

\( f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2} \)

Where:

  • \(\pi \approx 3.14159\)
  • \(e \approx 2.71828\) (Euler's number)

3. Cumulative Distribution Function (CDF)

\( F(x) = P(X \leq x) = \int_{-\infty}^{x} \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{t - \mu}{\sigma}\right)^2} \, dt \)

Z-scores are widely used in statistics, finance, quality control, and education to standardize data, assess probabilities, and determine percentiles.

Frequently Asked Questions (FAQs)

1. What is a Z-score?

A Z-score measures how many standard deviations a value is from the mean in a normal distribution.

2. How is the Z-score calculated?

Use the formula \( Z = \frac{x - \mu}{\sigma} \), where \( x \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

3. What does a positive Z-score mean?

A positive Z-score indicates the value is above the mean of the distribution.

4. What does a negative Z-score mean?

A negative Z-score indicates the value is below the mean of the distribution.

5. Why use a Z-score?

Z-scores standardize data, allowing comparison across different normal distributions and use with Z-tables for probability calculations.

6. What is the standard normal distribution?

It’s a normal distribution with a mean of 0 and a standard deviation of 1, used to standardize Z-scores.

7. Can I use Z-scores for non-normal distributions?

Z-scores are most meaningful for normal or near-normal distributions. For other distributions, results may be misleading.

8. How do I interpret a Z-score of 2?

A Z-score of 2 means the value is 2 standard deviations above the mean, corresponding to approximately the 97.7th percentile in a standard normal distribution.

9. What happens if the standard deviation is zero?

If the standard deviation is zero, the Z-score is undefined, as there’s no variability in the data.

10. Can I use this to find percentiles?

Yes! The Z-score can be used with a Z-table or statistical software to find the percentile (probability) associated with a value.

11. What are real-world uses of Z-scores?

Z-scores are used in education (grading), finance (risk assessment), quality control, and research to standardize and compare data.

12. How accurate is this calculator?

The calculator uses precise mathematical formulas and is highly accurate for practical purposes.

13. Can I use this for academic research?

Yes, it’s suitable for academic use. For high-precision needs, consider statistical software like R or Python’s SciPy.

14. How does the Z-score relate to the standard normal distribution?

The Z-score transforms a value from any normal distribution to the standard normal distribution (mean=0, sd=1) for easier probability calculations.

15. What is a Z-table?

A Z-table lists probabilities for Z-scores in a standard normal distribution, used to find percentiles or probabilities.

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