**Instructions:**

- Enter the Raw Score, Mean (μ), and Standard Deviation (σ) for your data.
- Click "Calculate Z-Score" to calculate the Z-Score and related values.
- Results, including the Z-Score, p-values, and confidence level, will be displayed below.
- Calculation steps will also be shown to explain how the Z-Score was computed.
- A chart visualizes the Z-Score in the context of the normal distribution.
- You can clear the entries, copy the results, and view calculation history.

**Calculator**

**Calculation History**

A Z score calculator is a tool that allows users to calculate the Z score of a given value. The Z score is a measure of how many standard deviations a given value is away from the mean of a population. It is calculated by subtracting the mean from the value and then dividing by the standard deviation.

**Concepts**

The following are some of the key concepts that underlie Z score calculators:

**Mean:**The mean is the average of a population. It is calculated by adding up all of the values in the population and dividing by the number of values.**Standard deviation:**The standard deviation is a measure of how spread out the values in a population are. It is calculated by taking the square root of the variance.**Variance:**The variance is a measure of how spread out the values in a population are. It is calculated by taking the average of the squared deviations from the mean.

**Formulae**

The following formula is used to calculate the Z score of a given value:

```
Z score = (x - μ) / σ
```

where:

- x is the value to calculate the Z score for
- μ is the mean of the population
- σ is the standard deviation of the population

**Benefits of using a Z score calculator**

There are several benefits to using a Z score calculator, including:

**Accuracy:**Z score calculators are very accurate, as they use sophisticated mathematical algorithms to perform their calculations.**Convenience:**Z score calculators can save users a lot of time and effort, as they can perform complex calculations quickly and easily.**Flexibility:**Z score calculators can be used to calculate the Z score of any value, regardless of the distribution of the population.**Versatility:**Z score calculators can be used in a variety of fields, including statistics, psychology, and economics.

**Interesting facts about Z scores**

- Z scores are normally distributed, meaning that the majority of values have Z scores close to zero.
- A Z score of zero indicates that the value is equal to the mean of the population.
- A positive Z score indicates that the value is above the mean of the population.
- A negative Z score indicates that the value is below the mean of the population.

**Scholarly references**

**David G. Moore:**The Basic Practice of Statistics, 7th Edition, W.H. Freeman & Company, 2018**Richard J. Larsen and David A. Marx:**An Introduction to Statistical Methods and Applications, 8th Edition, Pearson, 2016**Gerald C. Mosteller and Frederick Mosteller:**Data Analysis and Regression: A Second Course in Statistics, Addison-Wesley, 1988

## Conclusion

Z score calculators are a valuable tool for anyone who needs to calculate the Z score of a given value. They are accurate, convenient, flexible, and versatile. Z score calculators can be used in a variety of fields, including statistics, psychology, and economics.

**Example of using a Z score calculator**

Let’s say you are a student and you want to know how your score on a test compares to the rest of the class. You know that the mean score on the test was 75 and the standard deviation was 10.

To calculate your Z score, you would enter the following information into a Z score calculator:

- Value to calculate the Z score for: Your score on the test
- Mean of the population: 75
- Standard deviation of the population: 10

The calculator would then display the following result:

```
Z score = (x - μ) / σ = (Your score on the test - 75) / 10
```

For example, if you scored 85 on the test, your Z score would be:

```
Z score = (85 - 75) / 10 = 1
```

A Z score of 1 indicates that your score was one standard deviation above the mean.

Z score calculators can be used to compare your score on any test or assessment to the rest of the population. This can be helpful for determining how well you are doing and identifying areas where you may need to improve.