# Z-Score Calculator

 Data Point, x Population Mean, μ Standard Deviation, σ
Please provide either the Z-score or the probability (P-value) to convert between the two for a normal distribution.
 Z Score P-value (probability < Z)

This Z-score calculator computes the Z-score and probability of a normal distribution. It also provides a diagram showing the position of the Z-score on a normal distribution curve. Additionally, it can convert between Z-score and probability. To use this calculator, simply select the type of calculation from the tab, enter the values, and click the 'Calculate' button.

The Z-score, also known as a standard score, quantifies how many standard deviations an element is from the mean of its distribution. Essentially, it converts data from different scales to a common scale with a mean of zero and a standard deviation of one. This standardization allows for the direct comparison of scores from different data sets and is useful for making statistical decisions.

## Formula for Calculating the Z-Score

The Z-score is calculated using the following formula:

Z =
 x - μ σ
where:
x is the value to be standardized,
μ is the mean of the data set,
σ is the standard deviation of the data set.

This formula adjusts the score x by subtracting the mean μ and dividing the result by the standard deviation σ. The outcome is a score (Z-score) that tells how far and in what direction the original value lies from the mean, measured in units of standard deviation.

Example 1: Consider a set of test scores with a mean (μ) of 75 and a standard deviation (σ) of 10. A student scores 85 on this test. The Z-score would be calculated as follows:

Z =
 85 - 75 10
= 1

This Z-score means that the student's score is 1 standard deviation above the mean. If the score data follow a normal distribution, the probability associated with a Z-score of 1 is 0.841, which means approximately 84.1% of students scored below 85, and about 15.9% of students scored above 85.

Example 2: Imagine a study with a BMI mean of 22 and a standard deviation of 2. An individual in the study has a BMI of 18. The Z-score is calculated by:

Z =
 18 - 22 2
= -2

This result indicates that the individual's BMI is 2 standard deviations below the mean. If the BMI data follow a normal distribution, the probability associated with a Z-score of -2 is 0.023, which means that approximately 2.3% of people have a BMI lower than 18, and about 97.7% of people have a BMI higher than 18.

### Interpreting the Z-Score

The Z-score tells us not only the direction (above or below) but also the degree to which a data point diverges from the mean. Here are some key points for interpretation:

 Z = 0: The score is identical to the mean. Z > 0: The score is above the mean. Z < 0: The score is below the mean. |Z| > 1: The score is more than one standard deviation away from the mean. |Z| > 2: The score is considered unusual, being more than two standard deviations away from the mean. |Z| > 3: The score is considered an outlier, being more than three standard deviations away from the mean.

Z-scores are especially useful in situations where comparison between different data sets is necessary. For instance, in standardized testing, Z-scores are used to compare the performance of students from different classrooms or schools. In finance, Z-scores help assess the risk of investment portfolios by comparing the volatility of various assets against the market or sector averages.