ZScore Calculator
This Zscore calculator computes the Zscore and probability of a normal distribution. It also provides a diagram showing the position of the Zscore on a normal distribution curve. Additionally, it can convert between Zscore and probability. To use this calculator, simply select the type of calculation from the tab, enter the values, and click the 'Calculate' button.
The Zscore, 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 ZScore
The Zscore is calculated using the following formula:
Z = 

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 (Zscore) 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 Zscore would be calculated as follows:
Z = 
 = 1 
This Zscore 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 Zscore 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 Zscore is calculated by:
Z = 
 = 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 Zscore 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 ZScore
The Zscore 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. 
Zscores are especially useful in situations where comparison between different data sets is necessary. For instance, in standardized testing, Zscores are used to compare the performance of students from different classrooms or schools. In finance, Zscores help assess the risk of investment portfolios by comparing the volatility of various assets against the market or sector averages.