Z-score

A Z-score, alternatively recognized as a standard score, signifies the number of standard deviations a specific data point deviates from the mean of its distribution. Representing a fundamental statistical concept, Z-scores play an integral role in contrasting data points from diverse distributions or transforming a distribution into a standard normal distribution.

Calculating the Z-score

To derive the Z-score for a particular data point, the ensuing formula proves handy:
Z = (X - μ) / σ

In this equation:

Z denotes the Z-score
X refers to the value of the data point
μ represents the mean of the distribution
σ stands for the standard deviation of the distribution

Applications of Z-scores

Standardizing Data: Turning raw scores into Z-scores serves to standardize data. This process paves the way for an easier comparison of values across multiple distributions or scales. Upon standardization, data aligns itself with a standard normal distribution, boasting a mean of 0 and a standard deviation of 1.
Identifying Outliers: The utilization of Z-scores proves instrumental in uncovering potential outliers in a dataset. Data points linked to Z-scores that noticeably deviate from the mean (either greater than +3 or less than -3) could be flagged as outliers, necessitating further scrutiny.
Calculating Percentiles: Z-scores play an active role in the calculation of percentiles. These percentiles delineate the relative standing of a data point within a distribution. With the aid of a Z-score table or calculator, determining the percentile rank of a data point becomes feasible.

Limitations of Z-scores

Despite the many benefits of Z-scores in statistical analyses, there exist certain limitations that warrant attention:
Assumption of Normality: Z-scores operate under the presumption that data adheres to a normal distribution. If the underlying distribution deviates from normality, Z-scores might not accurately portray the data's relative position within the distribution.
Influence of Extreme Values: In distributions plagued with extreme values or outliers, the mean and standard deviation could be skewed. This distortion might lead to Z-scores that provide potentially deceptive information.