Z-score is a statistical measure that expresses the distance of a particular value from the mean of a group of values in terms of standard deviations. It is calculated by subtracting the mean from the value and dividing the result by the standard deviation. The formula for calculating the z-score of a data point x, with mean μ and standard deviation σ, is z = (x - μ) / σ (Grybauskas & Pilinkiene, 2018).
Z-scores in Construction
In construction works, the Z-score can be used to examine the performance of construction workers in plastering items in significant projects. As engineers, we can measure the workers' performance by the number of square feet that the worker can finish per hour. Accordingly, the mean and standard deviation of the performance metric for all individuals are as follows: This will give us a baseline against which we can compare each individual's performance.
We calculate each individual's z-score for the performance metric by subtracting their raw score from the group mean and dividing the result by the standard deviation. This will give us a standardized score representing how many standard deviations the individual's performance is above or below the group mean. Individuals with positive z-scores perform better than the group average, while those with negative z-scores perform worse. From these calculations, we can seek the help of qualified workers in projects called "Emergencies," which need to accomplish more progress in less time.
Different z-scores
A z-score of 0 indicates that the individual's score is strictly at the group's mean. This means their performance is average relative to the rest of the group (Laerd Statistics, 2018). A z-score of +1 indicates that the individual's score is one standard deviation above the group's mean. This means their performance is better than approximately 84% of the group. A z-score of +2 indicates that the individual's score is two standard deviations above the group's mean.
This means their performance is better than approximately 98% of the group. A z-score of -1 indicates that the individual's score is one standard deviation below the group's mean. This means their performance could be better than approximately 16% of the group. A z-score of -2 indicates that the individual's score is two standard deviations below the group's mean. This means their performance could be better than approximately 2% of the group.
Using z-scores with an unnormal distribution:
Using z-scores with a non-normal distribution can be problematic because z-scores assume that the data is normally distributed, with a mean of zero and a standard deviation of one. If the data is not normally distributed, the z-scores may not accurately reflect the position of individual data points relative to the rest of the data set. However, it is still possible to calculate z-scores for data that is not normally distributed, but it requires additional steps. One approach is to transform the data into a normal distribution using methods such as the Box-Cox transformation or log transformation. Once the data is transformed into a normal distribution, z-scores can be calculated as usual (Minitab 18 Support, 2019).
Conclusion
The Z-score indicates how many standard deviations the data point is from the mean. A positive z-score indicates that the data point is above the mean, while a negative z-score indicates that the data point is below the mean. A z-score of 0 indicates that the data point equals the mean (Iowa State University, n.d.). The z-score is a valuable tool for comparing values from different distributions, as it standardizes the values and allows for meaningful comparisons. Generally, it is best to use z-scores only for normally distributed data. However, if you are using z-scores for non-normal data, it is essential to be aware of the limitations of z-scores and to interpret them with caution.
References
Grybauskas, A. & Pilinkiene, V. (2018). Real Estate Market Stability: Evaluation of the Metropolitan Areas by Using Factor Analysis and Z-Scores. Engineering Economics, 29(2), 158–167. Retrieved from EBSCO multi-search
Iowa State University. (n.d.). What is a Z score? Retrieved from http://www2.econ.iastate.edu/classes/crp274/swenson/CRP272/What%20is%20a%20Z%20score.pdf
Laerd Statistics. (2018). Standard score. Retrieved from https://statistics.laerd.com/statistical-guides/standard-score.php
Minitab 18 Support. (2019). What is a Z-value? Retrieved from https://support.minitab.com/en-us/minitab/18/help-and-how-to/statistics/basic-statistics/supporting-topics/tests-of-means/what-is-a-z-value/