Z-Score Formula

Statistics is the branch of science that deals with collection, sorting and arrangement of data in such a way as to derive an inference regarding the subject of investigation. Most often, various tools of mathematics are employed to form hypotheses from the collected data and derive conclusions from these hypotheses. That's why, despite the ongoing debate on whether to categorize Statistics as an art or science, most people consider it as a separate discipline altogether which can be categorized as a mathematical science. Standard scores, also known as z-scores, z-values, standardized variables and normal score, are numerical figures that indicate the number of standard deviations, above or below, that a particular observation has with relation to a given mean value. Let's take a look at what is z-score and how is the z-score formula used to calculate the standardized variables of a given set of data.

What is a Z-Score?

The z-score figure is a pure number which is characterized by a complete lack of association with any sort of physical dimension. As such, it's dimension is always 1. This score is used to carry out the z-test which is part of standardized testing. Z-test is a statistical test wherein the probability distribution of a function of the sample involved in the null hypothesis is estimated via media, a normal distribution. Such a z-score is arrived at by calculating the difference between the value of the untransformed original datum (raw score) and the mean of the given population sample. For this, the population mean is deducted from the raw score and the difference so derived is divided by the standard deviation of the population. The z-score formula goes as follows:-

ɀ = (ϰ-µ)/σ

where,
ϰ = raw score
μ = mean of the given population sample
σ = standard deviation of given population sample

Altman Z-Score Formula

The Altman formula for Z-Score is a method of arriving at a z-score that is calculated for forecasting bankruptcy. This formula derives its name from its proponent, Edward Altman, who was an Assistant Professor at the New York University in the department of Finance. He published his research in 1968 which demonstrated how it is mathematically possible to calculate whether or not a firm is liable to go bankrupt in the next two years. The formula for calculating the Altman z score for bankruptcy is as follows:-

Z = 1.2T₁ + 1.4T₂ + 3.3T₃ + 0.6T₄ + 0.999T₅

Here,

Z = Z score
T₁ = Working capital of firm / Total assets of firm
T₂ = Retained earnings of firm / Total assets of firm
T₃ = EBIT / Total Assets of firm
T₄ = Equity value / Total liabilities of firm
T₅ = Total Sales / Total Assets of firm

Based upon the value of Z that is derived by using the above formula, the following inferences can be derived about the firm:-

• Z value is less than 2.99 :The firm is safe and is unlikely to go bankrupt in the next two years.
• Z value is between 1.8 and 2.99: The firm may head towards bankruptcy in the next two years but if corrective measures are taken now, it can be saved. This value also indicates that the firm may actually skirt bankruptcy in the next two years by a hair's breadth.
• Z value is less than 1.8: Danger zone! The firm is certainly headed towards bankruptcy in the next two years.

Z-Score to Percentile Calculation

Here is a way of calculating a specific percentile value by using the z score figure:-

X = Mean + ɀ + σ

Where X = nth percentile value

The reverse, that is z-score from percentile, can also be calculated by manipulating this formula if you have the percentile value with you. This can result in the following percentile to z-score formula:-

ɀ = (Mean + σ) - X

That, precisely, clarifies the basic concepts about a z-score and how to arrive at the value by using z-score formula. Hope this helps you understand this particular section of statistical data interpretation. For lengthy calculations involving percentile and z-scores, you can also make use of z-score tables and the good news is that you can access these types of statistical resources for practical application and various tutorials for statistics help online and for free! So, you no longer need to spend sleepless nights, hovering over heavy statistics books and pulling your hair out over complicated statistical exercises. Wish you all the best!