What Is Z Score – Its Formula, Calculation Measure and Interpretation
Z-Score is a statistical measure that indicates the robustness of a company. It measures how much of an outlier a data point is. Z-Scores can also be used to compare disparate data sets and derive useful results from them. That’s why it is often used by traders and investors.
This article breaks down the core concepts of the Z-Score, its variation (Altman Z-Score), calculation, interpretation and more. Read on!
What is a Z-Score?
A Z-Score (also called a ‘standard score’) is a statistic that shows, for a given set of data points, the number of standard deviations a given data point is from the mean. For example, if a given data point has a Z-Score of 3, that means that the data point is three standard deviations away from the mean.
A Z-Score can be positive or negative. A negative Z-Score indicates that the value of the data point is lower than the mean and vice versa.
What is Altman Z-Score?
In 1967, a finance professor named Edward Altman developed a variation of the Z-Score, now known as the Altman Z-Score. This score is based on five key ratios that can be worked out using a company’s annual reports. It aims to measure the probability that a company might go bankrupt.
How Does Z-Score Work?
To better understand how Z-Scores work, we first need to understand what the mean and standard deviation are.
The mean (also called the ‘average’) of a dataset is a so-called ‘measure of central tendency’. In other words, it serves to usefully condense a data set into a single “central” value. It is calculated by adding the values of all the data points and dividing the resulting sum by the total number of data points.
The standard deviation of a dataset is a measure of the dispersion of that dataset around its mean. It is calculated by taking the square root of the variance of the relevant dataset. The variance, in turn, is calculated as follows:
- For each data point, take the difference between it and the mean, and square this difference. Then, take the average of all these squared differences: the resulting value is the variance.
- Once you have these two quantities, you can plug them into the formula for Z-Score.
Why Are Z-Scores Important?
Z-Scores come in handy in a variety of situations. Some of their benefits are:
- They make it easier to understand the position of a data point in a distribution.
- They enable comparisons between dissimilar variables.
- They allow outliers to be identified.
- They make it possible to calculate probabilities and percentiles using the standard normal distribution (a special normal distribution whose mean is 0 and the standard deviation is 1).
In addition, traders can use the Z-Score to determine the level of volatility within a given stock.
As for Altman Z-Scores, they are important to investors because they can serve as a proxy for a company’s viability in the long run.
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Z-Score Formula
The formula for the Z-Score of a data point is:
Z = (x – μ) / σ
In this Z-Score formula,
x is the value of the data point,
μ is the mean of the relevant dataset,
σ is the standard deviation of the relevant dataset.
How to Calculate the Z-Score?
The conventional Z-Score can be calculated using the formula given above. As for the Altman Z-Score, it can be calculated using the following formula:
Z = 1.2A + 1.4B + 3.3C + 0.6D + 1.0E
Here, the letters A-E represent various financial ratios:
A = Working capital / Total assets
B = Retained earnings / Total assets
C = Earnings before interest and tax / Total assets
D = Market value of equity / Total liabilities
E = Sales / Total assets
How to Interpret the Z-Score?
As discussed above, the basic indication provided by a Z-Score is the number of standard deviations by which a given data point differs from the mean of the relevant dataset.
A common way to usefully interpret Z-Scores is to create what is called a Standard Normal Distribution (SND). This is also known as the Z-Score distribution or probability distribution. An SND is simply the distribution of the Z-Scores of a given dataset.
Any normal distribution can be transformed into an SND. An SND has two special characteristics: its mean is 0, and the standard deviation is 1. Thus, one standard deviation of the initial dataset always maps to a Z-Score of +1 or -1.
SNDs are useful because they allow comparing different datasets with different means and standard deviations. Such an SND-based Z-Score interpretation is very frequently used in statistics.
Let’s now talk about the interpretation of Altman Z-Scores. It should be noted that over the decades, the interpretation has changed.
An Altman Z-Score of less than 1.8 used to be indicative of a high probability of a company going bankrupt, while a score higher than 3 indicated a low probability of such a situation arising. Thus, investors could use these values as criteria to decide whether they should purchase a given company’s stock or not. In fact, if a company ever approached an Altman Z-Score of 1.8, that was enough reason for shareholders to sell the relevant company’s stock.
However, in more recent years, the Altman Z-Score that indicates financial trouble for a company has shifted to 0.
It is interesting to note that prior to the 2008 financial crisis, Altman calculated the median Altman Z-Score of companies to be 1.81. As a consequence, he believed that the credit market would crash; this belief was borne out.
What Are Z Score Confidence Intervals?
The definition of a confidence interval is: “a confidence interval is the range of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter.”
Let’s understand what this means. If your 95% confidence interval for a dataset is X-Y, then what that means is that you can be 95% sure that any randomly selected data point from that dataset will lie between X and Y.
Similar confidence intervals can be derived for any probability (e.g. 50% or 99%). Such confidence intervals are referred to as Z-Score confidence intervals in the context of SNDs.
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Final Word
Thus, it should be clear that the more statistically oriented Z-Score, as well as the more financially oriented Altman Z-Score, are two important measures that are used in different contexts. While Z-Scores can help traders determine the volatility within a given stock, the Altman Z-Score can help investors make investment decisions.
FAQs
Ans: A Z-Score (also called a ‘standard score’) is a statistic that shows, for a given set of data points, the number of standard deviations a given data point is from the mean.
Ans: There are many free Z-Score calculators available online. Typically, a Z-Score calculator will require you to input the raw score, the population mean, and the standard deviation.
Ans: The Z-Score of a data point indicates how far it is from the mean (in terms of standard deviations), while the standard deviation of a dataset indicates the degree of variability within that dataset.
Ans: Since a company’s Altman Z-Score is based on published financial data, it can be inaccurate and unhelpful if the data is misrepresented.
Ans: There is nothing intrinsically better about a higher Z-Score. To a trader, a higher Z-Score might indicate that a stock is volatile: this may be a good or bad thing, depending on the trader’s goal.
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This article has been prepared on the basis of internal data, publicly available information and other sources believed to be reliable. The information contained in this article is for general purposes only and not a complete disclosure of every material fact. It should not be construed as investment advice to any party. The article does not warrant the completeness or accuracy of the information and disclaims all liabilities, losses and damages arising out of the use of this information. Readers shall be fully liable/responsible for any decision taken on the basis of this article.
