## What is Z-Score?

A z-score is the distance between data points and the mean using standard deviations. Z-scores may be either positive or negative. The sign indicates whether the observation falls below or above the mean. A z-score +2 means that the data point is two standard deviations higher than the mean. Conversely, a score of -2 indicates that it is two standard deviations lower than the mean. A z-score equals the mean. Statisticians refer to z scores as standard scores. I’ll use these terms interchangeably.

The following benefits can be obtained by standardizing raw data and transforming them into Z-scores:

Understanding the relationship between a data point and a distribution is important.

Comparing observations of variables that are not the same.

Identify outliers

Use the standard normal distribution to calculate probabilities and percentiles.

This post will cover all the uses of z scores, as well as using z tables and z score calculators, and I’ll show you how to do all this in Excel.

How do I find a Z-score?

Calculating z-scores requires that you take the raw measurements and subtract the mean. Then divide the result by the standard deviation.

Z-scores are based on the original data distribution. Therefore, z-scores are calculated based on the distribution of the original data. The z-scores are based on the standard normal distribution, with a median of 0 and a standard deviation of 1. Skewed data can produce similar z-scores.

This post includes graphs of z scores using the standard normal distribution. They bring the concepts to life. Also, z-scores work best when data are normally distributed. Be aware, however, that z-scores can be invalid if your data is not normal.

Z-scores are based on the original data distribution. Therefore, z-scores are calculated based on the distribution of the original data. The z-scores are based on the standard normal distribution, with a median of 0 and a standard deviation of 1. Skewed data can produce similar z-scores.

This post includes graphs of z scores using the standard normal distribution. They bring the concepts to life. Also, z-scores work best when data are normally distributed. Be aware, however, that z-scores can be invalid if your data is not normal.

### Z-Scores vs. Standard Deviation

The standard deviation is simply a measure of the variability in a data set. The standard deviation is calculated first by determining the difference between each point of data and the mean. These differences are then added together, averaged, and squared. The variance is then calculated. The square root of the variance is called the standard deviation.

The Z-score is, in contrast, the number of standard deviations a data point has from the mean. The Z-score for data points below the mean is negative. Most large data sets have 99% values having a Z-score between 3 and 3. This means that they are within three standard deviations from the mean.

The standard deviation is simply a measure of the variability in a data set. The standard deviation is calculated first by determining the difference between each point of data and the mean. These differences are then added together, averaged, and squared. The variance is then calculated. The square root of the variance is called the standard deviation.

The Z-score is, in contrast, the number of standard deviations a data point has from the mean. The Z-score for data points below the mean is negative. Most large data sets have 99% values having a Z-score between 3 and 3. This means that they are within three standard deviations from the mean.

Careful calculation and interpretation of the Z-score are necessary. False accounting practices are possible, for example. The Z-score can be inaccurate if there are misrepresentations or coverups by companies in financial trouble.

The Z-score is not very useful for companies with little or no earnings. These companies will not score high regardless of their financial health. The Z-score does not address cash flows. It only indicates it through the net working capital ratio.

Z-scores may fluctuate from quarter to quarter if a company has one-off write-offs. These events can alter the final score and could falsely indicate that a company is at the edge of bankruptcy.

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#### Use Standard Scores to Compare Different Types Of Variables

You can use Z-scores to compare data points from different populations with standard deviations and means and then place them on a common scale. This standard scale allows you to compare observations for different variables, which would otherwise prove difficult. Standard scores are also known as z-scores. The process of standardizing raw data is called standardization. This allows you to compare data points between variables with different distributions.

Also, apples and oranges can be compared. Aren’t statistics amazing?

It’s not difficult to imagine that we will need to compare apples and oranges. We’ll be comparing their weights. We have a 110-gram Apple and a 100g Orange.

It’s easy to see that the apple weighs slightly less than the orange when comparing their raw values. Let’s now compare their z scores. We need to know their standard deviations and the means for different populations. Consider the following properties for apples and oranges:

Apples

Oranges

Mean weight grams

100

140

Standard Deviation

15

25

Let’s find out the Z-scores of our orange and apple!

Apple = (110-100), / 15 = 0.6667

Orange = (100-140), / 25 = = -1.6

A positive z-score = 0.667 indicates that the apple is heavier than the average. Although it’s not an extreme value but is higher than the average, it’s still a high Z-score. The orange, on the other hand, has a significantly negative Z-score (-1.6). It is well below the average weight of oranges. These standard scores are in the standard normal distribution.

The orange is smaller than the average apple, but our apple is slightly heavier than the average. We used z-scores to determine where each fruit is located within their distribution and how they compare.

#### Use Z-scores for Detecting Outliers

Z-scores are used to quantify the unusuality of an observation. Any raw data values significantly higher than the average are considered outliers and unusual. We are looking for absolute z-scores that are high.

The standard cutoff values are z-scores greater than three or more extreme for outliers. Below is the standard normal distribution plot showing the distribution of Z-scores. Z-scores above the cutoff is so rare that you cannot see the shading below the curve.