Characteristics of Z-Score

What is Z-Score?

The Z-score, also known as the standard score, tells us the deviation of a data point from the mean by expressing it in terms of standard deviations above or below the mean. It gives us an idea of how far a data point is from the mean. Hence, the Z-Score is measured in terms of standard deviation from the mean. For example, a Z-score of 2 indicates the value is 2 standard deviations away from the mean. To use a z-score, we need to know the population mean (μ) and also the population standard deviation (σ)....

How to Calculate Z-Score?

We are given the population mean (μ), the population standard deviation (σ), and the observed value (x) in the problem statement substituting the same in the Z-score equation yields us the Z-Score value. Depending upon whether the given Z-Score is positive or negative, we can use positive Z-Table or negative Z-Table available online or on the back of your statistics textbook in the appendix....

Characteristics of Z-Score

The magnitude of the Z-score reflects how far a data point is from the mean in terms of standard deviations.An element having a z-score of less than 0 represents that the element is less than the mean.Z-scores allow for the comparison of data points from different distributions.An element having a z-score greater than 0 represents that the element is greater than the mean.An element having a z-score equal to 0 represents that the element is equal to the mean.An element having a z-score equal to 1 represents that the element is 1 standard deviation greater than the mean; a z-score equal to 2, 2 standard deviations greater than the mean, and so on.An element having a z-score equal to -1 represents that the element is 1 standard deviation less than the mean; a z-score equal to -2, 2 standard deviations less than the mean, and so on.If the number of elements in a given set is large, then about 68% of the elements have a z-score between -1 and 1; about 95% have a z-score between -2 and 2; about 99% have a z-score between -3 and 3. This is known as the Empirical Rule, and it states the percentage of data within certain standard deviations from the mean in a normal distribution as demonstrated in the image below...

Calculate Outliers Using the Z-Score Value

We can calculate outliers in the data using the z-score value of the data points. The steps to consider an outlier data point are as :...

Implementation of Z-Score in Python

We can use Python to calculate the z-score value of data points in the dataset. Also, we will use the numpy library to calculate mean and standard deviation of the dataset....

Application of Z-Score

Z-scores are often used for feature scaling to bring different features to a common scale. Normalizing features ensures that they have zero mean and unit variance, which can be beneficial for certain machine learning algorithms, especially those that rely on distance measures.Z-scores can be used to identify outliers in a dataset. Data points with Z-scores beyond a certain threshold (usually 3 standard deviations from the mean) may be considered outliers.Z-scores can be used in anomaly detection algorithms to identify instances that deviate significantly from the expected behaviour.Z-scores can be applied to transform skewed distributions into more normal distributions.When working with regression models, Z-scores of residuals can be analysed to check for homoscedasticity (constant variance of residuals).Z-scores can be used in feature scaling by looking at their standard deviations from the mean....

Z-Scores vs. Standard Deviation

Z- Score Standard DeviationTransform raw data into a standardized scale. Measures the amount of variation or dispersion in a set of values. Makes it easier to compare values from different datasets because they take away the original units of measurement. Standard Deviation retains the original units of measurement, making it less suitable for direct comparisons between datasets with different units. Indicate how far a data point is from the mean in terms of standard deviations, providing a measure of the data point’s relative position within the distribution Expressed in the same units as the original data, providing an absolute measure of how spread out the values are around the mean...

Why are Z-scores Called Standard Scores?

Z-scores are also known as standard scores because they standardize the value of a random variable. This means that the list of standardized scores has a mean of 0 and a standard deviation of 1.0. Z-scores also allow for the comparison of scores on different kinds of variables. This is because they use relative standing to equate scores from different variables or distributions....

Z-Score in Statistics – FAQs

What is the significance of positive and negative Z-Scores?...