What is the Z-Score/Standard Score?
In statistics, the z-score, which is also called the standard score, is a measure of how many standard deviations a data point is from the mean. A positive z-score means the data point is greater than the mean, and a negative z-score means a data point is lesser than the mean.
The z-score formula for a population's data is given as:
z = (x - μ)⁄σ
Where z is the z-score, x is the raw score, μ is the population mean, and σ is the population standard deviation.
The z-score formula for a sample of data is given as:
z = (x - x̄)⁄S
Where z is the z-score, x is the raw score, x̄ is the sample mean, and S is the sample standard deviation.
If you would like to quickly calculate a z-score, try our z-score calculator.
When to Apply the Z-Score
We primarily use the z-score to standardize data points within their respective data set. For example, let's say the average car has a mass of 1500 kg and the standard deviation of cars' masses is 300 kg. If a particular car weighs 1950 kg, it is 450 kg greater than the average car mass.
If we use the population z-score formula, that particular car has a z-score of 1.5. The z-score standardizes the data point's distance from the mean, giving us a better understanding of how far it is from the average, within the context of the data set it comes from.
The Normal Distribution and Z-Table
We can also use the z-score in combination with the normal distribution and z-table to calculate cumulative percent (also known as percentile ranking). If a data set is normally distributed, a particular data point's z-score can tell us what percentage of the data falls below that data point via the cumulative z-table.
For example, the semester grades for an entire class are normally distributed. A student's grade has a z-score of 2.5. If we consult the cumulative z-table, we find that a z-score of 2.5 corresponds to a value of 0.9938, which means the cumulative percent is 99.38%.
99.38% of the class's grades are lower than that student's grade, which means that student is in the 99.38th percentile of their class.