Z-Score
z = (x - ฮผ) / ฯ
The z-score measures how many standard deviations a data point is above or below the population mean. It standardizes values from any normal distribution to the standard normal distribution (mean 0, standard deviation 1), enabling direct comparison across different scales and units.
Variables
The number of standard deviations from the mean
The individual data point being standardized
The mean of the population distribution
The standard deviation of the population
Example Calculation
Scenario
Adult male heights are normally distributed with ฮผ = 70 inches and ฯ = 3 inches. A man is 76 inches tall. What is his z-score?
Given Data
Calculation
z = (x - ฮผ) / ฯ = (76 - 70) / 3 = 6 / 3
Result
z = 2.0
Interpretation
This man's height is 2.0 standard deviations above the mean. Using a z-table, approximately 97.7% of adult males are shorter than him, placing him in the top 2.3% of the height distribution.
When to Use This Formula
- โStandardizing values from different distributions for comparison
- โFinding probabilities using the standard normal table
- โIdentifying outliers (values with |z| > 2 or |z| > 3)
- โComputing confidence intervals and conducting hypothesis tests when ฯ is known
Common Mistakes
- โUsing the sample standard deviation (s) in place of the population standard deviation (ฯ) when ฯ is actually known
- โForgetting that z-scores can be negative when the value is below the mean
- โApplying z-scores to data that is not approximately normally distributed
- โConfusing z-scores with t-scores, which are used when ฯ is unknown
Calculate This Formula Instantly
Snap a photo of any problem and get step-by-step solutions.
Download StatsIQFAQs
Common questions about this formula
A negative z-score means the data point falls below the population mean. For example, z = -1.5 means the value is 1.5 standard deviations below the mean. Negative z-scores are not inherently bad; they simply indicate the direction relative to the average.
Once you compute the z-score, look it up in a standard normal (z) table or use a calculator. The table gives the cumulative probability P(Z < z). For example, z = 1.96 corresponds to P(Z < 1.96) = 0.975, meaning 97.5% of values fall below that point.