Z-Score Calculator for Normal Distribution

Imagine you’re a student, Lisa, preparing for a statistics exam. Your teacher asks:

"If the class average on the final test was 80, with a standard deviation of 5, and you scored 90, how exceptional is your score?"

Lisa wonders: “Is a 90 way above average or just slightly better than most of my classmates?”

This is exactly what the Z-Score Calculator answers. It helps you understand how a specific value compares to the rest of the data in a normal distribution.

What is a Z-Score?

A Z-Score tells you how many standard deviations a particular value is away from the mean.

• If Z = 0 → the value is exactly at the mean.
  
• If Z = 1 → the value is one standard deviation above the mean.
  
• If Z = -2 → the value is two standard deviations below the mean.
  

This is widely used in statistics, finance, psychology, and quality control, anywhere you need to compare a value to a population or dataset.

Why is the Z-Score Important?

1. Standardizes values – Makes it easy to compare different datasets.
   
2. Identifies outliers – Helps detect unusual or extreme values.
   
3. Calculates probabilities – Find the chance of a value occurring within a dataset.
   
4. Supports decision-making – Used in exams, finance, research, and business analytics.
   

Foundation for advanced stats – Z-scores are essential for t-tests, ANOVA, and other statistical tests.

Z-Score Formula

Formula: Z = (X - μ) / σ

Where:

• X = your value
• μ (mu) = mean of the dataset
• σ (sigma) = standard deviation

Example

Lisa scored 90 on a test where:

• Mean (μ) = 80
• Standard deviation (σ) = 5

Z = (90 - 80) / 5 = 10 / 5 = 2

This means Lisa scored 2 standard deviations above the mean, which is quite exceptional.

How the Calculator Works (Step by Step)

1. Enter the Mean (μ) – The average value of your dataset.
   
2. Enter the Standard Deviation (σ) – Measures how spread out the data is.
   
3. Enter the X value – The data point you want to evaluate.
   
4. Select Calculation Type – You can calculate:
   
   • Z-Score → Shows how far X is from the mean in SD units.
     
   • Probability → Shows the likelihood of a value being below or above X.
     
5. Press Calculate – The tool instantly gives:
   
   • Z-Score
     

Graphical representation – Shows where X lies on the normal curve.