Z-Score Distribution

Understanding Z-Score

The Z-score (also called the standard score) indicates how many standard deviations a data point is from the mean.

The formula for the Z-score is:

Z = (X - μ) / σ
    

Where:

• X = individual data point
• μ = mean of the data
• σ = standard deviation

Example Calculation

Q: What is the Z-score of a student who scored 85 on a test where the class average is 75 with a standard deviation of 5?

Using the formula:

Z = (X - μ) / σ
Z = (85 - 75) / 5 = 2.0
    

This means the student's score is 2 standard deviations above the mean.

Interpreting Z-Scores

• A Z-score of 0 means the value is exactly at the mean.
• A positive Z-score means the value is above the mean.
• A negative Z-score means the value is below the mean.

Z-Score Table (Standard Normal Distribution)

The Z-score allows us to calculate probabilities using the standard normal table:

Z-Score   Probability
----------------------
-2.0      0.0228
-1.0      0.1587
 0.0      0.5000
 1.0      0.8413
 2.0      0.9772
    

For example, a Z-score of 1.0 corresponds to a probability of 0.8413, meaning 84.13% of the data lies below this value.

Visual Representation

The standard normal distribution curve:

• 68% of data falls between -1σ and +1σ.
• 95% falls between -2σ and +2σ.
• 99.7% falls between -3σ and +3σ.

Applications of Z-Scores

• Standardization: Used in machine learning to normalize data.
• Outlier detection: Identifies extreme values in datasets.
• Statistical significance: Helps determine whether a value is unusual.