Z Score: Standard Score and Normal Distribution

Z Score

Definition

A Z score, also known as a standard score, is a statistical measurement that indicates how many standard deviations an observation is away from the mean of a normal distribution.

Formula

The Z score formula is:

Z = (X - μ) / σ

Where:

• Z = Z score
• X = observed value
• μ = mean of the distribution
• σ = standard deviation of the distribution

Interpretation

A Z score of:

• 0 indicates that the observed value is equal to the mean
• Greater than 0 indicates that the observed value is above the mean
• Less than 0 indicates that the observed value is below the mean
• Greater than 1 or less than -1 indicates that the observed value is one standard deviation away from the mean
• Greater than 2 or less than -2 indicates that the observed value is two standard deviations away from the mean
• Greater than 3 or less than -3 indicates that the observed value is three standard deviations away from the mean

Uses

Z scores are used in various statistical applications, including:

• Identifying outliers in a dataset
• Comparing values from different normal distributions
• Calculating percentiles and percent ranks
• Standardizing values for comparison purposes

Z Score

• A Z score is a statistical measurement that indicates how many standard deviations an observation is away from the mean of a normal distribution.

Formula

• Z score formula: Z = (X - μ) / σ
• Where Z = Z score, X = observed value, μ = mean of the distribution, and σ = standard deviation of the distribution

Interpretation

• Z score of 0: observed value is equal to the mean
• Z score greater than 0: observed value is above the mean
• Z score less than 0: observed value is below the mean
• Z score greater than 1 or less than -1: observed value is one standard deviation away from the mean
• Z score greater than 2 or less than -2: observed value is two standard deviations away from the mean
• Z score greater than 3 or less than -3: observed value is three standard deviations away from the mean

Uses

• Identify outliers in a dataset
• Compare values from different normal distributions
• Calculate percentiles and percent ranks
• Standardize values for comparison purposes