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🎲intro to statistics review

Standardized Normal Variable

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

A standardized normal variable, also known as a z-score, is a dimensionless quantity derived by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation. This transformation converts the original variable into a new variable that follows a standard normal distribution with a mean of 0 and a standard deviation of 1.

5 Must Know Facts For Your Next Test

  1. Standardizing a variable by converting it to a z-score allows for direct comparisons between variables with different units or scales.
  2. The standardized normal variable follows a standard normal distribution, which means that approximately 68% of the values will fall within 1 standard deviation of the mean, 95% will fall within 2 standard deviations, and 99.7% will fall within 3 standard deviations.
  3. Z-scores can be used to determine the probability of a data point occurring within a normal distribution, which is essential for making inferences and decisions based on the data.
  4. Standardized normal variables are commonly used in hypothesis testing, as the test statistic often follows a standard normal distribution under the null hypothesis.
  5. Transforming variables into standardized normal variables can also help in the interpretation and visualization of data, as the z-scores provide a common scale for comparison.

Review Questions

  • Explain the purpose of standardizing a variable by converting it to a z-score.
    • The purpose of standardizing a variable by converting it to a z-score is to transform the original variable into a new variable that follows a standard normal distribution with a mean of 0 and a standard deviation of 1. This transformation allows for direct comparisons between variables with different units or scales, as the z-scores provide a common scale for comparison. Additionally, the standardized normal variable can be used to determine the probability of a data point occurring within the normal distribution, which is essential for making inferences and decisions based on the data.
  • Describe the characteristics of the standard normal distribution and how they relate to the standardized normal variable.
    • The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. When a variable is standardized by converting it to a z-score, the resulting standardized normal variable also follows this standard normal distribution. This means that approximately 68% of the values will fall within 1 standard deviation of the mean, 95% will fall within 2 standard deviations, and 99.7% will fall within 3 standard deviations. These characteristics of the standard normal distribution are crucial for interpreting and making inferences about the standardized normal variable.
  • Analyze the role of standardized normal variables in hypothesis testing and data interpretation.
    • Standardized normal variables play a crucial role in hypothesis testing because the test statistic often follows a standard normal distribution under the null hypothesis. This allows researchers to use the properties of the standard normal distribution to determine the probability of the observed test statistic, which is essential for making decisions about the null hypothesis. Additionally, transforming variables into standardized normal variables can aid in the interpretation and visualization of data, as the z-scores provide a common scale for comparison. This can help researchers identify patterns, outliers, and make more meaningful comparisons between different variables or data sets.