Find The Standardized Variable With Mean 16 And Standard Dev

Find The Standardized Variablezifxhas Mean 16 And Standard Deviation 4

Calculate the standardized variable z given that x has a mean of 16 and a standard deviation of 4. The standardized variable, often called a z-score, indicates how many standard deviations a data point x is from the mean. The formula for the z-score is:

z = (x - μ) / σ

where μ is the mean and σ is the standard deviation. Given the options, the correct formula to standardize a variable x in this context is:

  • Z = (x - 16) / 4

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The process of standardizing a variable, specifically calculating its z-score, plays a fundamental role in statistics by enabling comparison across different scales and distributions. Given a mean of 16 and a standard deviation of 4 for the variable x, the z-score for any specific data point x provides insight into its relative position within the data distribution. The formula used for calculating the z-score is z = (x - μ) / σ, where μ and σ denote the mean and standard deviation, respectively.

Applying this to the given parameters, the precise formula to compute the z-score for a data point x is z = (x - 16) / 4. This formula essentially measures how many standard deviations x lies away from the mean of 16. It is important to understand that a positive z-score indicates the value is above the mean, while a negative score indicates it is below.

This transformation enables statisticians to interpret data points on a standard normal distribution, which has a mean of 0 and a standard deviation of 1. Converting raw scores to z-scores facilitates the assessment of probabilities, comparisons across different datasets, and standardization of various statistical analyses. Accurate calculation of z-scores depends on proper application of this formula, which is essential for subsequent probability calculations and hypothesis testing.

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