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Suppose $X \sim N(12,6) .$ What is the $z$ -score of $x=2 ?$

   Suppose $X \sim N(12,6) .$ What is the $z$ -score of $x=2 ?$

Introductory Statistics

Introductory Statistics

Barbara Illowsky,… 1st Edition

Chapter 6, Problem 23

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Solved by Expert Bryan Meares

02/21/2020

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We know that $X$ follows a normal distribution with a mean of 12 and a standard deviation of 6. We are asked to find the $z$-score of $x=2$. Show more…

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Suppose $X \sim N(12,6) .$ What is the $z$ -score of $x=2 ?$

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Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution defined by its bell-shaped curve, characterized primarily by its mean and variance (or standard deviation). The mean determines the center of the distribution, while the variance (or standard deviation) measures the spread or how much the data is dispersed around the mean. This distribution is fundamental in statistics due to the Central Limit Theorem and its properties in representing many natural phenomena.

Z-Score

A z-score is a standardized score that indicates how many standard deviations an element is from the mean of its distribution. It is calculated by subtracting the mean from the observed value and then dividing by the standard deviation. This transformation allows for comparison across different normal distributions or different scales, essentially converting any normal distribution to the standard normal distribution with a mean of zero and a standard deviation of one.

Standardization

Standardization is the process of converting a random variable to a standard scale, typically by using the formula z = (x - ?) / ?, where x is the observed value, ? is the mean, and ? is the standard deviation. This process simplifies the analysis by allowing different datasets or variables to be compared on a common basis, and is especially useful when working with normally distributed data.

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