Calculating Z-Scores for Normally Distributed Random Variables

When dealing with normally distributed random variables, understanding how to calculate z-scores is crucial. A z-score, also known as a standard score, indicates the number of standard deviations a raw data point is from the mean of a distribution. This article aims to explain how to calculate z-scores using the given normal distribution parameters and to clarify any misconceptions about the notation.

Understanding the Notation

Given that ( X sim N(5, 6) ), this notation means that ( X ) is a normally distributed random variable with a mean ( mu 5 ) and a standard deviation ( sigma 6 ). It's essential to note that the variance is ( sigma^2 6 ), which is often written as ( sigma sqrt{6} ) instead of directly stating the variance.

Calculating the Z-Score

The z-score can be calculated using the formula:

[ z frac{x - mu}{sigma} ]

Where:

( z ) is the z-score ( x ) is the raw data point ( mu ) is the mean of the distribution ( sigma ) is the standard deviation of the distribution

Example Calculations

Let's apply the z-score formula to the given raw data points:

Part A: ( x 23 )

Given:

Mean ( mu 5 ) Standard deviation ( sigma 6 ) Raw data point ( x 23 )

Substitute the values into the z-score formula:

[ z frac{23 - 5}{6} frac{18}{6} 3 ]

Part B: ( x 2 )

Given:

Mean ( mu 5 ) Standard deviation ( sigma 6 ) Raw data point ( x 2 )

Substitute the values into the z-score formula:

[ z frac{2 - 5}{6} frac{-3}{6} -0.5 ]

Summary

The z-score for ( x 23 ) is ( z 3 ).

The z-score for ( x 2 ) is ( z -0.5 ).

Key Concepts

To better understand z-scores:

A z-score represents how many standard deviations a raw data point is from the mean. ( z ) scores for raw data ( x ) are calculated by subtracting the mean from the raw data and then dividing by the standard deviation. ( z ) scores are useful for comparing data from different normal distributions or for understanding the relative position of a data point within a distribution.

Conclusion

Careful attention to the notation and the application of the z-score formula is essential when working with normally distributed random variables. Understanding these concepts can help in making informed decisions and drawing accurate conclusions from data.