6.11:
z Scores and Area Under the Curve
A normal distribution can be converted into standard normal distribution by replotting the probability density as a function of the z score.
This simple conversion tells us how many standard deviations away from the mean each value lies, enabling direct comparison of data sets.
Additionally, it provides an easy way to find the probability of an event occurrence by calculating the cumulative area from the left to the z score value.
Consider the birth weight of babies in a hospital. What is the probability of having a birth weight below 4 kg?
First, the appropriate z score corresponding to 4 kg birth weight is calculated, which is 1.25.
Now, using the z table that provides the probability values, one can obtain the probability associated with 1.25.
In the left column of the z table, locate the first decimal,1.2.
Then locate the column for the second decimal place, 0.05.
The number at the intersection, 0.8944, gives the probability of babies having a birth weight of less than 4 kg.
6.11:
z Scores and Area Under the Curve
z scores are the standardized values obtained after converting a normal distribution into a standard normal distribution. A z score is measured in units of the standard deviation. The z score tells you how many standard deviations the value x is above (to the right of) or below (to the left of) the mean, μ. Values of x that are larger than the mean have positive z scores, and values of x that are smaller than the mean have negative z scores. If x equals the mean, then x has a z score of zero. The z score allows us to compare data that are normally distributed but scaled differently.
A standardized graph can help determine the probability function. The area under the density curve between two points corresponds to the probability that the variable falls between those two values. The area under the curve is always 1. One can also find the area for a particular z score by referring to the z score table, which shows the cumulative areas under the standard normal distribution from the left side of the curve.
This text is adapted from Openstax, Introductory Statistics, Section 6.1