What is the relationship between z-scores and percentages?
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to Z scores. Multiply the proportions by 100 to get percentages.
The relationship between z-scores and percentages is based on the concept of the standard normal distribution. A z-score represents the number of standard deviations a particular value is away from the mean of a distribution. It allows us to compare values from different distributions on a common scale.
To understand the relationship between z-scores and percentages, we can use a standard normal distribution table, also known as a z-table. This table provides the percentage or probability associated with each z-score.
The standard normal distribution is a bell-shaped curve with a mean (µ) of 0 and a standard deviation (σ) of 1. The z-score formula is:
z = (x - µ) / σ
where x represents a particular value, µ is the mean, and σ is the standard deviation.
When we convert a value to a z-score, we can use the z-table to find the corresponding percentage. The z-table provides the area under the curve for each z-score.
For example, suppose we have a z-score of 1.5. We can look up the corresponding percentage in the z-table for a z-score of 1.5. The z-score of 1.5 corresponds to a percentage of 93.32%. This means that the value we are interested in is greater than approximately 93.32% of the values in the distribution.
Conversely, we can also use the z-table to find the z-score associated with a given percentage. For instance, if we want to find the z-score that corresponds to a percentage of 80%, we can look up the value closest to 80% in the z-table. In this case, the closest value is 0.84. This means that approximately 80% of the values in the distribution are less than the value associated with a z-score of 0.84.
In summary, z-scores and percentages have a direct relationship. Given a z-score, we can find the corresponding percentage, and vice versa, by using the z-table. The z-table allows us to convert between z-scores and percentages for values in the standard normal distribution.