### Z-score to Percentile: Understanding the Relationship with Examples

In the world of statistics, understanding the position and significance of a particular data point within a dataset is crucial. This is where concepts like the Z-score and percentiles come into play. Both are tools that statisticians and researchers use to interpret data, but they offer different perspectives. To understand the relationship between Z-scores and percentiles, we must first understand each concept individually.

## What is a Z-score?

The Z-score, often termed as a standard score, measures how many standard deviations a data point (or score) is from the mean of a dataset. It offers a precise position of individual scores by standardizing the entire dataset, providing a clearer picture of how data is spread around the mean.

Z-score formula:

Where:

Z = Z-score

X = Individual data point

μ = Mean of the dataset

σ = Standard deviation of the dataset

Learn about Significance Level Vs. Confidence Level Vs. Confidence Interval

## What is a Percentile?

A percentile, on the other hand, indicates the relative standing of a data point within a dataset. For instance, if a score is in the 85th percentile, it means the score is higher than 85% of the other scores in the dataset.

## From Z-score to Percentile

Converting a Z-score to a percentile is a common practice, especially when one needs to convey the relative standing of a data point in more intuitive terms. To make this conversion, statisticians often use the standard normal distribution table, commonly known as the Z-table.

The Z-table provides the cumulative probabilities (or percentiles) associated with standard scores. Since the standard normal distribution is symmetrical around the mean (which has a Z-score of 0), a Z-score of 0 corresponds to the 50th percentile. Positive Z-scores will correspond to percentiles greater than 50%, while negative Z-scores will correspond to percentiles less than 50%.

## Examples

### Example 1:

Imagine a student scores 75 on a national mathematics test. The mean score for this test is 50, with a standard deviation of 10.

First, let’s calculate the Z-score:

A Z-score of 2.5 indicates that the student’s score is 2.5 standard deviations above the national average. Using the Z-table, this Z-score corresponds roughly to the 99th percentile. This means the student scored higher than approximately 99% of the other test-takers.

### Example 2:

Consider an employee who’s ranked 15th in sales out of 100 employees. When the sales rankings are normally distributed with a mean of 50 and a standard deviation of 15, we want to find out where this employee stands in terms of Z-score and percentile.

Using the formula:

The negative Z-score indicates the employee’s sales rank is below the mean. Consulting the Z-table for a Z-score of -2.33, we find a percentile of approximately 1%. This means the employee’s sales rank is higher than only about 1% of the other employees, suggesting they might need additional support or training.