Our previous discussion focused on the normal distribution. The density function of this distribution is represented by a symmetric bell-shaped curve where some random variable is more likely to be in the middle and less likely to be towards either tail. Two parameters describe the normal distribution and decide the center and spread of the curve, respectively: the mean and standard deviation .

The Empirical Rule states that 99.7% of the observed data fall within three standard deviations of the mean , or:

- 68% of the data will fall within one standard deviation of the mean
- 95% will fall within two
- 7% within three

Then, we touched upon standard normal distribution, a special case of the normal distribution. In the standardized form, the mean is set at 0 and the standard deviation is set at 1. This form is used to compare two normal distributions with different parameters .

## Z Scores and Converting Normal Distribution to Standard Normal Distribution

A Z-score measures how many standard deviations a data point is from the mean. This score allows for comparison between different normal distributions by standardizing values. To convert normal distribution into its standardized form, we must convert data points into Z-scores first. The Z-score is calculated using this formula:

Where:

*Z*is the Z-score*X*the data point from the original normal distribution- is the mean of the original normal distribution
- is the standard deviation of the original normal distribution

To demonstrate how this is used to compare two normal distributions, it’s best to work with an example.

### Example 1: Comparing ACT and SAT Scores

Let’s say that a college accepts both ACT and SAT scores, and it wants to find out which among the incoming first-year students scored the highest on these tests. The highest ACT scorer got 28, while the highest SAT scorer got 1350. These scores are not comparable, as they are the result of different exams. To compare them, we need to convert the values to Z scores and see which score is farther from the mean scores of their respective exams.

ACT

- Mean : 21
- Standard deviation : 5
- Score: 28

Applying the Z-score formula to the ACT score, we get:

SAT

- Mean : 1060
- Standard deviation : 210
- Score: 1350

Applying the Z-score formula to the SAT score, we get:

Remember that the Z-scores indicate how many standard deviations each score is from the mean of their respective distributions. According to our calculations:

- The highest ACT score (28) is 1.4 standard deviations above the mean or average ACT score.
- The highest SAT score (1350) is 1.38 standard deviations above the mean or average SAT score.

Since the Z-score for the highest ACT score (1.4) is slightly higher than the Z-score for the highest SAT score (1.38), the student with the highest ACT score performed better relative to their respective test population compared to the student with the highest SAT score.

## Example 2: Comparing Top Employee Performance

Let’s consider an example involving employees’ performance ratings in two different departments of a company. Department A’s top performer received 90 points, while Department B’s received 105 points. We want to compare the top performers from each department to determine who performed better relative to their peers.

Department A

- Mean : 75
- Standard deviation : 10
- Score: 90

Applying the Z-score formula to Department A’s top scorer, we get:

Department B

- Mean : 80
- Standard deviation : 15
- Score: 105

Applying the Z-score formula to Department B’s top scorer, we get:

Using the formula, we determined that:

- The top performer in Department A scored 1.5 standard deviations above the department’s mean performance score.
- The top performer in Department B scored 1.67 standard deviations above the department’s mean performance score.

Since the Z-score for the top performer in Department B (1.67) is higher than the Z-score for the top performer in Department A (1.5), the employee in Department B performed better relative to their peers compared to the employee in Department A.

## Understanding and Applying Z-Scores

By converting raw scores into Z-scores, businesses can make more informed decisions based on relative performance and deviations from the mean. Understanding how Z-scores work can be quite useful in many business situations and applications. It can be utilized to ensure product quality and determine whether a value is within acceptable quality limits, for instance. It can also be quite useful in comparing sales across different regions, evaluating the financial risks of different investments, and measuring customer satisfaction.

This concludes our lessons on the fundamentals of inferential statistics. Our next focus will be on hypothesis testing.