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Let's dive into the concept of z-scores, a valuable tool for understanding how a particular data point compares to the average in a dataset. The z-score indicates how many standard deviations a data point is from the mean.
The z-score formula is:

• \( z = \frac{x - \mu}{\sigma} \)

Here, \(x\) represents the data point of interest, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
This formula allows us to transform any data point into a standardized value.
When you calculate a z-score, you're essentially ranking a specific value in relation to all other values in a normal distribution.
If the z-score is positive, the data point is above the mean; if negative, it’s below the mean.
This standardization makes it possible to compare data from different distributions.

Probability in connection to normal distribution measures the likelihood of an event occurring within a specified range of values.
In terms of the normal curve, this is often about finding the area under the curve.
Normal distributions are symmetrical, so probabilities for values around the mean are relatively high. As you move away from the mean, probabilities decrease.
To find the probability of an event given a normal distribution, we convert the value of interest to a z-score, then use a z-table or technology to find out the probability associated with this z-score.
This method enables us to determine:

• The probability of a value being less than a given x-value (left-tailed probability).
• The probability of a value being greater than a given x-value (right-tailed probability).

Overall, understanding probability in a normal distribution context helps us to quantify uncertainty in various contexts such as finance and quality control.

A percentile is a measure used to express the standing of one value in relation to a dataset. Imagine requesting the height of students in a school and ordering it from the shortest to tallest.
A student's height in the 90th percentile means they are taller than 90% of their peers.
Percentiles help compare values across different datasets or distributions by giving a rank based on a percentage of the dataset.
To find what value represents a certain percentile in a normal distribution, we typically revert the z-score to a data point.
This is done using:

• \( x = \mu + z \cdot \sigma \)

For instance, finding the 90th percentile involves determining the z-score corresponding to the 90% mark in the distribution.
The percentile concept is widely used in statistics and everyday life, particularly in evaluations and assessments, as it provides a standard measure for understanding relative positions within a dataset.