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Z-score calculation is a way to determine how many standard deviations an individual data point is from the mean in a normal distribution. It provides a method to standardize scores on a uniform scale, allowing for comparison of different data points.
The formula used to compute a Z-score is:

• \( Z = \frac{X - \mu}{\sigma} \)

Where:

• \( X \) is the individual data point,
• \( \mu \) is the mean of the distribution, and
• \( \sigma \) is the standard deviation.

Let's take an example:
Suppose the stock price of a company is \\(40, and we want to calculate the Z-score. If the average stock price is \\)30 and the standard deviation is \\(8.20, we substitute these values into our formula:

• \( Z = \frac{40 - 30}{8.2} \approx 1.22 \)

A Z-score of 1.22 indicates that a stock price of \\)40 is 1.22 standard deviations above the mean. This standardization helps in knowing where the data point stands concerning the entire data set.

Probability in the context of a normal distribution tells us how likely it is for a certain event to occur. Each Z-score corresponds to a probability, which represents the likelihood that a randomly selected data point is equal to or less than that Z-score.
Suppose you calculated a Z-score of 1.22 for a stock price of \\(40. This Z-score relates to a cumulative probability of approximately 0.8888 if you're consulting standard Z-tables. This means there is an 88.88% chance that a stock price will be less than \\)40.
Therefore, to find the probability that a stock price is at least \\(40, you subtract this cumulative probability from 1:

• \( P(X \geq 40) = 1 - 0.8888 = 0.1112 \)

This calculation indicates an 11.12% probability that a company's stock price will be \\)40 or more.
These probability calculations inform decisions and expectations by illuminating how extreme or common a particular observation really is.

The percentile rank in a normal distribution shows you what percentage of data points lie below a particular score. Percentiles are practical, especially when comparing scores across different data sets or distributions.
For example, the 90th percentile indicates that a value is higher than 90% of the data points in the distribution.
To find the stock price that places a company in the top 10% of the S&P 500, we utilize the 90th percentile. A Z-score that corresponds to the 90th percentile is about 1.28. Inserting this Z-score back into the Z-score equation helps us solve for the stock price:

• \( X = Z \times \sigma + \mu = 1.28 \times 8.2 + 30 \approx 40.50 \)

This calculation means that a stock price of approximately \$40.50 is required for a company to be in the top 10% in terms of stock prices. Using percentile ranks thus allows businesses and analysts to understand where they stand within a large, potentially competitive field.