Hello there! I'm a seasoned expert in statistical analysis, and I'd be delighted to explain how the z-score is calculated. The z-score is a statistical measure that describes a value's relationship to the mean of a group of numbers. It is also known as the standard score, and it's used to standardize the data from a distribution. This standardization is particularly useful when comparing data sets with different units or scales.

The process of calculating the z-score involves three main steps: identifying the raw score, the mean of the distribution, and the standard deviation of the distribution. Let's delve into each of these steps in detail.

### Step 1: Identify the Raw Score
The raw score is the actual value you want to standardize. This could be a test score, a height measurement, or any other numerical value that you're interested in comparing to a larger set of data.

### Step 2: Determine the Mean of the Distribution
The mean, often referred to as the average, is calculated by adding up all the values in your data set and then dividing by the number of values. It's a measure of the central tendency of your data.

### Step 3: Calculate the Standard Deviation
The standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.

### The Z-Score Formula
Once you have the raw score, the mean, and the standard deviation, you can calculate the z-score using the following formula:

\[ z = \frac{(X - \mu)}{\sigma} \]

Where:
- \( z \) is the z-score,
- \( X \) is the raw score,
- \( \mu \) is the mean of the distribution,
- \( \sigma \) is the standard deviation of the distribution.

### Interpreting the Z-Score
The z-score tells you how many standard deviations an element is from the mean. A z-score of 0 means the value is exactly at the mean. A positive z-score indicates the value is above the mean, while a negative z-score indicates it is below the mean.

### Properties of Z-Scores
- The mean of a set of z-scores is always 0.
- The standard deviation of a set of z-scores is always 1.
- Z-scores follow a standard normal distribution, which is a type of continuous probability distribution for a real-valued random variable.

### Example
Let's say you have a data set of test scores with a mean of 80 and a standard deviation of 10. If a student scored 90 on the test, you would calculate the z-score as follows:

\[ z = \frac{(90 - 80)}{10} = \frac{10}{10} = 1 \]

This means the student's score is 1 standard deviation above the mean.

### Uses of Z-Scores
Z-scores are used in a variety of fields, including:
- Statistics: To compare data from different distributions.
- Finance: To measure the risk of an investment compared to the market as a whole.
- Science: To compare experimental results with control groups.

### Conclusion
Understanding how to calculate and interpret z-scores is fundamental to many areas of data analysis. They provide a standardized way to compare scores and understand the significance of deviations from the mean. Whether you're a student, a researcher, or a professional working with data, knowing how to work with z-scores can be incredibly valuable.

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Resultantly, these z-scores have a distribution with a mean of 0 and a standard deviation of 1. The formula for calculating the standard score is given below: As the formula shows, the standard score is simply the score, minus the mean score, divided by the standard deviation.read more >>