As a statistical expert with a focus on data analysis, I often work with various measures to understand and interpret data. One such measure is the standard score, also known as the z-score. The z-score is a standardized value that indicates how many standard deviations an element is from the mean. It is a crucial tool in statistics, as it allows for the comparison of data points across different scales.

To calculate the z-score for a given data point in a sample, you need to follow a systematic approach. Here's a step-by-step guide:


1. Determine the Mean (μ): The first step is to calculate the mean of your sample data. The mean, often denoted as μ (mu), is the average value of the data set. It is calculated by summing all the values in the data set and then dividing by the total number of values.


2. Calculate the Variance (σ²): Variance is a measure of how much the data points in the set differ from the mean. To find the variance, you subtract the mean from each data point, square the result (to make it positive), sum these squared differences, and then divide by the number of data points (for a sample variance, you would divide by the number of data points minus one).


3. Compute the Standard Deviation (σ): The standard deviation is the square root of the variance. It is a measure that indicates the amount of variation or dispersion in a set of values. A low standard deviation means that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range.


4. Identify the Data Point (X): The next step is to identify the specific data point for which you want to calculate the z-score.


5. Calculate the Z-Score: Finally, to find the z-score, subtract the mean from the data point (X - μ) and then divide the result by the standard deviation (σ). The formula for the z-score is:
\[
z = \frac{X - \mu}{\sigma}
\]

The z-score will tell you how many standard deviations the data point is from the mean. A z-score of 0 means the data point is exactly at the mean. A positive z-score indicates the data point is above the mean, while a negative z-score means it is below the mean.

The process of calculating the z-score is fundamental in statistical analysis. It is used in hypothesis testing, quality control, and in the standardization of test scores, among other applications. Understanding the z-score can help you make informed decisions, interpret data more effectively, and compare data across different scales.

Now, let's proceed to the translation.

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To find the Z score of a sample, you'll need to find the mean, variance and standard deviation of the sample. To calculate the z-score, you will find the difference between a value in the sample and the mean, and divide it by the standard deviation.read more >>