Understanding the Z-Score: A Statistical Measure for Analyzing Data Points in Relation to the Mean

The z-score in a normal probability distribution determines the number of standard deviations that a particular value, x, is from the mean

The z-score (also known as the standard score) is a measure in statistics that indicates how many standard deviations a data point, x, is from the mean of a normal distribution

The z-score (also known as the standard score) is a measure in statistics that indicates how many standard deviations a data point, x, is from the mean of a normal distribution. It allows us to compare and analyze data points relative to the rest of the data set.

The formula to calculate the z-score is given as:

z = (x – μ) / σ

Here,
z represents the z-score,
x is the data point,
μ is the mean of the distribution, and
σ is the standard deviation of the distribution.

By calculating the z-score, we can determine the position of the data point in relation to the mean. If the z-score is negative, it means the data point is below the mean, and if it is positive, it means the data point is above the mean. The larger the absolute value of the z-score, the farther away the data point is from the mean.

For example, let’s say we have a normal distribution with a mean (μ) of 50 and a standard deviation (σ) of 10. If we have a data point x = 60, we can calculate the z-score as follows:

z = (60 – 50) / 10
= 10 / 10
= 1

This z-score tells us that the data point lies exactly 1 standard deviation above the mean. Therefore, it is 1 standard deviation away from the average value of the distribution.

Z-scores can be used to compare different data points from the same distribution or to compare data points from different distributions. They are also useful in determining the probability of getting a certain value or range of values in a normal distribution.

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