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A z-score, often known as a standard score, is a numerical measurement that describes a value's relationship to the mean of a group of values. It's quite helpful in standardizing different data points from various distributions. By transforming values into z-scores, you can understand how many standard deviations an element is from the mean. Here's the formula:

• \(Z = \frac{X-\mu}{\sigma}\)

In this equation:

• \(X\) is the raw score or the original data point.
• \(\mu\) represents the mean of the dataset.
• \(\sigma\) is the standard deviation of the dataset.

The z-score tells you how many standard deviations an element \(X\) is from the mean \(\mu\). For example, if your z-score is 1, it means that the data point is one standard deviation above the mean. If it's -1, it's one standard deviation below the mean. This concept is particularly useful in biostatistics, where data distributions are often compared.

The standard normal distribution is a special type of normal distribution where the mean is 0 and the standard deviation is 1. Understanding this distribution is key when converting raw data into z-scores. It allows us to use a standardized scale to measure probabilities and make comparisons between different groups. Every data point transformed into a z-score under this distribution can be plotted on a standard normal curve. A few important properties of the standard normal distribution:

• The graph of the distribution is symmetrical about its mean, which is 0.
• About 68% of all data points fall within one standard deviation of the mean, 95% fall within two standard deviations, and 99.7% fall within three standard deviations.

This distribution is incredibly useful for determining the probability of scores within any range in a dataset. That's because any dataset transformed using z-scores can be analyzed using the standard normal distribution. In biostatistics, this property lets researchers compare data sets of varying scales.

In probability calculation, especially using a standard normal distribution, we often need to find the likelihood that a particular z-score falls under a certain value. When we want to know \(\operatorname{Pr}(Z < 0.5)\), we're searching for the probability that the z-score is less than 0.5 using the standard normal curve. This can be done easily with a standard normal distribution table or a calculator that provides cumulative probabilities for z-scores.Here's how it works:

• Locate 0.5 in the z-score table.
• The table value gives the cumulative probability from the most extreme left up to z-score 0.5.

In this particular exercise, the cumulative probability, \(\operatorname{Pr}(Z < 0.5)\), is approximately 0.6915. That means there is a 69.15% chance that a randomly selected z-score from our distribution will be less than 0.5. Such calculations are frequently used in biostatistics to evaluate the significance of results or the likelihood of various events.