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Understanding z-score calculations is pivotal when diving into the world of normal distribution in statistics. The z-score, also known as the standard score, quantifies how many standard deviations an observation is from the mean of the distribution. To calculate a z-score, you'd use the formula: \[\begin{equation} z = \frac{{(X - \mu)}}{{\sigma}} \end{equation}\]where 
\(X\) is the observed value, \(\mu\) is the mean of the distribution, and \(\sigma\) is the standard deviation. For example, with a mean systolic blood pressure of 120 and a standard deviation of 8, the z-score for a blood pressure of 130 is \[\begin{equation} z = \frac{{(130 - 120)}}{{8}} = 1.25 \end{equation}\]This indicates that 130 is 1.25 standard deviations above the mean. Z-scores can be positive or negative, depending on whether the observation lies above or below the average. By converting raw scores into z-scores, we can then easily compare different data points within the same distribution or even across different distributions.

The percentile range in a normal distribution represents the position of a score relative to the rest of the data. It tells us what percentage of observations fall below a certain point. A normal distribution has a symmetric, bell-shaped curve where the mean, median, and mode are all equal.Interpreting the middle 60% of a population's systolic blood pressure, for instance, requires identifying the range of values that cover the central portion of the distribution. A z-score identifies the corresponding percentile; for the middle 60%, we look at the 20th to the 80th percentile. Using a standard normal distribution table (or z-table), we find that the z-scores corresponding to these percentiles are approximately -0.84 and 0.84. We can then convert these z-scores back to actual blood pressures by applying the z-score formula in reverse:\[\begin{equation} X = \mu + (z \times \sigma) \end{equation}\]For a mean pressure of 120 and standard deviation of 8, we get an estimated range of blood pressures from approximately 113.28 to 126.72.

Interpreting a normal distribution allows us to make predictions and judgments about the data. With a symmetrical bell shape, the distribution's mean, median, and mode coincide at the center, dividing the curve into two equal halves. In a perfect normal distribution, about 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and nearly all—99.7%—within three standard deviations.

Application in Healthcare

In a healthcare context, interpreting the normal distribution can lead to informed decisions. If we consider systolic blood pressures in a population, healthcare professionals use the area under the curve to determine the percentage of people who may require additional monitoring or interventions. For example, knowing that a certain blood pressure marks the cutoff for the top 15% of the population helps healthcare providers identify individuals at potential risk. Using the 85th percentile as a benchmark, we find the corresponding z-score to be about 1.04, which translates to a blood pressure of 128.32 using the reverse z-score formula. This kind of interpretation allows healthcare workers to focus resources on those more likely to require medical attention based on their blood pressure readings.