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When dealing with normal distributions, probability calculations help us determine the likelihood of a certain event within the distribution. A normal distribution is characterized by its bell-shaped curve and is defined by the parameters: mean (\( \mu \)) and standard deviation (\( \sigma \)).
In continuous distributions, the probability of a single exact point is always 0 since there are infinitely many points. Instead, we calculate probabilities over intervals. For example, the probability that a value is less than a certain point is found by calculating the area under the curve to the left of that point. This requires the use of a standardized method like the z-score, which we will cover later.
By understanding these basics of probability, you can interpret and predict outcomes in real-life scenarios involving normally distributed data.

The standard deviation is a measure of how spread out the values are in a dataset. In the context of normal distribution, it shows how much variation exists from the average or mean value. A smaller standard deviation indicates that the values tend to be close to the mean, while a larger one suggests they are spread over a wider range.
For any normal distribution, approximately

• 68% of data falls within plus or minus one standard deviation from the mean,
• 95% within two standard deviations,
• 99.7% within three standard deviations.

This is known as the empirical rule or 68-95-99.7 rule.
In our exercise, with a standard deviation of 5, we can predict a significant amount of chloride concentration values will fall between 99 and 109 mmol/L.

The z-score is a way of determining how many standard deviations away a particular value is from the mean of a distribution. It's a crucial concept in normal distribution as it allows for the comparison and calculation of probabilities.
The formula for calculating the z-score is:
\[ z = \frac{x - \mu}{\sigma} \]
Where:

• \( x \) is the given data point,
• \( \mu \) is the mean,
• \( \sigma \) is the standard deviation.

Using a z-score table, or standard normal distribution table, we can find the probability of a value and compare it to other values within a distribution.
In our example, the z-score for a chloride concentration of 105 is 0.2, meaning it is 0.2 standard deviations above the mean, allowing us to compute the probability of observing such a value.

Percentiles give us valuable information about the relative standing of a value within the dataset. A percentile is a value below which a given percentage of observations in a group fall. They are especially useful in measuring the extreme values in a distribution.
For instance, the 50th percentile, or median, is the middle value of a dataset. In normal distribution, percentiles can help determine the cut-off for extreme events, such as the most extreme 0.1% of values.
To determine these percentile values in our problem, we employ z-scores that correspond to percentiles from the z-table. Such as in the exercise, z-scores of -3.2905 and 3.2905 represent the lower and upper extremes of 0.1% of data respectively.
These transformations give us insights into where the most unusual concentrations lie, crucial for understanding the distribution's tails and outliers.