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A z-score is a way of expressing a data point's position in terms of standard deviations from the mean of a data set.
It helps us understand how far away a specific data point is from the mean when data is distributed normally. To calculate the z-score, we use the formula: \[ Z = \frac{(X - \mu)}{\sigma} \] where:

• X is the data point of interest,
• \( \mu \) represents the mean of the data,
• \( \sigma \) is the standard deviation.

For example, if we want to know the z-score for a carbon monoxide exposure of 20 ppm, given a mean of 18.6 ppm and a standard deviation of 5.7 ppm, we substitute these into the formula: \( Z = \frac{(20 - 18.6)}{5.7} \approx 0.25 \).
This tells us that 20 ppm is 0.25 standard deviations above the mean. Calculating z-scores is critical because it lets us use standard normal distribution tables or software to find probabilities.

The cumulative distribution function (CDF) is a tool that helps us understand the probabilities of different outcomes for a random variable.
In a normal distribution, the CDF gives us the probability that a random variable takes a value less than or equal to a particular point.
When we're interested in knowing the proportion of data points above (or below) a specific value, converting that value to a z-score and then using the CDF is essential. For a z-score calculated earlier, such as \( Z \approx 0.25 \), the CDF gives us the probability of being up to that z-score.
Often, we want the complementary CDF, which is 1-CDF, to know the probability of exceeding a certain value.
This is the approach taken in our example, where we found the probability of the carbon monoxide exposure being more than 20 ppm and 25 ppm by using 1-CDF to get \( 1-CDF(0.25) = 0.4013 \) and \( 1-CDF(1.12) = 0.1314 \). This means that about 40.13% and 13.14% of the time the exposure levels exceed 20 ppm and 25 ppm respectively.

Standard deviation is a measurement that quantifies the amount of variation or dispersion in a set of data values.
It tells us how much the values in a dataset deviate from the mean, on average.
In a normal distribution, the standard deviation informs us about the width or spread of the distribution.
A smaller standard deviation indicates that the data points tend to be close to the mean, whereas a larger standard deviation means they are spread out over a wider range.

• For example, if you have a small standard deviation, values are tightly packed around the mean (more predictable).
• If you have a large standard deviation, values are spread out (less predictable).

Understanding standard deviation allows us to engage effectively with normal distribution to assess probabilities, as it is one of the key components used in the z-score calculation formula: \[ \sigma = \sqrt{\frac{1}{N-1} \sum_{i=1}^{N}(X_i - \mu)^2} \]Thus, knowing how standard deviation works gives deeper insights into the reliability and spread of your data.