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The Z-score is a statistical measure that indicates how many standard deviations a data point is from the mean. It is used to standardize data points within a normal distribution, allowing comparison between different datasets. The formula for calculating the Z-score is:
\[ Z = \frac{X - \mu}{\sigma} \]
where:

• \(X\) is the data point
• \(\mu\) is the mean of the dataset
• \(\sigma\) is the standard deviation

By calculating the Z-score, we transform a data point into a standardized value that represents its position within a normal distribution, which simplifies probability calculations and comparisons.

Probability is the measure of the likelihood that an event will occur, ranging from 0 to 1. In the context of a normal distribution, it represents the area under the curve for a given range of values. For instance, finding the probability that a pregnancy lasts at least 240 days involves calculating the area from 240 onwards under the normal distribution curve.
This probability helps in understanding the proportion of a dataset falling within specific boundaries. In continuous distributions, the probability of a single exact value is zero since there are infinitely many possible values between any two values in a continuous range.

Standard deviation is a measure reflecting the amount of variation or dispersion in a dataset. It indicates how spread out the numbers in a dataset are in relation to the mean. In a normal distribution, most data points are within one standard deviation of the mean. Mathematically, standard deviation is represented as:
\[ \sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(X_i - \mu)^2} \]
where:

• \(N\) is the number of observations
• \(X_i\) is each individual observation
• \(\mu\) is the mean of the dataset

A smaller standard deviation indicates that data points tend to be closer to the mean, whereas a larger one suggests they are spread over a wider range.

Continuous distributions describe variables that can take on an infinite number of possible values within a given range. Unlike discrete distributions, they deal with measurements rather than counts. A normal distribution is a classic example where the values can be any real number along a continuum.
A key characteristic of continuous distributions is that probabilities are calculated over intervals, since the probability of obtaining any specific value is technically zero. Intervals can be specified in ranges such as \(P(X \geq 240)\), meaning the probability that the variable \(X\) is greater than or equal to 240.

• These distributions are often represented by a probability density function (PDF) which shows the relative likelihood for a variable.
• The area under the curve of a PDF for a continuous distribution sums up to 1, representing the total probability space.

Understanding continuous distributions is crucial for working with data that naturally exist on a continuum, such as time, temperature, and in this case, pregnancy length.