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Standard deviation is a fundamental concept in statistics that measures the amount of variation or dispersion in a set of values. It is symbolized by the Greek letter sigma (\( \text{σ} \)). Essentially, it tells us how spread out the numbers in a data set are from their average value, known as the mean. To put it simply, a smaller standard deviation indicates that the values are close to the mean, while a larger standard deviation suggests that the values are more spread out.

In the context of a normal distribution, standard deviation defines the width of the curve. For a normal distribution, about 68% of the data fall within one standard deviation from the mean, approximately 95% within two standard deviations, and nearly 99.7% within three standard deviations. This is often referred to as the Empirical Rule or the 68-95-99.7 rule.

Understanding standard deviation is crucial because it gives context to the mean, allowing one to understand variability within the data set.

A z-score is a numerical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. The calculation of a z-score is done by subtracting the mean from the value in question and then dividing that result by the standard deviation. The formula is represented as:
\[ Z = \frac{{X - \mu}}{{\sigma}} \]
where \(X\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. Z-scores allow us to determine how unusual or usual a value is within a data set; the further away from 0 a z-score is, the more unusual it is.

In standard normal distribution, which has a mean of 0 and a standard deviation of 1, z-scores are particularly valuable. They enable us to compare scores from different normal distributions, as they are now on the same scale, and to calculate probabilities using the standard normal distribution table, which is a critical aspect in statistics to find out how likely a particular occurrence is.

The probability density function (PDF) is used to specify the probability of a random variable falling within a particular range of values. In essence, it is a function that describes the likelihood for a continuous random variable to take on a particular value. For the normal distribution, the PDF is represented by the bell-shaped curve – it is symmetric around the mean and decays rapidly as we move away from the center.

The area under the curve of a PDF in a given interval represents the probability that the random variable falls within that interval. Therefore, when looking for the probability that a random value is between two points, such as between 90 and 110 in exam scores, we would shade the region under the curve between these two values. This shaded area corresponds directly to the probability we are interested in finding.

In practice, to find the probability associated with specific values on a normal distribution, we often use z-scores to convert the values and then use the standard normal distribution table if the distribution is not standard (i.e., not a mean of 0 and a standard deviation of 1). The probability density function is thus integral to understanding how likely different outcomes are within a distribution, which is vital for hypothesis testing, forecasting, decision making, and many other statistical analyses.