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The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetrical around its mean, representing a common pattern for the way data is naturally distributed in many situations. This distribution is characterized by its familiar bell-shaped curve and is determined by two parameters: the mean \(\mu\) and the standard deviation \(\sigma\). In our case with IQ scores, mean \(\mu = 100\) represents the average IQ score, and the graph of the normal distribution would be centered at this value.

The curve's spread is indicated by the standard deviation, which measures how much individual data points deviate from the mean. The larger the standard deviation, the more spread out the data is. The normal distribution has the property that approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three. This is referred to as the Empirical Rule, which is fundamental for interpreting data in fields such as psychology, finance, and many natural and social sciences.

Standard deviation is a crucial concept in statistics, representing the amount of variation or dispersion from the average. In simpler terms, it indicates how spread out the data points in a data set are. When the standard deviation is low, it suggests that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range.

For our Stanford-Binet IQ scores exercise, the given standard deviation is \(\sigma = 15\). This means that the typical deviation of children's IQ scores from the average score (100) is 15 points. As standard deviation is a key part of finding probabilities in a normal distribution, understanding its value allows us to establish ranges for expected outcomes and to compute the likelihood of various results.

The Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. Essentially, it is a way of standardizing scores on the same scale to compare them in a meaningful way.

To calculate the Z-score, we subtract the mean from the individual data point and then divide the result by the standard deviation \( Z = \frac{X - \mu}{\sigma} \). In the context of our exercise, the Z-score for an IQ of 115 is calculated to be 1, meaning that an IQ of 115 is one standard deviation above the mean IQ score of 100. The Z-score helps us understand where an individual score lies in the distribution—whether it's above, below, or at the mean—and it subsequently allows us to calculate the probability of a randomly selected child having an IQ score within certain bounds.

Probability calculation in the context of normal distribution involves finding the likelihood of a data point falling within a particular range. Here, we're specifically interested in probabilistic outcomes under the curve of the bell-shaped graph of the normal distribution. Once we have the Z-score, we can use it to calculate the probability by looking up the corresponding value on a standard normal distribution table, often called a Z-table, which gives us the area to the left of the Z-score.

For instance, a Z-score of 1 corresponds to the 84.13th percentile, which means there's an 84.13% chance that a randomly selected child will have an IQ score below 115. By employing the Z-table, we transform the Z-score into a more understandable percentage or probability, concluding our statistical investigation and providing insight into the distribution of outcomes within our dataset.