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Understanding (normal distribution) helps in interpreting data precisely. For a set of data, a z-score quantifies how many standard deviations an element is away from the mean. It's calculated using the formula \[ z = \frac{X - \mu}{\sigma} \]where:

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

In simple terms, a z-score provides a way to compare different data sets. Imagine you took two different exams, and your teacher told you your test score was two standard deviations above the average on both. You'll know you did equally well compared to those who took each test.Z-scores are extremely useful for understanding probabilities. A positive z-score indicates a value above the mean, negative indicates below. This index helps in detecting outliers or comparing scores across different distributions with potentially different means and standard deviations.

The standard deviation (SD) is a statistical measure of variability or diversity, capturing how much individual observations differ from the mean. In essence, SD is the spread of a set of numbers. In a normal distribution, roughly 68% of the data lies within 1 standard deviation (\( +1, -1 \)) of the mean. Approximately 95% sits within 2 SDs (\( +2, -2 \)), and about 99.7% is covered within 3 SDs (\( +3, -3 \)). This is commonly referred to as the 68-95-99.7 rule.Why does this matter?

• If the SD is low, the values are close to the mean, showing consistency.
• If the SD is high, the values are spread out, indicating variability.

By understanding SD, you can see how varied your scores in the IQ test are compared to others. Smaller standard deviation implies scores are concentrated tightly around the mean score, while a larger one would show a wider span of scores.

Probability calculation in the context of the standard normal distribution involves determining the likelihood of a specific range of outcomes. For our normal distribution, with a mean of 100 and a standard deviation of 15, calculating probabilities becomes intuitive with z-scores.In this exercise, we sought to find the probability that an IQ score falls between the z-scores of -2 and 2.
To do this, we look at each z-score in the standard normal distribution table:

• Z = -2: Converts to a probability of approximately 0.0228.
• Z = 2: Converts to a probability of approximately 0.9772.

The probability for the range between these two z-scores is calculated by subtracting the probability of -2 from the probability of 2:\[ P(-2 < Y < 2) = P(z < 2) - P(z < -2) \]\[ = 0.9772 - 0.0228 = 0.9544 \] Meaning, there's a 95.44% chance a score will fall between these z-scores. This helps in understanding how typical or atypical an observation is, showing most scores hover around the average in the distribution.