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When discussing statistics, percentiles are crucial for understanding the distribution of data. A percentile shows the relative standing of a value within a dataset. For instance, the 15th percentile of IQ scores means that 15% of the scores fall below this value.
To compute percentiles in a normal distribution, we use the Z-score, which represents the number of standard deviations a data point is from the mean. Given a normal distribution with a mean of 100 and a standard deviation of 4, finding the Z-score for the 15th percentile involves looking up a standard normal distribution table to find a Z-score, which in this case is approximately -1.04.
The formula to calculate the specific data point corresponding to a percentile is: \[ X = ext{mean} + Z \cdot ext{standard deviation} \] For the 15th percentile: \[ X = 100 + (-1.04) \times 4 = 95.84 \] Therefore, individuals at the 15th percentile have an IQ of approximately 95.84, meaning most people score higher than this level. Conversely, the 98th percentile indicates a score higher than 98% of the dataset.

The Interquartile Range (IQR) is a measure of statistical dispersion and is very useful when analyzing the spread of data. Specifically, it is the range between the 75th percentile (Q3) and the 25th percentile (Q1). The IQR helps in understanding the central spread thus highlighting outliers.
In the IQ model, to find the IQR, we first determine the 75th and 25th percentiles. Using standard normal distribution tables, the Z-score for the 75th percentile is approximately 0.674, while the Z-score for the 25th percentile is -0.674.
Using the formula: \[ X = ext{mean} + Z \cdot ext{standard deviation} \] For the 75th percentile, we have: \[ X_{75} = 100 + 0.674 \times 4 = 102.696 \] For the 25th percentile, it is: \[ X_{25} = 100 - 0.674 \times 4 = 97.304 \] The IQR is then calculated as: \[ IQR = X_{75} - X_{25} = 102.696 - 97.304 = 5.392 \] Thus, the IQR of the IQ scores is approximately 5.39, revealing the spread of the middle 50% of the data.

Standard deviation is a critical concept in statistics that provides insight into the amount of variation or dispersion in a set of data. It essentially measures how spread out the numbers in a dataset are around the mean.
In a normal distribution like the IQ model we are considering, the standard deviation ( \(\sigma\) ) is the square root of the variance. This model has a mean of 100 and a variance of 16; therefore, the standard deviation is: \[ \sigma = \sqrt{16} = 4 \] Each standard deviation "step" around the mean represents an interval that includes around 68% of the data for one step (known as the empirical rule). This makes standard deviation a vital tool for predicting the outcomes within a dataset.
Understanding standard deviation helps with interpreting how typical or atypical a particular score, like an IQ score, is when compared to the average score. If an IQ score falls beyond two standard deviations from the mean in either direction, it is considered relatively unusual.