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The standard deviation is a measure of how spread out numbers are in a data set. In a normal distribution, it tells us about the average distance from the mean. Imagine you have a bunch of numbers and they all clump closely around an average value. Standard deviation helps you understand if they generally stick close to this mean or if they vary widely. For example, a standard deviation of 6 cm in the context of height indicates that most young American females' heights will fall within 6 cm of the average height of 166 cm.
An important property of the normal distribution is that around 68% of the data will lie within one standard deviation of the mean, while approximately 95% will fall within two standard deviations. This helps in predicting how concentrated the data values are around the mean.

• Standard deviation is denoted by the symbol \( \sigma \).
• In a perfectly normal distribution, the mean, median, and mode are all the same.
• A smaller standard deviation means data is tightly packed around the mean, whereas a larger one indicates more spread out data.

The Z-score is a measure that describes a value's position relative to the mean of a group of values. It is measured in terms of standard deviations from the mean. A Z-score can help determine how unusual or typical a certain value is in the context of a normal distribution. The formula for calculating a Z-score is given by:\[ Z = \frac{X - \mu}{\sigma}\]where \( X \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
For example, with an average height of 166 cm and a standard deviation of 6 cm, a height of 172 cm has a Z-score of 1, calculated as:\[ Z = \frac{172 - 166}{6} = 1\]This tells us that 172 cm is one standard deviation above the mean. Z-scores make it simple to compare different data points within a data set or across different data sets by normalizing the differences.

• Z-scores can be positive or negative.
• Positive Z-scores mean the data point is above the mean, while negative scores indicate it is below the mean.
• A Z-score of 0 indicates the value is exactly at the mean.

Cumulative probability is the probability that a random variable is less than or equal to a certain value. In a normal distribution, finding cumulative probability involves looking at the area under the curve up to that point. This is where Z-scores are used in conjunction with standard normal distribution tables or calculators to find these probabilities.
For instance, when calculating what percentage of heights are greater than 172 cm, we first find the cumulative probability for 172 cm using its Z-score, which was 1. From the standard normal distribution table, we find that \( P(Z \leq 1) = 0.8413 \). This is the cumulative probability or the proportion of the population that is 172 cm or shorter. To get the percentage taller than 172 cm, we calculate \( 1 - 0.8413 = 0.1587 \), meaning approximately 15.87% of females are taller.

• Cumulative probability can be found by integrating the probability distribution function up to a specific value.
• It helps in understanding the probability of a variable falling within a certain range.
• For values beyond the range, subtract the cumulative probability from 1 to find the probability of exceedance.