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The term "mean" refers to the average value of a set of numbers. In the context of a normal distribution, the mean, denoted as \(\mu\), is a measure that represents the central tendency of the distribution. In simpler terms, it indicates where the bulk of the numbers lie, or the center of the curve of the distribution.
To calculate the mean of a data set, sum all the values and then divide by the total number of values. Mathematically, if you have a dataset \(X = \{x_1, x_2, \dots, x_n\}\), the mean \(\mu\) is given by: \[ \mu = \frac{x_1 + x_2 + \cdots + x_n}{n} \] In a normal distribution, the mean not only helps describe the distribution's central point but also acts as an axis of symmetry. This means that the distribution curve is ideally the same on both sides of the mean. If we say \(X \sim N(3, 5)\), this indicates that the average value of \(X\) is 3, placing the center of our curve at 3. This is critical for understanding and predicting outcomes based on the distribution.

Variance is a measure of how spread out the numbers in a data set are. For a normal distribution, it is denoted by \(\sigma^2\). Variance tells us how much the values in the distribution deviate from the mean. A low variance indicates that the values are clustered close to the mean, whereas a high variance indicates that the values are spread out over a wider range.
To calculate variance, follow these steps:

• Find the mean \(\mu\).
• Subtract the mean from each value to find the deviation of each value from the mean.
• Square each deviation.
• Find the average of these squared deviations.This results in the variance:

\[ \sigma^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \cdots + (x_n - \mu)^2}{n} \] In our context, \(X \sim N(3, 5)\) gives us a variance of 5. This means that the data points are spread around the mean with a certain level of dispersion. In practical terms, this tells us about the amount of uncertainty or variability in the data.

Standard deviation is another way to measure the spread of a distribution. It is the square root of variance and is denoted with the symbol \(\sigma\). Because the standard deviation is in the same unit as the data set, it provides a clearer sense of the spread of values.

Calculating the standard deviation involves these steps:

• Take the variance \(\sigma^2\).
• Calculate its square root to find \(\sigma\).

In formula terms:\[\sigma = \sqrt{\sigma^2}\]Thus, for our exercise, we have \(X \sim N(3, 5)\) with \(\sigma^2 = 5\). Therefore, the standard deviation \(\sigma\) is:\[\sigma = \sqrt{5} \]This numerical value tells us the average distance of each data point from the mean. Standard deviation is especially useful when comparing the spread between different data sets, enabling a deeper understanding of the variability present.