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Standard deviation is a fundamental measure in statistics that quantifies the amount of variation or dispersion in a set of data values. In a normal distribution, standard deviation plays a crucial role in understanding the spread of a dataset. An easy way to think about standard deviation is to consider it as a measure of how much individual data points differ from the mean.

• If the standard deviation is small, data points tend to be close to the mean.
• If the standard deviation is large, data points are spread out over a wider range of values.

In the context of a normal distribution, about 68% of data falls within one standard deviation of the mean, approximately 95% within two standard deviations, and about 99.7% within three. This is known as the empirical rule or 68-95-99.7 rule. In our exercise, we use this concept to find that approximately 95.44% of the data lies within two standard deviations of the mean.

Probability is a way to measure the likelihood that an event will occur. It is a key concept in statistics and is essential for dealing with uncertainty. Probability values range from 0 to 1, where:

• 0 indicates an impossible event.
• 1 indicates a certain event.

In our exercise, the task is to find the probability that a random variable lies within two standard deviations of the mean in a normal distribution. Essentially, we're determining how likely it is for data to fall within a specific range. For example, the probability we calculated, 95.44%, tells us how likely it is that a randomly selected value falls within that range. This is done using the Z-table to find probabilities associated with standard normal variables, as completed in the solution.

A Z-table, also known as the standard normal distribution table, is an invaluable tool in statistics for finding probabilities related to the standard normal distribution. This table helps us determine the probability that a standard normal variable is less than or equal to a particular value, a process known as finding the cumulative probability. To use a Z-table, follow these steps:

• Calculate the Z-score, which standardizes a value from any normal distribution to the standard normal distribution.
• Locate the Z-score on the Z-table to find the probability associated with it.

In our example, the Z-table shows that the probability of a Z-score less than 2 is 0.9772, and the probability of a Z-score less than -2 is 0.0228. Subtracting these gives us the probability that a Z-score lies between -2 and 2, corresponding to being within two standard deviations of the mean.

A random variable is a variable that takes on different values based on the outcomes of a random phenomenon. In probability and statistics, random variables are used to quantify outcomes of random processes and are characterized by their probability distributions.For a continuous random variable, like in standard normal distribution, it can take any value within a given range. In our normal distribution example:

• The mean (\( \mu \) ) indicates the central or expected value of the distribution.
• The standard deviation (\( \sigma \) ) indicates how spread out the values are from the mean.

In the exercise, we aim to find the probability that the random variable lies within two standard deviations from the mean. The steps we used include standardizing the variable using Z-scores, which convert any normal random variable to a Z-score, enabling us to use the Z-table for probability calculation.