91Ó°ÊÓ

A z-score is a useful concept in statistics primarily because it tells us how far a particular data point is from the mean in terms of the standard deviation. Think of it as a way to standardize scores on a common scale. To calculate a z-score, use the formula:
\[ z = \frac{X - \mu}{\sigma} \]
Where:

• \(X\) is the data point in question, in this case, the inches of rainfall.
• \(\mu\) represents the mean, or average, rainfall.
• \(\sigma\) is the standard deviation of the dataset.

The z-score helps us understand the probability related to a data point. For instance, knowing the z-score allows you to use a table known as the standard normal distribution table to see how common or rare that data point is compared to the dataset overall.
It converts a value to a score that can be quickly interpreted, helping us to see if 5 inches of rain (or another amount) is a common occurrence or an outlier.

Calculating probability in a normal distribution involves determining the likelihood of different outcomes. Once you have the z-score, you can use the standard normal distribution table to find probabilities. These tables show the probability of a random variable falling to the left of a given z-score.
Here’s how you can find the probability related to a certain amount of rainfall:

• First, calculate the z-score for the rainfall you're interested in (for example, whether it will exceed 5 inches).
• Then, look up that z-score in the standard normal distribution table to find the probability up to that point.
• For probabilities that need to show more than a certain z-score, subtract the table value from 1.

For example, the table tells us that a z-score of 1.875 corresponds to a probability of 0.9693. This means that 96.93% of the time, the rainfall is less than 5 inches. To find the probability that it rains more than 5 inches, subtract this value from 1.

The standard deviation is a key concept in understanding the spread or variability of a dataset. In a normal distribution, it helps determine how much variation exists from the average (mean).
Here's what you need to know about standard deviation:

• It's denoted by \(\sigma\) and calculated as the square root of the variance.
• The larger the standard deviation, the more spread out the data points are from the mean.
• In the context of this problem, it tells us how variable the precipitation is over several Aprils.

A small standard deviation suggests that the data points are close to the mean, while a large standard deviation indicates that they are spread over a wider range. This intrinsic property is what allows the normal distribution to be defined efficiently.
For instance, if the standard deviation is 0.8 inches, it means most April rainfalls are within 0.8 inches above or below the average of 3.5 inches.