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Normal distribution, often referred to as the bell curve due to its characteristic shape, is a fundamental concept in statistics that describes how data is distributed around a mean or average value. In the case of Chicago's winter daily temperatures, the data is said to be normally distributed around the mean temperature of 28 degrees. This implies that the temperature values are more likely to fall closer to 28 degrees and gradually less likely as they move further away from the mean.

As a property of a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This is known as the empirical rule or the 68-95-99.7 rule, which is vital for understanding the spread of data in relation to the mean.

Standard deviation serves as a gauge of variability in a dataset. In essence, it quantifies how spread out the values are in relation to the mean. A smaller standard deviation indicates that the data points are clustered closely around the mean, whereas a larger standard deviation signifies more spread.

For the Chicago winter temperature example, the standard deviation is 8 degrees, meaning that most of the winter daily temperatures are expected to fall within a range of 8 degrees above or below the mean of 28 degrees. Understanding the standard deviation helps us predict how temperature values are scattered and anticipate the likelihood of extreme weather days.

The area under the curve in the context of normal distribution represents the probability of a value falling within a certain range. Since the total area under the normal distribution curve equals 100%, each segment of the curve represents a portion of that total probability.

To determine what percentage of winter days in Chicago have a daily temperature of 35 degrees or warmer, we use the area to the right of the corresponding Z-score on the curve. The area to the left represents all the days with temperatures below 35 degrees. By finding the complement of this area, we're left with the percentage of days that meet or exceed the 35 degrees mark.

A standard normal distribution table, also known as a Z-table, is a reference tool used to find the probability (area under the curve) for a Z-score in a standard normal distribution. The Z-score is a standardized score that tells us how many standard deviations away a particular value is from the mean.

In practical terms, once the Z-score is calculated, such as 0.875 for the 35-degree temperature in Chicago, you look up this Z-score in the table to find the corresponding probability. The table usually lists the area to the left of a Z-score. Therefore, to find the probability of days warmer than 35 degrees, we subtract this value from 100% as demonstrated in the exercise solution.