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A normal distribution is a probability distribution that is symmetric around its mean, showing that data near the mean is more frequent in occurrence than data far from the mean. Imagine the shape of a bell curve, gently sloping upwards and downwards from the mean. This bell-shaped curve is a hallmark of normal distribution.

In the context of the problem, our data set is normally distributed with a mean of 30. This means that if you collected a lot of data from this distribution, most of the values would cluster around 30. Additionally, the symmetry means values further away from the mean occur less frequently. This is why normal distribution is so helpful in statistics: it provides a simple model for a wide range of phenomena.

• Mean (\(ar{x}\)): This is the average of all the data points. Here, it's given as 30.
• Symmetry: The distribution is perfectly symmetrical around the mean.
• Bell Shape: Most of the data points lie close to the mean, creating a bell-shaped curve.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation means that the values tend to be close to the mean, while a high standard deviation means that the values are spread out over a wider range.

In our exercise, the standard deviation of the data is 5. This number helps us understand how spread out the numbers are around the mean of 30.

Let's break it down:

• \(\sigma = 5\): Standard deviation means typical variability from the mean. A smaller standard deviation would mean data points are closer to 30.
• Spans of Standard Deviation: Each increment of standard deviation (1σ, 2σ, etc.) represents a step of 5 units away from the mean.
• Practical Utility: It helps in defining intervals such as 25 to 35, 20 to 40, etc., to contain predictable amounts of data as per normal distribution.

The 68-95-99.7 rule, also known as the empirical rule, is a handy guideline for understanding normal distributions. It tells us how much of our data lies within one, two, and three standard deviations from the mean.

Consider these key points:

• 68% of data falls within one standard deviation (between 25 and 35 in this example).
• 95% of data is contained within two standard deviations (from 20 to 40 here).
• 99.7% of data will be within three standard deviations (which covers from 15 to 45 in our given problem).

This rule is significant because it allows us to understand and predict the dispersion of data in a bell-shaped normal distribution simply by knowing the mean and standard deviation. It makes it easy to see how concentrated or spread out the data can be.