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In statistics, the term _"mean"_ represents the average value of a set of numbers. In the context of a uniform distribution, where every outcome is equally likely, the mean gives us a central value around which the data is distributed.
For data that is uniformly distributed between two values, the mean can be calculated using the straightforward formula:

• \[ \mu = \frac{a + b}{2} \]

where \( a \) is the minimum value, and \( b \) is the maximum value.
This formula works because it weights each possible outcome equally, essentially finding the midpoint of the distribution.
Using the example of lengths ranging from 590.0 mm to 590.9 mm, the mean length is calculated as follows:

• First, we add the two boundary values: 590.0 + 590.9 = 1180.9.
• Next, we divide by 2 to find the average: \[ \mu = \frac{1180.9}{2} = 590.45 \]

This result, 590.45 mm, represents the mean of the distribution, offering a central point that characterizes the entire range of uniformly distributed lengths.

Variance is a measure that tells us how much the values in a set differ from the mean. For a uniform distribution, calculating variance helps us understand the spread of the data across the distribution's range.
The specific formula for variance \( \sigma^2 \) in a uniform distribution is:

• \[ \sigma^2 = \frac{(b - a)^2}{12} \]

Here, \( a \) and \( b \) represent the minimum and maximum values of the distribution, respectively.
The term \((b - a)^2\) gives us the squared range of the distribution, and dividing by 12 accounts for the equal likelihood of all data points, spreading the range evenly.
In our glass plate examples, substituting the values 590.0 mm and 590.9 mm into the formula gives us:

• First, find the range: \( 590.9 - 590.0 = 0.9 \).
• Square the range: \( 0.9^2 = 0.81 \).
• Divide by 12 to find the variance: \[ \sigma^2 = \frac{0.81}{12} = 0.0675 \]

Thus, the variance is 0.0675 \( \text{mm}^2 \), indicating how closely packed the lengths are around the mean.

When data is said to be uniformly distributed, this means each value within a specified range is equally probable. Essentially, no outcome within the range is more or less likely than any other.
Uniform distribution is often visualized as a flat, horizontal line in a probability distribution graph, where the height of the line is constant.
In real-world applications, uniform distributions can occur in scenarios as diverse as the rolling of a fair die or the measurement of consistently manufactured products.
For example, if you are gauging the lengths of glass plates measured every tenth of a millimeter from 590.0 to 590.9, each potential value like 590.0, 590.1, and so on up to 590.9 is measured with equal likelihood.
This means:

• There is no favoritism in the spread of data points.
• Each interval within the given range holds the same significance.
• The distribution creates a scenario where any random observation within the range has the same chance of selection.

Understanding this even disbursement can be useful in many practical and theoretical contexts, providing a straightforward model for calculations such as mean and variance.