91Ó°ÊÓ

The mean is an average that tells us the central value of a dataset. In the case of a **uniform distribution** with discrete values like the glass lengths, calculating the mean is straightforward. We simply sum all the possible values and divide by the number of values.
The sum of all lengths from 590.0 mm to 590.9 mm is essentially a sequence of 10 measurements. Using the formula for the mean of a uniform distribution for discrete values:

• Mean, \( \bar{x} = \frac{\sum x_i}{n} \)
• Where \( x_i \) are the individual values, and \( n \) is the total number of values

Plugging the values from our problem, we emit 590.45 mm as the average length. This mean gives us a clear point of comparison for understanding the dataset's balance and central tendency.

Variance helps us understand the spread or dispersion of a set of values around the mean. For **uniform distributions**, it measures how each value differs from the average. The formula for variance in a discrete uniform distribution is slightly adapted:

• Variance, \( \sigma^2 = \frac{(b-a+1)^2 - 1}{12} \)
• Where \( a \) and \( b \) represent the endpoints of our data range

In the context of glass lengths from 590.0 to 590.9 mm, the variance calculation results in 0.1508 mm². This measure allows us to grasp the level of variation within the dataset, informing us how much individual measurements differ from each other and the mean. The lower the variance, the closer the lengths are to each other and the mean.

In statistics, **discrete values** refer to data that can only take on certain specific values. These values are often countable, and in our example, they are lengths measured in tenths of millimeters, ranging from 590.0 to 590.9 mm.
Each of these measurements is taken as it stands without any possibilities in between, thus creating 10 distinct data points. Working with discrete values makes certain statistical calculations, like mean and variance, straightforward because each value stands alone. This kind of understanding is crucial when dealing with datasets that consist of specific, individual units like our glass lengths.

**Statistical methods** provide the frameworks and tools to analyze data sets effectively. When dealing with data such as uniform distributions, these methods allow us to quantify summaries of data sets, such as the mean and variance. Through these calculations, we derive insights into data’s central tendency and variability.
Using statistical methods with uniform distributions facilitates clear decision-making and understanding, which can be pivotal in fields ranging from quality control to research. By applying formulas to calculate essential statistics, we transform raw data into useful insights that can guide businesses and scientific endeavors.
In our glass length scenario, the mean provides a yardstick to gauge the central point, while variance offers clues about consistency across the measurements. These statistical tools form the backbone of interpreting data trends and patterns in real-world applications.