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The sample mean is a fundamental statistical measure used to determine the central tendency of a data set. When you have a series of numbers, like our piston ring measurements, the sample mean gives you a single value that represents the average of those numbers. To calculate it, add up all the values and then divide by the number of values.

Here's the formula for the sample mean:

• Compute the sum of all measurements.
• Divide by the number of measurements.

The result is called the sample mean, represented by \( \bar{x} \). In our case, the sample mean of piston ring diameters was calculated as follows:

• Add: 74.001 + 74.003 + 74.015 + 74.000 + 74.005 + 74.002 + 74.005 + 74.004 = 592.035
• Divide: \( \frac{592.035}{8} = 74.004375 \)

Thus, the sample mean is 74.004375 mm. It's a useful measure because it gives us a straightforward, simplified overview of the data's center.

The sample standard deviation is another important statistical tool that measures the amount of variation or dispersion in a data set. In simple terms, it tells us how spread out the measurements are from their average value (the sample mean). A smaller standard deviation indicates that the measurements are closely clustered around the mean, while a larger one suggests more spread.

Here’s how to compute it:

• Determine the deviation of each measurement from the sample mean.
• Square each deviation to eliminate negative values.
• Sum all squared deviations.
• Divide this sum by one less than the number of measurements (\( n-1 \)), a step known as the degrees of freedom.
• Take the square root of the result to get the standard deviation.

For our exercise, the squared deviations were calculated and summed to 0.00009375. Dividing by 7 (since there are 8 measurements, \( n-1 = 7 \)), we have:

• \( s = \sqrt{\frac{0.00009375}{7}} \approx 0.00366 \) mm

Such a small standard deviation suggests that our piston ring measurements are very consistent, showing minimal variability among them.

A dot diagram, or dot plot, is a simple and effective way to visualize the frequency of each value in a data set. It consists of a simple number line where each measurement is represented by a dot above its value. When a value is repeated, dots are stacked vertically.

This visualization helps in quickly identifying clusters, gaps, peaks, and outliers. It’s particularly helpful for smaller data sets, as it gives a clear, visual representation of the distribution.

In our exercise with the piston ring diameters, you would:

• Draw a horizontal number line covering the range of the data set.
• Place a dot above the number line for each measurement. Repeat for duplicate values, like 74.005.

To comment on the data, you can see the values are clustered near the mean, with no extreme outliers, indicating stable manufacturing quality. The dot plot shows how often each specific measurement was reached, allowing a quick visual snapshot of data distribution.