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The mean is a way to find the average of a set of numbers. To calculate the mean, we first sum up all the data points and then divide by the number of data points. This gives us a single representative value for the entire dataset.
In the exercise, we are given the masses of 12 quarters, and the sum of these masses is 66.100 grams. By dividing this sum by 12, which is the total number of quarters, we arrive at the mean value:
\[ \text{Mean} = \frac{66.100}{12} = 5.5083 \]
This mean tells us that, on average, each quarter weighs approximately 5.5083 grams. The mean is useful for providing an overall picture of what each value would be if they were all the same.

The median is the middle value in a list of numbers. It represents the point at which half of the numbers are above and half are below. To find the median, the data must first be arranged in increasing order.
For the given data sorted as: 5.536, 5.539, 5.548, 5.549, 5.551, 5.552, 5.552, 5.554, 5.620, 5.632, 5.683, 5.684, we observe that there are 12 numbers.
When the dataset has an even number of observations, as in this case, the median is the average of the two middle numbers. Here, the 6th and 7th numbers are both 5.552.
Thus, the median is calculated as:
\[ \text{Median} = \frac{5.552 + 5.552}{2} = 5.552 \]
This median indicates that it divides the dataset into two equal halves, providing insight into the distribution of the data.

The range is a measure of how spread out the values in a dataset are. It is the difference between the highest and lowest values, giving a sense of the dispersion in the data.
For the masses of the quarters, the maximum value is 5.684 grams, and the minimum value is 5.536 grams. By subtracting the minimum from the maximum, we find the range:
\[ \text{Range} = 5.684 - 5.536 = 0.148 \]
A smaller range indicates that the values are closely clustered together, while a larger range would suggest more variability. In this case, the range is quite small, implying that the masses of the quarters are similar to one another.

Standard deviation and variance are statistical measures that reveal how much the values in a data set deviate from the mean.

• Variance quantifies the average of the squared deviations from the mean. It highlights the extent of variability within the dataset.
• Standard deviation, being the square root of variance, provides a more intuitive measure of dispersion in the same units as the data itself.

To compute the variance for the quarters' masses, first calculate each data point's deviation from the mean (5.5083), square these deviations, and then average them:
Sum of squared differences from the mean is 0.101571. Dividing by the number of observations, the variance is:
\[ \text{Variance} = \frac{0.101571}{12} \approx 0.008464 \]
Taking the square root of the variance gives the standard deviation:
\[ \text{Standard Deviation} = \sqrt{0.008464} \approx 0.092 \]
These metrics show the consistency in the quarter masses, with a smaller standard deviation signifying that the data points are closely clustered around the mean.