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The concept of range is one of the simplest measures of variation. It tells us the spread of the data set by calculating the difference between the highest and lowest values. In the context of our exercise, the annual costs of tuition and fees reported are: 49,138, 47,890, 47,510, 47,343, 46,962, 46,944, 46,930, 46,902, 46,870, and 46,785 dollars. To find the range, we subtract the smallest value (46,785 dollars) from the largest value (49,138 dollars):
Range = 49,138 - 46,785 = 2,353 dollars.
This indicates that there is a 2,353 dollar difference between the most and least expensive tuitions among these colleges.

Variance measures the average squared differences from the mean, giving us a sense of how spread out the values are in a data set. For our data set, the colleges' tuitions are re-ordered as follows: 46,785, 46,870, 46,902, 46,930, 46,944, 46,962, 47,343, 47,510, 47,890, and 49,138 dollars.
First, we calculate the mean (average), \(\bar{x}\), of these values:
\[ \bar{x} = \frac{46,785 + 46,870 + 46,902 + 46,930 + 46,944 + 46,962 + 47,343 + 47,510 + 47,890 + 49,138}{10} = 47,327.4 \text{ dollars} \]
Next, we find the squared differences from the mean for each data point, sum those squared differences, and divide by one less than the number of data points (since we are dealing with a sample).
\[ \text{Variance} (\sigma^2) = \frac{(46,785 - 47,327.4)^2 + (46,870 - 47,327.4)^2 + \text{...} + (49,138 - 47,327.4)^2}{10 - 1} = 615,759.82 \text{ dollars}^2 \]
This variation value gives us an overall measure of how much the tuitions differ from the mean.

While variance gives us a useful measure of data spread, it is in squared units which can be difficult to interpret with the original data. This is where the standard deviation comes in handy as it is in the same units as the original data.
Standard deviation is the square root of the variance.
\[ \sigma = \sqrt{615,759.82} = 784.69 \text{ dollars} \]
Hence, the standard deviation for our data set of tuition costs is 784.69 dollars. This means that, on average, the tuition costs deviate from the mean by about 784.69 dollars, providing a more intuitive understanding of the variation in tuition costs for these top 10 expensive colleges.