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The range measures the spread of the data by finding the difference between the largest and smallest values. It gives a quick sense of how spread out the values are. To calculate the range, simply subtract the smallest value from the largest. In our data set, the smallest value is 5 dollars, and the largest value is 2500 dollars.

The range is calculated as:

\( \text{Range} = 2500 - 5 = 2495 \text{ dollars} \).

While the range is easy to calculate, it can be heavily influenced by outliers, making it less reliable for highly variable data sets.

Variance offers insights into how spread out the data points are around the mean. It is the average of the squared differences from the mean. Calculating variance involves several steps:

• First, find the mean.
• Next, compute the squared difference of each data point from the mean.
• Sum these squared differences.
• Finally, divide by the number of data points minus one.

The formula for variance is:

\( s^2 = \frac{\textstyle \frac{\textstyle \textstyle{{\textstyle \textstyle}\textstyle x_i} - \text{mean})^2 }{\textstyle}{n-1} \).

The variance in our data set is approximately 82696.88 dollars squared. Because it uses squared units, comparing variance directly with the original data values can be difficult.

Standard deviation is the square root of variance and provides a measure of spread in the same units as the original data. It makes interpretation easier compared to variance. To find the standard deviation, simply take the square root of the variance.

The formula to calculate standard deviation is:

\( s = \text{sqrt}(s^2) \).

In our data set, the standard deviation is approximately 287.52 dollars. Like the variance, standard deviation gets heavily influenced by outliers, showing greater spread when they are present.

Outliers are data points that are significantly higher or lower than the rest of the data. They can skew the results of various statistical measures, making it essential to identify them. The Interquartile Range (IQR) method is one common way to do this. Calculate the first (Q1) and third quartiles (Q3) to determine the IQR:

\( \text{IQR} = Q3 - Q1 \).

Then, find the outlier thresholds:

• Lower Bound = Q1 - 1.5 * IQR.
• Upper Bound = Q3 + 1.5 * IQR.

Values outside these bounds are considered outliers. In our data set, Q1 is 55 dollars and Q3 is 500 dollars, so the IQR is 445 dollars.

The thresholds are:

\( \text{Lower Bound} = 55 - 1.5 \times 445 = -612.5 \)

and

\( \text{Upper Bound} = 500 + 1.5 \times 445 = 1167.5 \).

Any data point above 1167.5 dollars is an outlier. In our case, 1500 and 2500 dollars are outliers.

The Interquartile Range (IQR) measures the middle 50% of the data. It provides a measure of variability that is more resistant to outliers than the range. Calculating the IQR involves the following steps:

• Find the first quartile (Q1), which is the median of the first half of the data.
• Find the third quartile (Q3), which is the median of the second half of the data.
• Subtract Q1 from Q3 to get the IQR.

In our data set, Q1 is 55 dollars, and Q3 is 500 dollars. Thus, the IQR is:

\( \text{IQR} = 500 - 55 = 445 \text{ dollars} \).

The IQR is particularly useful for identifying outliers and understanding the distribution of the central portion of the data.