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The mean, also known as the average, is a measure of central tendency. It is calculated by adding up all the values in a data set and then dividing by the number of values. In our problem, we added costs of marriage proposal packages which totaled 9608 dollars. With 25 values in the data set, the mean cost comes out to be approximately 384.32 dollars.
The formula used is: \[\text{Mean} = \frac{\text{Sum}}{\text{Number of Values}}\] The mean offers a quick glimpse into the 'average' cost but it can be significantly affected by very high or very low values in the data set. This makes it very sensitive to outliers.

The median is the middle value in a data set when the numbers are listed in ascending order. For an odd number of values, it is the middle value. For an even number, it is the average of the two middle values. In our data set of 25 values, the middle (13th) value when sorted is 200 dollars.
The median is calculated as follows:
1. Arrange the data in ascending order.
2. Identify the middle value.
The median provides a better measure of central tendency in skewed data sets, as it is not affected by outliers like the mean.

The mode is the value that appears most frequently in a data set. In this exercise, we found that 50 and 500 dollars each appear 4 times, making them both modes of the data set. When a data set has two modes, it is described as bimodal. The mode helps identify the most common value(s) in the data set, offering insight into frequent trends.
To determine the mode:
1. Count the frequency of each value.
2. Identify the value(s) with the highest frequency.
Mode is particularly useful in categorical data where you wish to know the most common category.

The midrange is calculated as the average of the maximum and minimum values in a data set. It is defined by the formula: \[\text{Midrange} = \frac{\text{Min} + \text{Max}}{2}\] For our data, the minimum value is 39, and the maximum value is 2500. Thus, the midrange is: \[\text{Midrange} = \frac{39 + 2500}{2} = 1269.5\] While the midrange offers a quick snapshot of the data range, it is highly sensitive to extreme values and does not offer a robust measure of central tendency.

Outliers are values in a data set that are significantly higher or lower than most other values. They can distort statistical analyses, so it's crucial to identify them. We use the Interquartile Range (IQR) method to detect outliers. The steps are:
1. Calculate Q1 (the 25th percentile) and Q3 (the 75th percentile). In this data set, Q1 is 75 and Q3 is 500.
2. Calculate the IQR: \[\text{IQR} = Q3 - Q1 = 500 - 75 = 425\]
3. Values below \[\text{Q1} - 1.5 \times \text{IQR} = 75 - 1.5 \times 425 = -562.5\] and above \[\text{Q3} + 1.5 \times \text{IQR} = 500 + 1.5 \times 425 = 1137.5\] are considered outliers.
For our data, values 1500 and 2500 are identified as outliers. Identifying outliers helps to refine data analysis and ensures more accurate statistical interpretations.