Question

# Spirit Airlines kept track of the number of empty seats on flight $308$ (DEN-DTW) for $10$ consecutive trips on each weekday except Friday. (a) Sort the data for each day. (b) Find the mean, median, mode, midrange, geometric mean, and 10 percent trimmed mean (i.e., dropping the first and last sorted observations) for each day. (c) Do the measures of center agree? Explain. (d) Note strengths or weaknesses of each statistic of center for the data.Monday: $6,1,5,9,1,1,6,5,5,1$Tuesday: $1,3,3,1,4,6,9,7,7,6$Wednesday: $6,0,6,0,6,10,0,0,4,6$Thursday: $1,1,10,1,1,1,1,1,1,1$

Solution

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To find the sample mean, denoted by $\bar{x}$, we use the following formula:

$\bar{x} = \frac{\sum_{i=1}^n x_i}{n}, \tag{1}$

where

• $x_i$ are the $x$-values,
• $n$ is the number of items in the sample.

The median is the number in the middle of a set of numbers if $n$ is odd and the average of the two middle terms' values if $n$ is even. Keep in mind that the data need to be sorted in increasing order.

The mode is the most common number in the data set.

The midrange is defined as the average of the lowest and highest values, i.e.,

$\text{Midrange} = \frac{x_{\text{min}} + x_{\text{max}}}{2}, \tag{2}$

where

• $x_{\text{min}}$ is the lowest value,
• $x_{\text{min}}$ is the highest value.

The geometric mean, denoted by $G$, is a multiplicative average, calculated by multiplying the data values and then taking the $n$th root of the result, i.e.,

$G = \sqrt[n]{x_1 \cdot x_2 \cdots x_n}. \tag{3}$

This can be computed only if all the values are positive.

To calculate the trimmed mean, we remove the highest and lowest $k$ percent of the observations in the sorted data array and then apply the usual mean formula on the rest.