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Describing Data Using Numerical Measuresinformation contained in a set of data. To make your descriptive toolkit complete, you need to become familiar with key descriptive measures that are widely used to fully describe data.
You will need to combine the graphical tools discussed in Chapter 2 with the numerical measures introduced in this chapter.
3.1 Measures of Center and Location
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Chapter 3 99TABLE 3.1 San Carlo Hotel Data
Week Rooms Rented Revenue Complaints
1 22 $1,870 0
2 13 $1,590 2
3 10 $1,760 1
4 16 $2,345 0
5 23 $4,563 2
6 13 $1,630 1
7 11 $2,156 0
8 13 $1,756 0
x1 = Total number of rooms rented
x2 = Total dollar revenue from the room rentals
x3 = Number of customer complaints that came from guests each Sunday
These data are shown in Table 3.1. They are a population because they include all data that interest the owner.
Figure 3.1 shows the frequency histogram for the number of rooms rented. If the manager wants to describe the data further, she can locate the center of the data by finding the balance point for the histogram. Think of the horizontal axis as a plank and the histogram bars as weights proportional to their area. The center of the data would be the point at which the plank would balance. As shown in Figure 3.1, the balance point seems to be about 15 rooms.
Eyeing the histogram might yield a reasonable approximation of the center. However, computing a numerical measure of the center directly from the data is preferable. The most frequently used measure of the center is the mean. The population mean for number of rooms rented is computed using Equation 3.1 as follows:
m = ax
N = 22 + 13 + 10 + 16 + 23 + 13 + 11 + 13 8
= 121 8 m = 15.125
Thus, the average number of rooms rented on Sunday for the past eight weeks is 15.125.
This is the true balance point for the data. Take a look at Table 3.2, where we calculate what is called a deviation 1xi - m2 by subtracting the mean from each value, xi.
The Excel 2016 function for the mean is
=Average122,13,10,16,23, 13,11,132
Number of Occurrences
5 to 10 11 to 15 16 to 20 21 to 25 Approximate Balance Point
Rooms Rented 5
4 3 2 1 0 FIGURE 3.1 Balance Point,
Rooms Rented at San Carlo Hotel
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Describing Data Using Numerical MeasuresNote that the sum of the deviations of the data from the mean is zero. This is not a coincidence. For any set of data, the sum of the deviations around the mean will be zero.
TABLE 3.2 Deviations Around the Mean Using Hotel Data x (x−µ) = Deviation
22 22 - 15.125 = 6.875
13 13 - 15.125 = -2.125 10 10 - 15.125 = -5.125
16 16 - 15.125 = 0.875
23 23 - 15.125 = 7.875
13 13 - 15.125 = -2.125 11 11 - 15.125 = -4.125 13 13 - 15.125 = -2.125
g1x-m2 = 0.000dSum of deviations equals zero.
EXAMPLE 3-1
Computing the Population Mean
United Airlines As the airline industry becomes increas- ingly competitive, in an effort to increase profits, many airlines are reducing flights. Therefore the supply of idled airplanes has increased. United Airlines, headquartered in Chicago, has decided to expand its fleet. Suppose United selects additional planes from a list of 17 possible planes, including such models as the Boeing 747-400, the Air Bus 300-B4, and the DC 9-10. At a recent meeting, the chief operating officer asked a mem- ber of his staff to determine the mean fuel consumption rate per hour of operation for the population of 17 planes.
s t e p 1 Collect data for the quantitative variable of interest.
The staff member was able to determine, for each of the 17 planes, the hourly fuel consumption in gallons for a flight between Chicago and New York City.
These data are recorded as follows:
Airplane Fuel Consumption
1 gal>hr 2
B747-400 3,529
L-1011-100/200 2,215
DC-10-10 2,174
A300-B4 1,482
A310-300 1,574
B767-300 1,503
B767-200 1,377
B757-200 985
B727-200 1,249
MD-80 882
B737-300 732
DC-9-50 848
B727-100 806
B737-100/200 1,104
F-100 631
DC-9-30-11 804
DC-9-10 764
HOW TO DO IT (Example 3-1)
Computing the Population Mean (when the available data constitute the population of interest)
1. Collect the data for the varia- ble of interest for all items in the population. The data must be quantitative.
2. Sum all values in the population 1Σ x2.
3. Divide the sum 1Σ x2. by the number of values (N) in the population to get the population mean. The formula for the population mean is
m = ax N
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Chapter 3 101s t e p 2 Add the data values.
ax = 3 ,5 2 9 + 2 ,2 1 5 + 2 ,1 7 4 + c + 7 6 4 = 2 2 ,6 5 9
s t e p 3 Divide the sum by the number of values in the population using Equation 3.1.
m = ax
N = 22,659
17 = 1,332.9
The mean number of gallons of fuel consumed per hour on these 17 planes is 1,332.9.
BUSINESS APPLICATION
Population Mean
The San Carlo Hotel (continued ) In addition to collecting data on the number of rooms rented on Sunday nights, the San Carlo Hotel manager collected data on the room-rental revenue generated and the number of complaints on Sunday nights. Excel can quite easily be used to calculate numerical measures such as the mean. Because these data are the popu- lation of all nights of interest to the hotel manager, she can compute the population mean, m, revenue per night. The population mean is m = +2,208.75, as shown in the Excel output in Figure 3.2. Likewise, the mean number of complaints is m = 0.75 per night. (Note that other measures are shown in the figure. We will discuss several of these later in the chapter.)
Now, for these eight Sunday nights, the manager can report to the hotel’s owner that the mean number of rooms rented is 15.13 (rounded up from 15.125). This level of busi- ness generated a mean nightly revenue of $2,208.75. The number of complaints averaged 0.75 (less than 1) per night. These values are the true means for the population and are, therefore, called parameters.
Excel Tutorial
Excel 2016 Instructions 1. Open file: San Carlo Hotel.
2. Select the Data tab.
3. Click on Data Analysis + Descriptive Statistics.
4. Define data range for the desired variables.
5. Check Summary Statistics.
6. Name new Output Sheet.
7. On Home tab, adjust decimal places as desired.
FIGURE 3.2 Excel 2016 Output Showing Mean Revenue for the San Carlo Hotel
Mean rooms rented =15.13 Mean revenue = $2,208.75 Mean complaints = 0.75