Data

BUSINESS AND ECONOMICS 11e

CHAPTER 1 CHAPTER 1

1.2 Data

Dataare the facts and figures collected, analyzed, and summarized for presentation and in- terpretation. All the data collected in a particular study are referred to as the data setfor the study. Table 1.1 shows a data set containing information for 25 mutual funds that are part of the Morningstar Funds500for 2008. Morningstar is a company that tracks over 7000 mutual funds and prepares in-depth analyses of 2000 of these. Their recommendations are followed closely by financial analysts and individual investors.

Elements, Variables, and Observations

Elementsare the entities on which data are collected. For the data set in Table 1.1 each in- dividual mutual fund is an element: the element names appear in the first column. With 25 mutual funds, the data set contains 25 elements.

Avariableis a characteristic of interest for the elements. The data set in Table 1.1 in- cludes the following five variables:

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Fund Type: The type of mutual fund, labeled DE (Domestic Equity), IE (Interna- tional Equity), and FI (Fixed Income)

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Net Asset Value ($): The closing price per share on December 31, 2007

1.2 Data 5

5-Year Expense

Fund Net Asset Average Ratio Morningstar

Fund Name Type Value ($) Return (%) (%) Rank

American Century Intl. Disc IE 14.37 30.53 1.41 3-Star

American Century Tax-Free Bond FI 10.73 3.34 0.49 4-Star

American Century Ultra DE 24.94 10.88 0.99 3-Star

Artisan Small Cap DE 16.92 15.67 1.18 3-Star

Brown Cap Small DE 35.73 15.85 1.20 4-Star

DFA U.S. Micro Cap DE 13.47 17.23 0.53 3-Star

Fidelity Contrafund DE 73.11 17.99 0.89 5-Star

Fidelity Overseas IE 48.39 23.46 0.90 4-Star

Fidelity Sel Electronics DE 45.60 13.50 0.89 3-Star

Fidelity Sh-Term Bond FI 8.60 2.76 0.45 3-Star

Gabelli Asset AAA DE 49.81 16.70 1.36 4-Star

Kalmar Gr Val Sm Cp DE 15.30 15.31 1.32 3-Star

Marsico 21st Century DE 17.44 15.16 1.31 5-Star

Mathews Pacific Tiger IE 27.86 32.70 1.16 3-Star

Oakmark I DE 40.37 9.51 1.05 2-Star

PIMCO Emerg Mkts Bd D FI 10.68 13.57 1.25 3-Star

RS Value A DE 26.27 23.68 1.36 4-Star

T. Rowe Price Latin Am. IE 53.89 51.10 1.24 4-Star

T. Rowe Price Mid Val DE 22.46 16.91 0.80 4-Star

Thornburg Value A DE 37.53 15.46 1.27 4-Star

USAA Income FI 12.10 4.31 0.62 3-Star

Vanguard Equity-Inc DE 24.42 13.41 0.29 4-Star

Vanguard Sht-Tm TE FI 15.68 2.37 0.16 3-Star

Vanguard Sm Cp Idx DE 32.58 17.01 0.23 3-Star

Wasatch Sm Cp Growth DE 35.41 13.98 1.19 4-Star

Source: Morningstar Funds500(2008).

TABLE 1.1 DATA SET FOR 25 MUTUAL FUNDS

file

WEB

Morningstar

Data sets such as Morningstar are available on the website for this text.

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5-Year Average Return (%):The average annual return for the fund over the past 5 years

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Expense Ratio:The percentage of assets deducted each fiscal year for fund expenses

•

Morningstar Rank:The overall risk-adjusted star rating for each fund; Morningstar ranks go from a low of 1-Star to a high of 5-Stars

Measurements collected on each variable for every element in a study provide the data. The set of measurements obtained for a particular element is called an observation. Referring to Table 1.1 we see that the set of measurements for the first observation (American Cen- tury Intl. Disc) is IE, 14.37, 30.53, 1.41, and 3-Star. The set of measurements for the sec- ond observation (American Century Tax-Free Bond) is FI, 10.73, 3.34, 0.49, and 4-Star, and so on. A data set with 25 elements contains 25 observations.

Scales of Measurement

Data collection requires one of the following scales of measurement: nominal, ordinal, interval, or ratio. The scale of measurement determines the amount of information con- tained in the data and indicates the most appropriate data summarization and statistical analyses.

When the data for a variable consist of labels or names used to identify an attribute of the element, the scale of measurement is considered a nominal scale. For example, refer- ring to the data in Table 1.1, we see that the scale of measurement for the Fund Type vari- able is nominal because DE, IE, and FI are labels used to identify the category or type of fund. In cases where the scale of measurement is nominal, a numeric code as well as non- numeric labels may be used. For example, to facilitate data collection and to prepare the data for entry into a computer database, we might use a numeric code by letting 1 denote Domestic Equity, 2 denote International Equity, and 3 denote Fixed Income. In this case the numeric values 1, 2, and 3 identify the category of fund. The scale of measurement is nominal even though the data appear as numeric values.

The scale of measurement for a variable is called an ordinal scale if the data ex- hibit the properties of nominal data and the order or rank of the data is meaningful. For example, Eastside Automotive sends customers a questionnaire designed to obtain data on the quality of its automotive repair service. Each customer provides a repair service rating of excellent, good, or poor. Because the data obtained are the labels—excellent, good, or poor—the data have the properties of nominal data. In addition, the data can be ranked, or ordered, with respect to the service quality. Data recorded as excellent indi- cate the best service, followed by good and then poor. Thus, the scale of measurement is ordinal. As another example, note that the Morningstar Rank for the data in Table 1.1 is ordinal data. It provides a rank from 1 to 5-Stars based on Morningstar’s assessment of the fund’s risk-adjusted return. Ordinal data can also be provided using a numeric code, for example, your class rank in school.

The scale of measurement for a variable is an interval scaleif the data have all the properties of ordinal data and the interval between values is expressed in terms of a fixed unit of measure. Interval data are always numeric. Scholastic Aptitude Test (SAT) scores are an example of interval-scaled data. For example, three students with SATmath scores of 620, 550, and 470 can be ranked or ordered in terms of best performance to poorest performance.

In addition, the differences between the scores are meaningful. For instance, student 1 scored 620⫺550⫽70 points more than student 2, while student 2 scored 550⫺470⫽80 points more than student 3.

The scale of measurement for a variable is a ratio scaleif the data have all the prop- erties of interval data and the ratio of two values is meaningful. Variables such as dis- tance, height, weight, and time use the ratio scale of measurement. This scale requires that a zero value be included to indicate that nothing exists for the variable at the zero point.

For example, consider the cost of an automobile. A zero value for the cost would indicate that the automobile has no cost and is free. In addition, if we compare the cost of $30,000 for one automobile to the cost of $15,000 for a second automobile, the ratio property shows that the first automobile is $30,000/$15,000⫽2 times, or twice, the cost of the second automobile.

Categorical and Quantitative Data

Data can be classified as either categorical or quantitative. Data that can be grouped by specific categories are referred to as categorical data. Categorical data use either the nom- inal or ordinal scale of measurement. Data that use numeric values to indicate how much or how many are referred to as quantitative data. Quantitative data are obtained using either the interval or ratio scale of measurement.

Acategorical variableis a variable with categorical data, and a quantitative variable is a variable with quantitative data. The statistical analysis appropriate for a particular vari- able depends upon whether the variable is categorical or quantitative. If the variable is categorical, the statistical analysis is limited. We can summarize categorical data by count- ing the number of observations in each category or by computing the proportion of the observations in each category. However, even when the categorical data are identified by a numerical code, arithmetic operations such as addition, subtraction, multiplication, and division do not provide meaningful results. Section 2.1 discusses ways for summarizing categorical data.

Arithmetic operations provide meaningful results for quantitative variables. For exam- ple, quantitative data may be added and then divided by the number of observations to compute the average value. This average is usually meaningful and easily interpreted. In general, more alternatives for statistical analysis are possible when data are quantitative.

Section 2.2 and Chapter 3 provide ways of summarizing quantitative data.

Cross-Sectional and Time Series Data

For purposes of statistical analysis, distinguishing between cross-sectional data and time series data is important.Cross-sectional dataare data collected at the same or approxi- mately the same point in time. The data in Table 1.1 are cross-sectional because they de- scribe the five variables for the 25 mutual funds at the same point in time.Time series data are data collected over several time periods. For example, the time series in Figure 1.1 shows the U.S. average price per gallon of conventional regular gasoline be- tween 2006 and 2009. Note that higher gasoline prices have tended to occur in the sum- mer months, with the all-time-high average of $4.05 per gallon occurring in July 2008.

By January 2009, gasoline prices had taken a steep decline to a three-year low of $1.65 per gallon.

Graphs of time series data are frequently found in business and economic publications.

Such graphs help analysts understand what happened in the past, identify any trends over time, and project future levels for the time series. The graphs of time series data can take on a variety of forms, as shown in Figure 1.2. With a little study, these graphs are usually easy to understand and interpret.

For example, Panel (A) in Figure 1.2 is a graph that shows the Dow Jones Industrial Average Index from 1997 to 2009. In April 1997, the popular stock market index was near 7000. Over the next 10 years the index rose to over 14,000 in July 2007. However, notice the sharp decline in the time series after the all-time high in 2007. By March 2009, poor economic conditions had caused the Dow Jones Industrial Average Index to return to the 7000 level of 1997. This was a scary and discouraging period for investors. By June 2009, the index was showing a recovery by reaching 8700.

1.2 Data 7

The statistical method appropriate for

summarizing data depends upon whether the data are categorical or quantitative.

The graph in Panel (B) shows the net income of McDonald’s Inc. from 2003 to 2009. The declining economic conditions in 2008 and 2009 were actually beneficial to McDonald’s as the company’s net income rose to an all-time high. The growth in McDonald’s net income showed that the company was thriving during the economic downturn as people were cutting back on the more expensive sit-down restaurants and seeking less-expensive alternatives offered by McDonald’s.

Panel (C) shows the time series for the occupancy rate of hotels in South Florida over a one-year period. The highest occupancy rates, 95% and 98%, occur during the months of February and March when the climate of South Florida is attractive to tourists. In fact, January to April of each year is typically the high-occupancy season for South Florida hotels. On the other hand, note the low occupancy rates during the months of August to October, with the lowest occupancy rate of 50% occurring in September. High tempera- tures and the hurricane season are the primary reasons for the drop in hotel occupancy during this period.

Date

Mar 06 Oct 06 Apr 07 Nov 07 Jun 08 Dec 08 Jul 09

Average Price per Gallon

$0

$0.50

$1.00

$1.50

$2.00

$2.50

$3.00

$3.50

$4.00

$4.50

FIGURE 1.1 U.S. AVERAGE PRICE PER GALLON FOR CONVENTIONAL REGULAR GASOLINE

Source:Energy Information Administration, U.S. Department of Energy, May 2009.

NOTES AND COMMENTS

1. An observation is the set of measurements ob- tained for each element in a data set. Hence, the number of observations is always the same as the number of elements. The number of measure- ments obtained for each element equals the num- ber of variables. Hence, the total number of data items can be determined by multiplying the num- ber of observations by the number of variables.

2. Quantitative data may be discrete or continu- ous. Quantitative data that measure how many (e.g., number of calls received in 5 minutes) are discrete. Quantitative data that measure how much (e.g., weight or time) are continuous be- cause no separation occurs between the possi- ble data values.

FIGURE 1.2 A VARIETY OF GRAPHS OF TIME SERIES DATA

1.2 Data 9

Month

Percentage Occupied

20 40 60 80 100

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

(C) Occupancy Rate of South Florida Hotels Year

1998 2000 2002 2004 2006 2008 2010

Dow Jones Industrial Average

5000 6000 7000 8000 9000 10000 11000 13000 12000 14000

(A) Dow Jones Industrial Average

4

2 5

0 3

1 6

Net Income ($ billions)

Year

2003 2004 2005 2006 2007 2008 2009

(B) Net Income for McDonald’s Inc.