Chapter 2 Data Collection Mc. Graw-Hill/Irwin Data Vocabulary Level of Measurement Time Series and Cross-sectional Data Sampling Concepts Sampling Methods Data Sources Survey Research © 2008 The Mc. Graw-Hill Companies, Inc. All rights reserved.
2 -2 Data Vocabulary • Data is the plural form of the Latin datum (a “given” fact). • In scientific research, data arise from experiments whose results are recorded systematically. • In business, data usually arise from accounting transactions or management processes. • Important decisions may depend on data.
2 -3 Data Vocabulary ¯ Subjects, Variables, Data Sets • We will refer to Data as plural and data set as a particular collection of data as a whole. • Observation – each data value. • Subject (or individual) – an item for study (e. g. , an employee in your company). • Variable – a characteristic about the subject or individual (e. g. , employee’s income).
2 -4 Data Vocabulary ¯ Subjects, Variables, Data Sets • Three types of data sets: Data Set Variables Typical Tasks Univariate One Histograms, descriptive statistics, frequency tallies Bivariate Two Scatter plots, correlations, simple regression Multivariate More than two Multiple regression, data mining, econometric modeling
2 -5 Data Vocabulary ¯ Subjects, Variables, Data Sets Consider the multivariate data set with 5 variables 8 subjects 5 x 8 = 40 observations
2 -6 Data Vocabulary ¯ Data Types • A data set may have a mixture of data types. Types of Data Attribute (qualitative) Verbal Label Coded X = economics X=3 (your major) (i. e. , economics) Numerical (quantitative) Discrete X=2 (your siblings) Continuous X = 3. 15 (your GPA)
2 -7 Data Vocabulary ¯ Attribute Data • Also called categorical, nominal or qualitative data. • Values are described by words rather than numbers. • For example, - Automobile style (e. g. , X = full, midsize, compact, subcompact). - Mutual fund (e. g. , X = load, no-load).
2 -8 Data Vocabulary ¯ Data Coding • Coding refers to using numbers to represent categories to facilitate statistical analysis. • Coding an attribute as a number does not make the data numerical. • For example, 1 = Bachelor’s, 2 = Master’s, 3 = Doctorate • Rankings may exist, for example, 1 = Liberal, 2 = Moderate, 3 = Conservative
2 -9 Data Vocabulary ¯ Binary Data • A binary variable has only two values, 1 = presence, 0 = absence of a characteristic of interest (codes themselves are arbitrary). • For example, 1 = employed, 0 = not employed 1 = married, 0 = not married 1 = male, 0 = female 1 = female, 0 = male • The coding itself has no numerical value so binary variables are attribute data.
2 -10 Data Vocabulary ¯ Numerical Data • Numerical or quantitative data arise from counting or some kind of mathematical operation. • For example, - Number of auto insurance claims filed in March (e. g. , X = 114 claims). - Ratio of profit to sales for last quarter (e. g. , X = 0. 0447). • Can be broken down into two types – discrete or continuous data.
2 -11 Data Vocabulary ¯ Discrete Data • A numerical variable with a countable number of values that can be represented by an integer (no fractional values). • For example, - Number of Medicaid patients (e. g. , X = 2). - Number of takeoffs at O’Hare (e. g. , X = 37).
2 -12 Data Vocabulary ¯ Continuous Data • A numerical variable that can have any value within an interval (e. g. , length, weight, time, sales, price/earnings ratios). • Any continuous interval contains infinitely many possible values (e. g. , 426 < X < 428).
2 -13 Level of Measurement ¯ Four levels of measurement for data: Level of Measurement Characteristics Example Nominal Categories only Eye color (blue, brown, green, hazel) Ordinal Rank has meaning Bond ratings (Aaa, Aab, C, D, F, etc. ) Interval Distance has meaning Temperature (57 o Celsius) Ratio Meaningful zero exists Accounts payable ($21. 7 million)
2 -14 Level of Measurement ¯ Nominal Measurement • Nominal data merely identify a category. • Nominal data are qualitative, attribute, categorical or classification data (e. g. , Apple, Compaq, Dell, HP). • Nominal data are usually coded numerically, codes are arbitrary (e. g. , 1 = Apple, 2 = Compaq, 3 = Dell, 4 = HP). • Only mathematical operations are counting (e. g. , frequencies) and simple statistics.
2 -15 Level of Measurement ¯ Ordinal Measurement • Ordinal data codes can be ranked (e. g. , 1 = Frequently, 2 = Sometimes, 3 = Rarely, 4 = Never). • Distance between codes is not meaningful (e. g. , distance between 1 and 2, or between 2 and 3, or between 3 and 4 lacks meaning). • Many useful statistical tests exist for ordinal data. Especially useful in social science, marketing and human resource research.
2 -16 Level of Measurement ¯ Interval Measurement • Data can not only be ranked, but also have meaningful intervals between scale points (e. g. , difference between 60 F and 70 F is same as difference between 20 F and 30 F). • Since intervals between numbers represent distances, mathematical operations can be performed (e. g. , average). • Zero point of interval scales is arbitrary, so ratios are not meaningful (e. g. , 60 F is not twice as warm as 30 F).
2 -17 Level of Measurement ¯ Likert Scales • A special case of interval data frequently used in survey research. • The coarseness of a Likert scale refers to the number of scale points (typically 5 or 7). “College-bound high school students should be required to study a foreign language. ” (check one) Strongly Agree Somewhat Agree Neither Agree Nor Disagree Somewhat Disagree Strongly Disagree
2 -18 Level of Measurement ¯ Likert Scales • A neutral midpoint (“Neither Agree Nor Disagree”) is allowed if an odd number of scale points is used or omitted to force the respondent to “lean” one way or the other. • Likert data are coded numerically (e. g. , 1 to 5) but any equally spaced values will work. Likert coding: 1 to 5 scale Likert coding: -2 to +2 scale 5 = Help a lot 4 = Help a little 3 = No effect 2 = Hurt a little 1 = Hurt a lot +2 = Help a lot +1 = Help a little 0 = No effect 1 = Hurt a little 2 = Hurt a lot
2 -19 Level of Measurement ¯ Likert Scales • Careful choice of verbal anchors results in measurable intervals (e. g. , the distance from 1 to 2 is “the same” as the interval, say, from 3 to 4). • Ratios are not meaningful (e. g. , here 4 is not twice 2). • Many statistical calculations can be performed (e. g. , averages, correlations, etc. ).
2 -20 Level of Measurement ¯ Likert Scales • More variants of Likert scales: How would you rate your marketing instructor? (check one) Terrible Poor Adequate Good Excellent How would you rate your marketing instructor? (check one) Very Bad Very Good
2 -21 Level of Measurement ¯ Ambiguity • Grades are usually coded numerically (A = 4, B = 3, C = 2, D = 1, F = 0) and are used to calculate a mean GPA. • Is the interval from 3. 0 to 4. 0 really the same as the interval from 1. 0 to 2. 0? • What is the underlying reality ranging from 0 to 4 that we are measuring? • Best to be conservative and limit statistical tests to those for ordinal data.
2 -22 Level of Measurement ¯ Ratio Measurement • Ratio data have all properties of nominal, ordinal and interval data types and also possess a meaningful zero (absence of quantity being measured). • Because of this zero point, ratios of data values are meaningful (e. g. , $20 million profit is twice as much as $10 million). • Zero does not have to be observable in the data, it is an absolute reference point.
2 -23 Level of Measurement ¯ Use the following procedure to recognize data types: Question If “Yes” Q 1. Is there a meaningful zero point? Ratio data (all statistical operations are allowed) Q 2. Are intervals between scale points meaningful? Interval data (common statistics allowed, e. g. , means and standard deviations) Q 3. Do scale points represent rankings? Ordinal data (restricted to certain types of nonparametric statistical tests) Q 4. Are there discrete categories? Nominal data (only counting allowed, e. g. finding the mode)
2 -24 Level of Measurement ¯ Changing Data by Recoding • In order to simplify data or when exact data magnitude is of little interest, ratio data can be recoded downward into ordinal or nominal measurements (but not conversely). • For example, recode systolic blood pressure as “normal” (under 130), “elevated” (130 to 140), or “high” (over 140). • The above recoded data are ordinal (ranking is preserved) but intervals are unequal and some information is lost.
2 -25 Time Series and Cross-sectional Data ¯ Time Series Data • Each observation in the sample represents a different equally spaced point in time (e. g. , years, months, days). • Periodicity may be annual, quarterly, monthly, weekly, daily, hourly, etc. • We are interested in trends and patterns over time (e. g. , annual growth in consumer debit card use from 1999 to 2006).
2 -26 Time Series and Cross-sectional Data ¯ Cross-sectional Data • Each observation represents a different individual unit (e. g. , person) at the same point in time (e. g. , monthly VISA balances). • We are interested in - variation among observations or in - relationships. • We can combine the two data types to get pooled cross-sectional and time series data.
2 -27 Sampling Concepts ¯ Sample or Census? • A sample involves looking only at some items selected from the population. • A census is an examination of all items in a defined population. • Why can’t the United States Census survey every person in the population? - Mobility - Illegal immigrants - Budget constraints - Incomplete responses or nonresponses
2 -28 Sampling Concepts Situations Where A Sample May Be Preferred: Infinite Population No census is possible if the population is infinite or of indefinite size (an assembly line can keep producing bolts, a doctor can keep seeing more patients). Destructive Testing The act of sampling may destroy or devalue the item (measuring battery life, testing auto crashworthiness, or testing aircraft turbofan engine life). Timely Results Sampling may yield more timely results than a census (checking wheat samples for moisture and protein content, checking peanut butter for aflatoxin contamination).
2 -29 Sampling Concepts Situations Where A Sample May Be Preferred: Accuracy Sample estimates can be more accurate than a census. Instead of spreading limited resources thinly to attempt a census, our budget of time and money might be better spent to hire experienced staff, improve training of field interviewers, and improve data safeguards. Cost Even if it is feasible to take a census, the cost, either in time or money, may exceed our budget. Sensitive Information Some kinds of information are better captured by a well-designed sample, rather than attempting a census. Confidentiality may also be improved in a carefully-done sample.
2 -30 Sampling Concepts Situations Where A Census May Be Preferred Small Population If the population is small, there is little reason to sample, for the effort of data collection may be only a small part of the total cost. Large Sample Size If the required sample size approaches the population size, we might as well go ahead and take a census. Database Exists If the data are on disk we can examine 100% of the cases. But auditing or validating data against physical records may raise the cost. Legal Requirements Banks must count all the cash in bank teller drawers at the end of each business day. The U. S. Congress forbade sampling in the 2000 decennial population census.
2 -31 Sampling Concepts ¯ Parameters and Statistics • Statistics are computed from a sample of n items, chosen from a population of N items. • Statistics can be used as estimates of parameters found in the population. • Symbols are used to represent population parameters and sample statistics.
2 -32 Sampling Concepts ¯ Parameters and Statistics Parameter or Statistic? Parameter Any measurement that describes an entire population. Usually, the parameter value is unknown since we rarely can observe the entire population. Parameters are often (but not always) represented by Greek letters. Statistic Any measurement computed from a sample. Usually, the statistic is regarded as an estimate of a population parameter. Sample statistics are often (but not always) represented by Roman letters.
2 -33 Sampling Concepts ¯ Parameters and Statistics • The population must be carefully specified and the sample must be drawn scientifically so that the sample is representative. ¯ Target Population • The target population is the population we are interested in (e. g. , U. S. gasoline prices). • The sampling frame is the group from which we take the sample (e. g. , 115, 000 stations). • The frame should not differ from the target population.
2 -34 Sampling Concepts ¯ Finite or Infinite? • A population is finite if it has a definite size, even if its size is unknown. • A population is infinite if it is of arbitrarily large size. • Rule of Thumb: A population may be treated as infinite when N is at least 20 times n (i. e. , when N/n > 20) N Here, N/n > 20 n
2 -35 Sampling Methods Probability Samples Simple Random Sample Use random numbers to select items from a list (e. g. , VISA cardholders). Systematic Sample Select every kth item from a list or sequence (e. g. , restaurant customers). Stratified Sample Select randomly within defined strata (e. g. , by age, occupation, gender). Like stratified sampling except strata are geographical areas (e. g. , zip codes). Cluster Sample
2 -36 Sampling Methods Nonprobability Samples Judgment Sample Use expert knowledge to choose “typical” items (e. g. , which employees to interview). Convenience Sample Use a sample that happens to be available (e. g. , ask co-worker opinions at lunch).
2 -37 Sampling Methods ¯ Simple Random Sample • Every item in the population of N items has the same chance of being chosen in the sample of n items. • We rely on random numbers to select a name. =RANDBETWEEN(1, 48)
2 -38 Sampling Methods ¯ Random Number Tables • A table of random digits used to select random numbers between 1 and N. • Each digit 0 through 9 is equally likely to be chosen. ¯ Setting Up a Rule • For example, Nil. Co wants to award cash prizes to 10 of its 875 loyal customers. • To get 10 three-digit numbers between 001 and 875, we define any consistent rule for moving through the random number table.
2 -39 Sampling Methods ¯ Setting Up a Rule • Randomly point at the table to choose a starting point. • Choose the first three digits of the selected fivedigit block, move to the right one column, down one row, and repeat. • When we reach the end of a line, wrap around to the other side of the table and continue. • Discard any number greater than 875 and any duplicates.
Start Here 2 -40 Table of 1, 000 Random Digits 82134 14458 66716 54269 31928 46241 03052 00260 32367 25783 07139 16829 76768 11913 42434 91961 92934 18229 15595 02566 45056 43939 31188 43272 11332 99494 19348 97076 95605 28010 10244 19093 51678 63463 85568 70034 82811 23261 48794 63984 12940 84434 50087 20189 58009 66972 05764 10421 36875 64964 84438 45828 40353 28925 11911 53502 24640 96880 93166 68409 98681 67871 71735 64113 90139 33466 65312 90655 75444 30845 43290 96753 18799 49713 39227 15955 46167 63853 03633 19990 96893 85410 88233 22094 30605 79024 01791 38839 85531 94576 75403 41227 00192 16814 47054 16814 81349 92264 01028 29071 78064 92111 51541 76563 69027 67718 06499 71938 17354 12680 26246 71746 94019 93165 96713 03316 75912 86209 12081 57817 98766 67312 96358 21351 86448 31828 86113 78868 67243 06763 37895 51055 11929 44443 15995 72935 99631 18190 85877 31309 27988 81163 52212 25102 61798 28670 01358 60354 74015 18556 19216 53008 44498 19262 12196 93947 90162 76337 12646 26838 28078 86729 69438 24235 35208 48957 53529 76297 41741 54735 34455 61363 93711 68038 75960 16327 95716 66964 28634 65015 53510 90412 70438 45932 57815 75144 52472 61817 41562 42084 30658 18894 88208 97867 30737 94985 18235 02178 39728 66398
2 -41 Sampling Methods ¯ With or Without Replacement • If we allow duplicates when sampling, then we are sampling with replacement. • Duplicates are unlikely when n is much smaller than N. • If we do not allow duplicates when sampling, then we are sampling without replacement.
2 -42 Sampling Methods ¯ Computer Methods Excel - Option A Enter the Excel function =RANDBETWEEN(1, 875) into 10 spread-sheet cells. Press F 9 to get a new sample. Excel - Option B Enter the function =INT(1+875*RAND()) into 10 spreadsheet cells. Press F 9 to get a new sample. Internet The web site www. random. org will give you many kinds of excellent random numbers (integers, decimals, etc). Minitab Use Minitab’s Random Data menu with the Integer option. These are pseudo-random generators because even the best algorithms eventually repeat themselves.
2 -43 Sampling Methods ¯ Randomizing a List • In Excel, use function =RAND() beside each row to create a column of random numbers between 0 and 1. • Copy and paste these numbers into the same column using “Paste Special | Values” (to paste only the values and not the formulas). • Sort the spreadsheet on the random number column.
2 -44 Sampling Methods ¯ Randomizing a List • The first n items are a random sample of the entire list (they are as likely as any others).
2 -45 Sampling Methods ¯ Systematic Sampling • Sample by choosing every kth item from a list, starting from a randomly chosen entry on the list. • For example, starting at item 2, we sample every k = 4 items to obtain a sample of n = 20 items from a list of N = 78 items. • Note that N/n = 78/20 4.
2 -46 Sampling Methods ¯ Systematic Sampling • A systematic sample of n items from a population of N items requires that periodicity k be approximately N/n. • Systematic sampling should yield acceptable results unless patterns in the population happen to recur at periodicity k. • Can be used with unlistable or infinite populations. • Systematic samples are well-suited to linearly organized physical populations.
2 -47 Sampling Methods ¯ Systematic Sampling • For example, out of 501 companies, we want to obtain a sample of 25. What should the periodicity k be? k = N/n = 501/25 20. • So, we should choose every 20 th company from a random starting point.
2 -48 Sampling Methods ¯ Stratified Sampling • Utilizes prior information about the population. • Applicable when the population can be divided into relatively homogeneous subgroups of known size (strata). • A simple random sample of the desired size is taken within each stratum. • For example, from a population containing 55% males and 45% females, randomly sample 120 males and 80 females (n = 200).
2 -49 Sampling Methods ¯ Stratified Sampling • Or, take a random sample of the entire population and then combine individual strata estimates using appropriate weights. • For a population with L strata, the population size N is the sum of the stratum sizes: N = N 1 + N 2 +. . . + NL • The weight assigned to stratum j is wj = Nj / n • For example, take a random sample of n = 200 and then weight the responses for males by w. M =. 55 and for females by w. F =. 45.
2 -50 Sampling Methods ¯ Cluster Sample • Strata consist of geographical regions. • One-stage cluster sampling – sample consists of all elements in each of k randomly chosen subregions (clusters). • Two-stage cluster sampling, first choose k subregions (clusters), then choose a random sample of elements within each cluster.
2 -51 Sampling Methods ¯ Cluster Sample • Here is an example of 4 elements sampled from each of 3 randomly chosen clusters (two-stage cluster sampling).
2 -52 Sampling Methods ¯ Cluster Sample • Cluster sampling is useful when - Population frame and stratum characteristics are not readily available - It is too expensive to obtain a simple or stratified sample - The cost of obtaining data increases sharply with distance - Some loss of reliability is acceptable
2 -53 Sampling Methods ¯ Judgment Sample • A nonprobability sampling method that relies on the expertise of the sampler to choose items that are representative of the population. • Can be affected by subconscious bias (i. e. , nonrandomness in the choice). • Quota sampling is a special kind of judgment sampling, in which the interviewer chooses a certain number of people in each category.
2 -54 Sampling Methods ¯ Convenience Sample • Take advantage of whatever sample is available at that moment. A quick way to sample. ¯ Sample Size • Sample size depends on the inherent variability of the quantity being measured and on the desired precision of the estimate.
2 -55 Data Sources ¯ Useful Data Sources Type of Data Examples U. S. general data U. S. economic data Statistical Abstract of the U. S. Economic Report of the President Almanacs Periodicals World Almanac, Time Almanac Economist, Business Week, Fortune Indexes Databases New York Times, Wall Street Journal Compu. Stat, Citibase, U. S. Census World data Web CIA World Factbook Google, Yahoo, msn
2 -56 Survey Research ¯ Basic Steps of Survey Research • Step 1: State the goals of the research • Step 2: Develop the budget (time, money, staff) • Step 3: Create a research design (target population, frame, sample size) • Step 4: Choose a survey type and method of administration
2 -57 Survey Research ¯ Basic Steps of Survey Research • Step 5: Design a data collection instrument (questionnaire) • Step 6: Pretest the survey instrument and revise as needed • Step 7: Administer the survey (follow up if needed) • Step 8: Code the data and analyze it
2 -58 Survey Research ¯ Survey Types Type of Survey Characteristics Mail You need a well-targeted and current mailing list (people move a lot). Low response rates are typical and nonresponse bias is expected (nonrespondents differ from those who respond). Zip code lists (often costly) are an attractive option to define strata of similar income, education, and attitudes. To encourage participation, a cover letter should clearly explain the uses to which the data will be put. Plan for follow-up mailings.
2 -59 Survey Research ¯ Survey Types Type of Survey Characteristics Telephone Random dialing yields very low response and is poorly targeted. Purchased phone lists help reach the target population, though a low response rate still is typical (disconnected phones, caller screening, answering machines, work hours, no-call lists). Other sources of nonresponse bias include the growing number of non-English speakers and distrust caused by scams and spams.
2 -60 Survey Research ¯ Survey Types Type of Survey Characteristics Interviewing is expensive and time-consuming, yet a trade-off between sample size for high-quality results may still be worth it. Interviews must be carefully handled so interviewers must be welltrained – an added cost. But you can obtain information on complex or sensitive topics (e. g. , gender discrimination in companies, birth control practices, diet and exercise habits).
2 -61 Survey Research ¯ Survey Types Type of Survey Characteristics Web surveys are growing in popularity, but are subject to nonresponse bias because those who participate may differ from those who feel too busy, don’t own computers or distrust your motives (scams and spam are again to blame). This type of survey works best when targeted to a well-defined interest group on a question of self-interest (e. g. , views of CPAs on new proposed accounting rules, frequent flyer views on airline security).
2 -62 Survey Research ¯ Survey Types Type of Survey Characteristics Direct Observation This can be done in a controlled setting (e. g. , psychology lab) but requires informed consent, which can change behavior. Unobtrusive observation is possible in some nonlab settings (e. g. , what percentage of airline passengers carry on more than two bags, what percentage of SUVs carry no passengers, what percentage of drivers wear seat belts).
2 -63 Survey Research ¯ Survey Guidelines Plan What is the purpose of the survey? Consider staff expertise, needed skills, degree of precision, budget. Design Invest time and money in designing the survey. Use books and references to avoid unnecessary errors. Quality Take care in preparing a quality survey so that people will take you seriously.
2 -64 Survey Research ¯ Survey Guidelines Pilot Test Buy-in Pretest on friends or co-workers to make sure the survey is clear. Improve response rates by stating the purpose of the survey, offering a token of appreciation or paving the way with endorsements. Expertise Work with a consultant early on.
2 -65 Survey Research ¯ Getting Advice • Consider hiring a consultant in the early stages. • Many resources are available to help - The American Statistical Association - The Research Industry Coalition - The Council of American Survey Research Organizations
2 -66 Survey Research ¯ Questionnaire Design • Use a lot of white space in layout. • Begin with short, clear instructions. • State the survey purpose. • Assure anonymity. • Instruct on how to submit the completed survey.
2 -67 Survey Research ¯ Questionnaire Design • Break survey into naturally occurring sections. • Let respondents bypass sections that are not applicable (e. g. , “if you answered no to question 7, skip directly to Question 15”). • Pretest and revise as needed. • Keep as short as possible.
2 -68 Survey Research ¯ Questionnaire Design Type of Question Example Open-ended question Briefly describe your job goals. Fill-in-the-blank How many times did you attend formal religious services during the last year? ____ times Check boxes Which of these statistics packages have you ever used? SAS Visual Statistics SPSS Mega. Stat Systat Minitab
2 -69 Survey Research ¯ Questionnaire Design Type of Question Example Ranked choices “Please evaluate your dining experience” Excellent Good Fair Poor Food Service Ambiance Cleanliness Overall
2 -70 Survey Research ¯ Questionnaire Design Type of Question Example Pictograms “What do you think of the President’s economic policies? ” (circle one) Likert scale Statistics is a difficult subject. Strongly Agree Slightly Agree Neither Agree Nor Slightly Strongly Disagree
2 -71 Survey Research ¯ Question Wording • The way a question is asked has a profound influence on the response. For example, 1. Shall state taxes be cut? 2. Shall state taxes be cut, if it means reducing highway maintenance? 3. Shall state taxes be cut, it is means firing teachers and police?
2 -72 Survey Research ¯ Question Wording • Make sure you have covered all the possibilities. For example, Are you married? Yes No • Overlapping classes or How old is your father? unclear categories are a 35 – 45 problem. For example, 45 – 55 – 65 or older
2 -73 Survey Research ¯ Coding and Data Screening • Responses are usually coded numerically (e. g. , 1 = male 2 = female). • Missing values are typically denoted by special characters (e. g. , blank, “. ” or “*”). • Discard questionnaires that are flawed or missing many responses. • Watch for multiple responses, outrageous or inconsistent replies or range answers. • Follow-up if necessary and always document your data-coding decisions.
2 -74 Survey Research ¯ Sources of Error Source of Error Characteristics Nonresponse bias Respondents differ from nonrespondents Selection bias Self-selected respondents are atypical Response error Respondents give false information Coverage error Incorrect specification of frame or population Responses influenced by interviewer Interviewer error Measurement error Sampling error Survey instrument wording is biased or unclear Random and unavoidable
2 -75 Survey Research ¯ Data File Format • Enter data into a spreadsheet or database as a “flat file” (n subjects x m variables matrix).
2 -76 Survey Research ¯ Advice on Copying Data • Using commas (, ), dollar signs ($), or percents (%) as part of the values may result in your data being treated as text values. • A numerical variable may only contain the digits 0 -9, a decimal point, and a minus sign. • To avoid round-off errors, format the data column as plain numbers with the desired number of decimal places before you copy the data to a statistical package.
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