Data Preparation Steps in Data Preparation Editing Coding Entering Data Data Tabulation Reviewing Tabulations Statistically adjusting the data (e.g. weighting) Editing Carefully checking survey data for Completeness (no omissions) Non-ambiguous (e.g. two boxes checked instead of one) Right informant (e.g. under age, when all supposed to be over 18) Consistency e.g. charging something on a credit card when the person does not own a credit card Accuracy (e.g. no numbers out of range) Most important purpose is to eliminate or at least reduce the number of errors in the raw data. Solutions 1. Ideally re-interview respondent 2. Eliminate all unacceptable surveys (case wise deletion) (if sample is large and few unacceptable) 3. In calculations only the cases with complete responses are considered (pair wise deletion) (means that some statistics will be based on different sample sizes) 4. Code illegible or missing answers into a a “no valid response” category 5. substitute a neutral value - typically the mean response to the variable, therefore the mean remains unchanged Coding • The process of systematically and consistently assigning each response a numerical score. • The key to a good coding system is for the coding categories to be mutually exclusive and the entire system to be collectively exhaustive. • To be mutually exclusive, every response must fit into only one category. • To be collectively exhaustive, all possible responses must fit into one of the categories. • Exhaustive means that you have covered the entire range of the variable with your measurement. Coding • Coding Missing Numbers: When respondents fail to complete portions of the survey. – Whatever the reason for incomplete surveys, you must indicate that there was no response provided by the respondent. – For single digit responses code as “9”, 2 digit code as “99” Coding Open-Ended Questions: When open-ended questions are used, you must create categories. – All responses must fit into a category – similar responses should fall into the same category. e.g. Who services your car? ______________ Possible categories: self, garage, husband, wife, friend, relative etc. • To make it collectively exhaustive add an “other” or “none of the above” category –Only a few i.e. < 10% should fit into this category Precoded Questionnaires: Sometimes you can place codes on the actual questionnaire, which simplifies data entry. This… Are you: Male Female How satisfied are you with our product? ___Very Satisfied ___Somewhat Satisfied ___Somewhat Dissatisfied ___Very Dissatisfied ___No opinion Becomes this… Are you: (1) Male (2) Female How satisfied are you with our product? _1__Very Satisfied _2__Somewhat Satisfied _3__Somewhat Dissatisfied _4__Very Dissatisfied _5__No opinion 1. Are you solely responsible for taking care of your automotive service needs ___ Yes ___ No 2. If No who performs the simple maintenance ___________ 3. If scheduled maintenance is done on your automobile, how do you keep track of what has been done •Not tracked •auto dealer records •mental recollection •other 4. How often is your automobile serviced? •Once per month •Once every three months •Once every six months •Once per year •Other _______________ Code Book Col. No Question No. Question Des. Range of permissible values 1 ID # N/A 001-200 (this also means the surveys themselves should be numbered) 2 1 Responsible for Maintenance 0= No. 1=yes, 9= blank 3 2 perform simple maintenance 0=husband, 1=boyfriend, 2=father, 3=mother, 4=relative, 5=friend, 6=other, 9=blank 4 3 How maintenance tracked 0=not tracked, 1=auto dealer records, 2=personal records, 3=mental recollection, 4=other, 9=blank 5 4 How often maintenance performed 1=Once per month, 2=3 month, 3=6 months , 4=year , 5=other, 9= blank In questions that permit multiple responses, each possible response option should be assigned a separate column 6. Which magazines do you read, choose all that apply. • Time • National Geographic • Readers Digest • Chatelaine • MacLean's Col. No Question No. Question Des. Range of permissible values 15 6 Time 0 =read, 1= not read 16 6 Readers Dig. 0 =read, 1= not read 17 6 MacLean's 0 =read, 1= not read 18 6 National Geo. 0 =read, 1= not read 19 6 Chatelaine 0 =read, 1= not read For rank order questions, separate columns are also needed 7. Please rank the following brands of toothpaste in order of preference (1-5) with 1 being the most important • Crest • Aquafresh • Pepsodent • Colgate • Arm & Hammer Col.# Q. No. Question Des. Range of permissible values 20 7 Crest rank 0 =blank, 1 = most important, 2 =2nd most important, 3 =third, 4=fourth, 5= fifth 21 7 Colgate rank 0 =blank, 1 = most important, 2 =2nd most important, 3 =third, 4=fourth, 5= fifth 22 7 Acquafresh rank 0 =blank, 1 = most important, 2 =2nd most important, 3 =third, 4=fourth, 5= fifth 23 7 A & H rank 0 =blank, 1 = most important, 2 =2nd most important, 3 =third, 4=fourth, 5= fifth 25 7 Pepsodent rank 0 =blank, 1 = most important, 2 =2nd most important, 3 =third, 4=fourth, 5= fifth Preparing the Data for Analysis Variable Re-specification • Existing data modified to create new variables • Large number of variables collapsed into fewer variables • E.g. If 10 reasons for purchasing a car are given they might be collapsed into four categories e.g. performance, price, appearance, and service • Creates variables that are consistent with research questions Entering Data • Problems can occur during data entry, such as transposing numbers and inputting an infeasible code(e.g out of range) – E.g. Score on range of 1-5 then 0, 6, 7, and 8 are unacceptable or out of range (might be due to transcription error) • Always check the data-entry work. Descriptive Statistics Five types of statistical analysis Descriptive What are the characteristics of the respondents? Inferential What are the characteristics of the population? Differences Are two or more groups the same or different? Associative Are two or more variables related in a systematic way? Predictive Can we predict one variable if we know one or more other variables? Descriptive Statistics Summarization of a collection of data in a clear and understandable way the most basic form of statistics lays the foundation for all statistical knowledge Measures of central tendency (mean, median, mode) Measures of dispersion (range, standard deviation, and coefficient of variation) Measures of shape (skewness and kurtosis) •If you use fewer statistics to describe the distribution of a variable, you Type of Measurement Type of descriptive analysis Two categories Nominal More than two categories Frequency table Proportion (percentage) Frequency table Category proportions (percentages) Mode Type of Measurement Type of descriptive analysis Ordinal Rank order Median Interval Arithmetic mean Ratio means Data Tabulation • Tabulation: The organized arrangement of data in a table format that is easy to read and understand. – A count of the number of responses to each question. • Simple Tabulation: tabulating of results of only one variable informs you how often each response was given. • Frequency Distribution: A distribution of data that summarizes the number of times a certain value of a variable occurs expressed in terms of percentages. Frequency Tables The arrangement of statistical data in a row-andcolumn format that exhibits the count of responses or observations for each category assigned to a variable • How many of certain brand users can be called loyal? • What percentage of the market are heavy users and light users? • How many consumers are aware of a new product? • What brand is the “Top of Mind” of the market? More on relative frequency distributions • Rules for relative frequency distributions: – – – – – Make sure each observation is in one and only one category. Use categories of equal width. Choose an appealing number of categories. Provide labels Double-check your graph. How did you find your last job? 643 Netw orking 213 print ad 179 Online recruitment site 112 Placement firm 18 Temporary agency 1.5 % Temporary agency 9.6 % Placement firm 15.4 % Online recruitment site 18.3 % print ad 55.2 % Netw orking 0 A histogram is a relative frequency distribution of a quantitative variable 100 200 300 400 500 600 700 A bar graph is a relative frequency distribution of a qualitative variable How many times per week do you use mouthwash ? 1__ 2__ 3__ 4__ 5__ 6__ 7__ 112223333344444445555566677 1 2 2 3 3 5 4 7 5 5 7 6 5 4 3 2 1 6 3 7 2 0 1 2 3 4 5 6 7 Normal Distribution - a b IQ The total area under the curve is equal to 1. I.e. It takes in all observations The area of a region under the normal distribution between any two values equals the probability of observing a value in that range when an observation is randomly selected from the distribution For example, on a single draw there is a 34% chance of selecting from the distribution a person with an IQ between 100 and 115 Normal Distributions Curve is basically bell shaped from - to symmetric with scores concentrated in the middle (i.e. on the mean) than in the tails. Mean, medium and mode coincide They differ in how spread out they are. The area under each curve is 1. The height of a normal distribution can be specified mathematically in terms of two parameters: the mean () and the standard deviation (). Skewed Distributions Occur when one tail of the distribution is longer than the other. Positive Skew Distributions have a long tail in the positive direction. sometimes called "skewed to the right" more common than distributions with negative skews E.g. distribution of income. Most people make under $40,000 a year, but some make quite a bit more with a small number making many millions of dollars per year The positive tail therefore extends out quite a long way Negative Skew Distributions have a long tail in the negative direction. called "skewed to the left." negative tail stops at zero • Kurtosis: how peaked a distribution is. A zero indicates normal distribution, positive numbers indicate a peak, negative numbers indicate a flatter distribution) Peaked distribution Flat distribution Thanks, Scott! Summary statistics –central tendency –Dispersion or variability A quantitative measure of the degree to which scores in a distribution are spread out or are clustered together; Measures of Central Tendency • Mode: the number that occurs most often in a string (nominal data) • Median: half of the responses fall above this point, half fall below this point (ordinal data) • Mean: the average (interval/ratio data) Mode the most frequent category users 25% non-users 75% Advantages: • meaning is obvious • the only measure of central tendency that can be used with nominal data. Disadvantages • many distributions have more than one mode, i.e. are "multimodal • greatly subject to sample fluctuations • therefore not recommended to be used as the only measure of central tendency. Median the middle observation of the data number times per week consumers use mouthwash 112223333344444445555566677 Frequency distribution of Mouthwash use per week Light user Mode Median Mean Heavy user The Mean (average value) sum of all the scores divided by the number of scores. a good measure of central tendency for roughly symmetric distributions can be misleading in skewed distributions since it can be greatly influenced by extreme scores in which case other statistics such as the median may be more informative formula = SX/N (population) X ¯ = Sxi/n (sample) where an ¯ X are the population & sample means and N and n are the number of scores. Normal Distributions with different Means - 1 0 2 Measures of Dispersion or Variability • Minimum, Maximum, and Range (Highest value minus the lowest value) • Variance • Standard Deviation (A measure’s distance from the mean) Distribution of Final Course Grades in MGMT 3220Y 25 Frequency 20 - 1 SD 15 + 1 SD 10 5 RANGE 0 Frequency F D C B A 3 10 20 23 12 Grade Variance • The difference between an observed value and the mean is called the deviation from the mean • The variance is the mean squared deviation from the mean • i.e. you subtract each value from the mean, square each result and then take the average. 2 = S(x¯ xi)2/n • Because it is squared it can never be negative Standard Deviation • The standard deviation is the square root of the variance 2/n S = S(xx ) ¯ i • Thus the standard deviation is expressed in the same units as the variables • Helps us to understand how clustered or spread the distribution is around the mean value. Measures of Dispersion Suppose we are testing the new flavor of a fruit punch Dislike 1 1. 2 3 4 5 Like Data x x 2. 3. x x 2/n 2 = S(xx ) ¯ i X= 4 2= 1 S=1 5 3 x 6. 5 3 x 4. 5. 3 5 2/n S = S(xx ) ¯ i Measures of Dispersion Dislike 1 2 3 4 1. 2. 5 Like Data x 5 4 x 3. x 5 4. x 5 5. x 5 6. 2/n 2 = S(xx ) ¯ i x X ¯ = 4.67 2=0.22 S = 0.47 4 2/n S = S(xx ) ¯ i Measures of Dispersion Dislike 1 1. 4 5 Like Data 1 x x 4. 5. 3 x 2. 3. 2 1 x 2/n 2 = S(xx ) ¯ i 5 X= ¯ 3 2=4 S=2 1 x 6. 5 x 5 2/n S = S(xx ) ¯ i Normal Distributions with different SD 2 - 1 3 Cross Tabulation • A statistical technique that involves tabulating the results of two or more variables simultaneously • informs you how often each response was given • Shows relationships among and between variables • frequency distribution for each subgroup compared to the frequency distribution for the total sample • must be nominally scaled Cross-tabulation • Helps answer questions about whether two or more variables of interest are linked: – Is the type of mouthwash user (heavy or light) related to gender? – Is the preference for a certain flavor (cherry or lemon) related to the geographic region (north, south, east, west)? – Is income level associated with gender? • Cross-tabulation determines association not causality. Dependent and Independent Variables • The variable being studied is called the dependent variable or response variable. • A variable that influences the dependent variable is called independent variable. Cross-tabulation • Cross-tabulation of two or more variables is possible if the variables are discrete: – The frequency of one variable is subdivided by the other variable categories. • Generally a cross-tabulation table has: – Row percentages – Column percentages – Total percentages • Which one is better? DEPENDS on which variable is considered as independent. Contingency Table • A contingency table shows the conjoint distribution of two discrete variables • This distribution represents the probability of observing a case in each cell – Probability is calculated as: Observed cases P= Total cases Cross tabulation GROUPINC * Gender Crosstabulation Gender Female GROUPINC income <= 5 5>Income<= 10 Count 9 19 % within GROUPINC 52.6% 47.4% 100.0% % within Gender 55.6% 18.8% 28.8% % of Total 15.2% 13.6% 28.8% 5 25 30 % within GROUPINC 16.7% 83.3% 100.0% % within Gender 27.8% 52.1% 45.5% 7.6% 37.9% 45.5% 3 14 17 % within GROUPINC 17.6% 82.4% 100.0% % within Gender 16.7% 29.2% 25.8% 4.5% 21.2% 25.8% 18 48 66 27.3% 72.7% 100.0% 100.0% 100.0% 100.0% 27.3% 72.7% 100.0% Count Count % of Total Total Total 10 % of Total income >10 Male Count % within GROUPINC % within Gender % of Total General Procedure for Hypothesis Test 1. Formulate H0 (null hypothesis) and H1 (alternative hypothesis) 2. Select appropriate test 3. Choose level of significance 4. Calculate the test statistic (SPSS) 5. Determine the probability associated with the statistic. • Determine the critical value of the test statistic. General Procedure for Hypothesis Test 6 a) Compare with the level of significance, b) Determine if the critical value falls in the rejection region. (check tables) 7 Reject or do not reject H0 8 Draw a conclusion 1. Formulate H1and H0 • The hypothesis the researcher wants to test is called the alternative hypothesis H1. • The opposite of the alternative hypothesis is the null hypothesis H0 (the status quo)(no difference between the sample and the population, or between samples). • The objective is to DISPROVE the null hypothesis. • The Significance Level is the Critical probability of choosing between the null hypothesis and the alternative hypothesis 2. Select Appropriate Test • The selection of a proper Test depends on: – Scale of the data • nominal • interval – the statistic you seek to compare • Proportions (percentages) • means – the sampling distribution of such statistic • Normal Distribution • T Distribution • 2 Distribution – Number of variables • Univariate • Bivariate • Multivariate – Type of question to be answered Example A tire manufacturer believes that men are more aware of their brand than women. To find out, a survey is conducted of 100 customers, 65 of whom are men and 35 of whom are women. The question they are asked is: Are you aware of our brand: Yes or No. 50 of the men were aware and 15 were not whereas 10 of the women were aware and 25 were not. Are these differences significant? Men Aware 50 Unaware 15 65 Women 10 25 35 Total 60 40 100 1. Formulate H1and H0 We want to know whether brand awareness is associated with gender. What are the Hypotheses H0: There is no difference in brand awareness based on gender H1: There is a difference in brand awareness based on gender Chi-square test results are unstable if cell count is lower than 5 2. Select Appropriate Test X2 (Chi Square) • Used to discover whether 2 or more groups of one variable (dependent variable) vary significantly from each other with respect to some other variable (independent variable). • Are the two variables of interest associated: – Do men and women differ with respect to product usage (heavy, medium, or light) – Is the preference for a certain flavor (cherry or lemon) related to the geographic region (north, south, east, west)? H0: Two variables are independent (not associated) H1: Two variables are not independent (associated) • Must be nominal level, or, if interval or ratio must be divided into categories Awareness of Tire Manufacturer’s Brand Men Women Total Aware 50/39 10/21 60 Unaware 15/26 65 25/14 35 40 100 Estimated cell Frequency E ij = R iC j n Ri = total observed frequency in the ith row Cj = total observed frequency in the jth column n = sample size Eij = estimated cell frequency 3. Choose Level of Significance Whenever we draw inferences about a population, there is a risk that an incorrect conclusion will be reached The real question is how strong the evidence in favor of the alternative hypothesis must be to reject the null hypothesis. The significance level states the probability of rejecting H0 when in fact it is true. In this example an error would be committed if we said that there is a difference between men and women with respect to brand awareness when in fact there was no difference i.e. we have rejected the null hypothesis when it is in fact true This error is commonly known as Type I error, The value of is called the significance level of the test Type I error • Significance Level selected is typically .05 or .01 • i.e 5% or 1% •In other words we are willing to accept the risk that 5% (or 1%) of the time the results we get indicate that we should reject the null hypothesis when it is in fact true. • 5% (or 1%) of the time we are willing to commit a Type 1 error • stating there is a difference between men and women with respect to brand awareness when in fact there is no difference 3. Choose Level of Significance • We commit Type error II when we incorrectly accept a null hypothesis when it is false. The probability of committing Type error II is denoted by . • In our example we commit a type II error when we say that. there is NO difference between men and women with respect to brand awareness (we accept the null hypothesis) when in fact there is Type I and Type II Errors Null is true Null is false Accept null Reject null Correctno error Type I error Type II error Correctno error Which is worse? • Both are serious, but traditionally Type I error has been considered more serious, that’s why the objective of hypothesis testing is to reject H0 only when there is enough evidence that supports it. • Therefore, we choose to be as small as possible without compromising . (accepting when false) • Increasing the sample size for a given α will decrease β (I.e. accepting the null hypothesis when it is in fact false) Awareness of Tire Manufacturer’s Brand Men Women Total Aware 50/39 10/21 60 Unaware 15/26 65 25/14 35 40 100 Estimated cell Frequency E ij = R iC j n Ri = total observed frequency in the ith row Cj = total observed frequency in the jth column n = sample size Eij = estimated cell frequency Chi-Square Test R iC j Estimated cell E = ij Frequency n Ri = total observed frequency in the ith row Cj = total observed frequency in the jth column n = sample size Eij = estimated cell frequency Chi-Square statistic x² = (Oi E i )² Ei x² = chi-square statistics Oi = observed frequency in the ith cell Ei = expected frequency on the ith cell Degrees of Freedom d.f.=(R-1)(C-1) 4. Calculate the Test Statistic Chi-Square Test: Differences Among Groups X + 2 = ( 15 ( 50 39 ) 39 26 ) 26 2 + 2 + ( 25 ( 10 21 ) 21 14 ) 14 2 2 2 = 3 .102 + 5 .762 + 4 .654 + 8 .643 = 2 = 22 .161 d . f . = ( R 1)( C 1) d . f . = ( 2 1)( 2 1) = 1 Chi-square test results are unstable if cell count is lower than 5 Degrees of Freedom the number of values in the final calculation of a statistic that are free to vary For example To calculate the standard deviation of a random sample, we must first calculate the mean of that sample and then compute the sum of the squared deviations from that mean While there will be n such squared deviations only (n - 1) of them are free to assume any value whatsoever. This is because the final squared deviation from the mean must include the one value of X such that the sum of all the Xs divided by n will equal the obtained mean of the sample. All of the other (n - 1) squared deviations from the mean can, theoretically, have any values whatsoever.. 5. Determine the Probabilityvalue (Critical Value) •The p-value is the probability of seeing a random sample at least as extreme as the sample observed given that the null hypothesis is true. • given the value of alpha, we use statistical theory to determine the rejection region. • If the sample falls into this region we reject the null hypothesis; otherwise, we accept it • Sample evidence that falls into the rejection region is called statistically significant at the alpha level. Critical value A critical value is the value that a test statistic must exceed in order for the the null hypothesis to be rejected. For example, the critical value of t (with 12 degrees of freedom using the .05 significance level) is 2.18. This means that for the probability value to be less than or equal to .05, the absolute value of the t statistic must be 2.18 or greater. critical value Significance level (.05) Test statistic /2 /2 -2.023 0 2.023 2.816 Significance from p-values -continued • How small is a “small” p-value? This is largely a matter of semantics but if the – p-value is less than 0.01, it provides “convincing” evidence that the alternative hypothesis is true; – p-value is between 0.01 and 0.05, there is “strong” evidence in favor of the alternative hypothesis; – p-value is between 0.05 and 0.10, it is in a “gray area”; – p-values greater than 0.10 are interpreted as weak or no evidence in support of the alternative. 5. Determine the Probability-value (Critical Value) Chi-square Test for Independence Under H0, the probability distribution is approximately distributed by the Chi-square distribution (2). Chi-square Reject H0 3.84 2 22.16 2 with 1 d.f. at .05 critical value = 3.84 6 a) Compare with the level of significance, b) Determine if the critical value falls in the rejection region. (check tables) 22.16 is greater than 3.84 and falls in the rejection area In fact it is significant at the .001 level, which means that the chance that our variables are independent, and we just happened to pick an outlying sample, is less than 1/1000 Or, in other words, the chance that we have a Type 1 error is less than .1% i.e. That there is a .1% chance that we reject the null hypothesis when it is true -- that there is no difference between men and women with respect to brand awareness, and say that there is, when in fact the null hypothesis is true: there is no difference. 7 Reject or do not reject H0 Since 22.16 is greater than 3.84 we reject the null hypothesis 8 Draw a conclusion Men and women differ with respect to brand awareness, specifically, men are more brand aware then women