2. Chapter 12Univariate statistical analysis: Arecap of inferential statistics 2
3. Review sampling• You want to see a new movie this weekend. So you get onto a website and checkout previews of what’s on.• Is this sampling?• How good a sample would this be> 3
4. Census vs Sampling 4
5. Learning Objectives• Understand and explain the need for data preparation techniques such as editing, coding, cleaning and statistically adjusting the data where required• Develop a data analysis strategy based on specific research objectives• Identify the factors influencing the selection of an appropriate data analysis strategy• Outline various analysis techniques
6. Data Preparation ProcessPrepare preliminary plan of data analysis Check questionnaires Edit Code Transcribe Clean data Statistically adjust the data Select a data analysis strategy
7. Questionnaire Checking• Review all questionnaires for completeness and interviewing quality• Unacceptable questionnaires include: – Parts of the questionnaire that are incomplete – Skip patterns may not have been followed – Little variances in responses – Pages missing – Late questionnaires – Respondents does not fit the selection criteria
8. Data Editing• A review of the questionnaires with the objective of increasing accuracy and precision.• Identify responses that are: – Illegible – Incomplete – Inconsistent – Ambiguous responses
9. Data Editing cont.• Treatment of unsatisfactory responses – Return to the field • Recontact the respondent – Assign missing values • If the number of unsatisfactory responses is small • Key variables are not missing – Discard unsatisfactory respondents (cases) • Proportion of unsatisfactory responses is small • Sample size is large • Unsatisfactory respondents do not differ from satisfactory respondents • Responses to key variables are missing
10. Data Coding• Assigning a code [number] to each possible response to each question [variable] – Structured questionnaires [pre-coded] – Unstructured questions [post-coding]• Category codes should be mutually exclusive and collectively exhaustive.• Category codes should be assigned for critical issues even if no one mentions them.
11. A Basic Questionnaire1. In a typical month, how many times would you say you visit a fast-food restaurant? (Tick one box only) None One Two Three Four Five Six or more2. On your last visit to a fast-food restaurant, what was the dollar amount you spent on food and beverages? Under $2.00 $6.01 - $10.00 More than $14.00 $2.01 - $6.00 $10.01 - $14.00 Don’t remember3. How many of these restaurants would you say you visited in the past two months? Tick as many as apply. KFC Pizza Hut Wendy’s Red Rooster McDonalds Other Hungry jacks Have not visited any of these establishments4. On a scale of 1 to 5, with 1 being strongly disagree to 5 being strongly agree, how would you rate fast-food restaurants on the following dimensions: I only visit those fast-food establishments that are conveniently located to my home 1 2 3 4 5 I prefer to visit fast-food restaurants that serve healthy/nutritious food 1 2 3 4 5 The price of food items is not important when visiting a fast-food restaurant 1 2 3 4 5 All fast-food restaurants should offer some type of child’s menu or kid’s meal 1 2 3 4 55. How many children do you have living at home? None One Two Three Four Five or more6. Which category does you total annual household income fall? Under $20,000 $20,000 - $39,999 $40,000 - $59,999 $60,000 or more
12. Coding the QuestionnaireVariable Variable CodingNumber Name Instruction (99=missing value)1 Number of visits per month 0=None 1=one 2= two 3=three 4=Four 5= five 6= six or more2 Amount spent 1= Under $2 2= $2.01 - $6.00 3= $6.01 - $10.00 4= $10.01 - $14.00 5= More than $14.00 6= Don’t remember3.1 Visited KFC 1=Yes, 0= No
13. Coding the Questionnaire cont.3.2 Visited Wendy’s 1=Yes, 0= No3.3 Visited McDonalds 1=Yes, 0= No3.4 Visited Hungry Jacks 1=Yes, 0= No3.5 Visited Pizza Hut 1=Yes, 0= No3.6 Visited Red Rooster 1=Yes, 0= No3.7 Visited Other establishment 1=Yes, 0= No3.8 Have not visited any establishment 1=Yes, 0= No4.1 Visit conveniently located stores 1= strongly disagree 2= disagree 3=neither agree/disagree 4=agree 5=strongly agree4.2 Prefer healthy fast food stores As above
14. Coding the Questionnaire cont.4.3 Price is important As above4.4 Children’s menu is important As above5 Number of children 0=None 1=one 2= two 3=three 4=Four 5= five or more6 Annual household income 1=under $20,000 2=$20,000 - $39,000 3=$40,000 - $59,000 4=$60,000 or more
15. Transcribing• Transferring coded data from the questionnaire to a computer to be used for analysis.• Variations to manual transcribing: – CATI or CAPI – Mark sense forms and optical scanning – UPC – Computerised sensory analysis systems• For verification of the entire dataset, re-enter the responses
16. Transcribing cont.
17. Data Cleaning• Consistency check – Out of range [see study status] – Logically inconsistent [e.g., does not own the product but is a heavy user] – Extreme values [indiscriminatingly responding the same way on all attributes]
18. Example: Out of Range Study Status Cumulative Frequency Percent Valid Percent PercentValid Full time student 923 91.8 91.8 91.8 Part time student 81 8.1 8.1 99.9 3.00 1 .1 .1 100.0 Total 1005 100.0 100.0
19. Data Cleaning cont.• Treatment of missing responses – Substitute a neutral value [substitute the ‘mean’ response of the variable] – Substitute an imputed response [use the respondent’s pattern of responses to other questions] – Casewise deletion [respondents with any missing values are discarded from the analysis] – Pairwise deletion [use only cases or respondents with complete responses for each calculation]
20. Statistically Adjusting the Data• Weighting – Each case is assigned a weight to reflect its importance relative to other cases, often used to make the sample more representative of a target population• Variable re-specification – Transformation of data to create new variables or modify existing variables to better suit the research objectives by summing several variables, log transformations, dummy variables [see next slide]• Scale transformation – Manipulation of scale values to ensure comparability with other scales or otherwise make the data suitable for analysis [when data is not normally distributed].
21. Variable re-specification: Composite variables•Aesthetics of awebsite•Measured using twoitems –“The website is visually pleasing” –“The website is visually appealing” –Combine these two items to create a new variable “Aesthetics of a website” – this new variable is used with further analysis in place of the two items.
22. Variable re-specification: Recode variables (to recode negatively-worded scale items)Role Overload Strongly Disagree Disagree Neither Agree Agree Strongly Disagree Somewhat agree nor Somewhat Agree disagreeI have too much work to do, to do everything 1 2 3 4 5 6 7wellThe amount of work I am asked to do is fair 1 2 3 4 5 6 7I never seem to have enough time to get 1 2 3 4 5 6 7everything done•Role overload is measured by 3 items.•Which item is reverse-coded?•We need to code this so all item are flowing in the samedirection.•We need to inform SPSS that 1=7, 2=6, 3= 5, 4=4, 5=3, 6=2,7=1 for the reverse coded item.
23. Variable re-specification: Recode variables•“Overall, I’m (to collapse a continuous variable) cont.satisfied with myjob” was measuredusing a seven-pointscale.•When we performdata analysis(particularly cross-tabs) we may wishto have fewercategories forbrevity.
24. Strategy for Data Analysis• Determine the type of data which is available [nominal, ordinal, interval, ratio]• Decide what needs to be discussed in order to tell ‘the story’• Choose techniques to best get information on specific parts of what has to be discussed• Run the results• Determine what the results mean, what patterns can be seen, what kind of statistical decisions should be made• Write about the results to explain what is going on to someone who does not like numbers and has never heard of statistics
25. Overview of Techniques• Descriptive Statistics – Frequency distribution and cross tabulations – Measures of central tendency [mean, median, mode] – Measures of dispersion [range, interquartile range, standard deviation] – Shape [skewness, kurtosis]• Inferential Statistics – Parametric tests [Z or t test, paired t test] – Non-parametric tests [Chi-square]
26. Descriptive and inferential statistics• Descriptive statistics are used to describe characteristics of a population.• Inferential statistics are used to make inferences about a population from a sample of that population. 26
27. Sample statistics and population parameters• Sample statistics are variables in a sample or measures computed from sample data.• Population parameters are variables in a population or measured characteristics of the population.• But, generally we do not know what these population parameters are and that is why we use samples. 27
28. Frequency distributions• Frequency distribution involves a process of recording the number of times a particular value of a variable occurs.• Percentage distribution is a distribution of relative frequency.• Probability is the long–run relative frequency with which an event will occur. 28
29. Frequency distributions 29
30. Measures of central tendency• Mean: arithmetic average• Median: the midpoint – The value below which half the values in a distribution fall.• Mode: the value that occurs most often. 30
31. Measures of dispersion• The tendency of observations to depart from the central tendency.• Range: distance between the smallest and largest values.• Deviation scores: how far any observation is from the mean. – Average deviation• Variance: measure of variability or dispersion – Its square root is the standard deviation. 31
32. Measures of dispersion• Standard deviation: quantitative index of a distribution’s spread. – Using square root of variance reverts to the original measurement units. 32
33. The normal distribution• A symmetrical, bell–shaped distribution that describes the expected probability distribution of many chance occurrences. – 99% of its values are within + 3 standard deviations from its mean. 33
34. The normal distribution• Standardised normal distribution has: – symmetry about its mean – infinite number of cases – area under the curve with probability density equal to 1 – mean of 0 and standard deviation of 1. Standardised value = Value to be transformed – Mean Standard deviation Z=X-µ σ 34
35. An example of standardised value• Toy manufacturer has mean sales of 9000 units and standard deviation of 500 units.• Wishes to know whether wholesalers will demand between 7500 and 9635 units. Z = X - µ = 7500 – 9000 = -3.00 σ 500 Z = X - µ = 9625 – 9000 = 1.25 σ 500• Referring to Table 12.8, we find that: – When Z = –3.00, the area under the curve = 0.499. – When Z = 1.25, the area under the curve = 0.394. – The total area under the curve = 0.499 + 0.394 = 0.893. – There is a 0.893 probability that sales will in that range. 35
36. The standardised normal table 36
37. Population, sample, and sampling distribution• Population distribution: a frequency distribution of the elements of a population.• Sample distribution: a frequency distribution of a sample.• Sampling distribution: a theoretical probability of sample means for all possible samples of a certain size drawn from a particular population. 37
38. Population, sample, and sampling distribution• Standard error of the mean: the standard error of the sampling distribution.• Sampling distribution is important because it addresses the question of ‘ What would happen if we were to draw a large number of samples, each having n elements, from a specified population?’ 38
39. Population, sample, and sampling distribution 39
40. Central–limit theorem• Central–limit theorem states that as the sample size increases, the distribution of the mean of a random sample taken from practically any population approaches a normal distribution. 40
41. Confidence intervals• A confidence interval estimate is based on the knowledge that the population mean is the sample mean plus or minus a small sampling error. – After calculating an interval estimate, we can determine how probable it is that the population mean will fall within this range of statistical values.• Confidence level is a percentage that indicates the long–run probability that the results will be correct. 41
42. Confidence intervals∀ µ=X+E where E = range of sampling error• E = Zc.l.SX where Zc.l. = value of Z at a specified confidence level (c.l.) and SX = standard error of the mean∀ µ = X + Zc.l.SX where SX = S , S = standard deviation and n = sample size √n• Thus, µ = X + Zc.l.S √n 42
43. An example of confidence intervals• Sporting goods store caters to working women who golf.• Survey showed the mean age is 37.5 years and standard deviation of 12.0 years.• Wishes to be 95% confident that the sample estimates will include the population parameter. µ = X + Zc.l. S = 37.5 + Zc.l. 12.0 √n √100• Including 95% of the area requires that 47.5% of the distribution on each side be included.• Referring to Table B.2 in Appendix B, we find that 0.475 corresponds to the Z-value 1.96. Thus: µ = 37.5 + (1.96)(1.2) = 37.5 + 2.352• 95% of the time µ is in range of 35.15 to 39.85 years. 43
44. Frequency Distributions• A count of the number of responses associated with different values of the variable Where did you hear about VUs Open Day? Cumulative Frequency Percent Valid Percent Percent Valid Radio 39 12.7 12.8 12.8 Newspaper 29 9.4 9.5 22.3 Internet site 25 8.1 8.2 30.5 Friend/Relation 52 16.9 17.0 47.5 School 160 51.9 52.5 100.0 Total 305 99.0 100.0 Missing System 3 1.0 Total 308 100.0
45. Frequency Distributions cont. Age of respondent Cumulative Frequency Percent Valid Percent PercentValid 18 or under 197 64.0 64.6 64.6 19 - 29 71 23.1 23.3 87.9 Over 29 37 12.0 12.1 100.0 Total 305 99.0 100.0Missing System 3 1.0Total 308 100.0
46. Bar Chart Produced from Frequency Distributions40% 38.00%35% 34.00%30%25%20% 18.00% The course offered15%10% 6.00%5% 4.00%0% Very Important Of some Of little Of absolutely important importance importance no importance
47. Frequencies for Multiple Response Questions• Example of a question using multiple-response formattingQ9.Which of the following people had an influence on your choice of university?Parents 01Friends 02Ex-VU student 03Teacher at high school 04Careers teacher at high school 05Colleagues 06Other 07
48. Frequencies for Multiple Response Questions Influence on choice of university (Value tabulated = 1) Pct of Pct of Dichotomy label Name Count Responses Cases Influenced by Parents Q9A 420 26.4 42.3 Influenced by friends Q9B 331 20.8 33.4 Influenced by student Q9C 149 9.4 15.0 Teacher at high school Q9D 158 9.9 15.9 Careers teacher at high school Q9E 259 16.3 26.1 Colleagues Q9F 88 5.5 8.9 Other Q9G 184 11.6 18.5 ------- ----- ----- Total responses 1589 100.0 160.2
49. Statistics Associated with Frequency Distributions: Measures of Location• Mean – ‘average’• Mode – The value that occurs most frequently. – Most appropriate for categorical data.• Median – Middle value in the data set when the data are arranged in ascending or descending order.
50. Mean Mode Median NominalType of data Interval Ordinal Interval Ratio Interval Ratio RatioInfluenced Yes No Noby outliers
51. Statistics Associated with Frequency Distributions: Measures of Variability• Range – The difference between the largest and smallest values of a distribution.• Interquartile range – The range of a distribution encompassing the middle 50 percent of the observations.• Variance and Standard deviation – Variance is the mean squared deviation of all the values from the mean. The standard deviation measures the average spread (deviation) from the mean and uses values which are consistent with the original observations.• Coefficient of variation – The standard deviation expressed as a percentage of the mean.
52. Table 1: Factors students consider when selecting University
53. Statistics Associated with Frequency Distributions•Measure of shapeskewnesssymmetry•Kurtosis
54. Cross-Tabulations• Describes two or more variables simultaneously
55. Expressing the data as percentages
56. Can also be presented graphically.
57. Notes on writing up results• Do not simply repeat the numbers in the table as part of the discussion• The discussion should focus on the patterns in the data• Percentages (rather than numbers) are more generalisable to the population,• However, keep in mind that because of sampling error the percentage in the population will not exactly match that of the sample• We rarely care about the sample itself, except what it tells us about the population, it is supposed to represent