University:
Massachusetts Institute of Technology
Course:
15.501/516 | Accounting
Academic year:

2023
Views:

70

Pages:

56
Author:

Zaniyah Shepherd

Measurement scales Measurement scales Nominal scale Bus #; Gender? Ordinal scale House numbers; Rank in class Interval scale Complex, maybe Grades Ratio scale Heart rate; any measure in % Scales & Statistics Can you take an average of a nominal scale? Gender Apartment number Ice-cream flavor Think about the meaning of the measurements! What scale is this? What scale is this? Outcomes What scales do we have? In most cases when asking people for their opinions researchers behave as if they have ratio scales, wishing they had interval scales but in fact in most cases we only have ordinal scales Luckily many of the statistical tools we use are not very sensitive to the violation of this assumption Granularity of scales It is important to use measurement scales that people find meaningful If people can distinguish only 3 (7) levels, there is not much use in a 100 point scale Descriptive statistics The goal of statistics is to Report data in meaningful ways Make predictions about future events 40 Course 15 Course 6 # of people 30 20 10 0 150 160 170 180 190 200 Weight Describing a state – Descriptive statistics – Capturing a picture of the data) – This was the origin of statistics – Started for gambling First some descriptive statistics How do you like 15.301? Excellent 100 15 75 5 1 50 6 25 12% 13% 0 Bad Good Very good Excelent 34% 41% Central tendencies • Representing central tendencies of distributions is a very efﬁcient way to understand something about it. • Mode • Median • Mean The Mode • The most “popular” frequent occurring instance in the sample. – This is the only central tendency that can be used with a nominal scale • The mode is sensitive to aggregation of categories – Age 18 vs age 18-21 • Sometimes there are multiple modes – Bimodal distributions The Median • The median is a value which 1/2 of the values are above it and 1/2 below • After sorting the values by magnitude, the median is at the (n+1)/2 location • 123, 85, 34, 20, 18, 15, 14 ? • 123, 85, 34, 20, 18, 15 ? • When data is grouped, calculating the mode is a bit more complex The Mean • • • • Mean = (Xi) /n The most important statistic Used for many other computations Stable – Smallest mean square deviations from it • Sensitive to extreme values • Not “well behaved’ in non-standard distributions Plots I Q1 Q1 Q2 100 100 75 75 50 50 25 25 0 0 2001 2002 2003 2004 Q2 2001 2002 2003 2004 Plots II 2001 Q1 2002 Q2 200 2003 2004 9% 14% 150 100 50% 28% 50 0 2001 2002 2003 2004 Location of central tendencies Normal Mean Mode Median 1.0 0.8 0.6 0.4 0.2 0 Location of central tendencies Bimodal 1.0 Mean Mode Median 0.8 0.6 0.4 0.2 0 Location of central tendencies Skew to right Mean Mode Median 0.5 0.4 0.3 0.2 0.1 0 Location of central tendencies Skew to left Mean Mode Median 0.5 0.4 0.3 0.2 0.1 0 Distribution descriptors • The Range – The range is (Max - Min) – – • Interquartile range – Calculating is similar to median – Q3-Q1 (1/2 of the observations) Variation I • Variance (^2) – (Xi-µ)^2/(n) – (Xi-µ)^2/(n-1) • Standard deviation () – Square root of variance – Standard deviation is in the same units as the distribution Variation II • Variance ( ^2) is: • insensitive to transformations consisting of adding a constant. • sensitive to transformations consisting of multiplying by a constant. When to use When should you use the: Mean Mode Median The Correlation • The relationship between 2 variables does not have to be linear – But in many cases they are • Positive and negative correlations What is a correlation? • What line to pick? – Sum of all deviations from the line is 0 – The sum of square deviations of the points from the line is minimal. Estimating correlations in scatter grams What is the correlation here? Estimating correlations in scatter grams What is the correlation here? Estimating correlations in scatter grams What is the correlation here? Estimating correlations in scatter grams What is the correlation here? Estimating correlations in scatter grams What is the correlation here? Estimating correlations in scatter grams What is the correlation here? Estimating correlations in scatter grams What is the correlation here? The correlations were: • • • • • • • 1 0.1 2 0.3 3 0.5 4 0.7 5 0.9 6 0.99 7 01 Types of measurements Two types of measures Subjective Introspection Intuitions Opinions Objective Behavioral What people actually do Two types of measures Examples: How much do you enjoy this class? Who came to class today? Subjective vs. objective: which is better? Why? Objective measures Objective measures Observing behavior (parking, men's room) Choice Payments, bids, etc Time in task Limited to things we can observe “Statistically expensive” in many cases (choice) Usually harder to get Partial objective measures We can ask people for the their opinions but ask them to use objective scales. How much time have you used for action X How much will you pay How much energy will you expand Statistically expensive? No Yes Subjective measures Subjective measures Intentions What would you do What would others do Attitudes How much do you like X? Verbal reports On-line introspections Quantitative subjective measures Quantitative objective Likert scales (1-7; 1-10; 1-100) Guttman scale A yes no response to a continuum of questions that are ordinal Hypothetical choice Ranking Subjective measures I Very popular Cheap Can tell us things we cannot observe (why..) Depend on interpretation Intuitions & not behavior Depends on the extend to which people can introspect Subjective measures II Social desirability demands How often do you cheat? Would you steal if you were sure no one will catch you? Would you rather watch “the piano” or Rambo? Demand characteristics (effects) Do you agree that red is a nice color? Types of measurement summary I Objective and subjective measurements Each has its advantages and disadvantages Beware of subjective measures if your goal is to predict behavior Types of measurement summary II Subjective vs. objective (behavioral measures) The main question is what are people able to report Sensitivity to scales, order, etc. Response language The constructive view Do people have preferences? Inferences of attitudes from questions / answers One more issue ... Fresh-Samantha Paradox Flavor Mean ranking % Market Carrot 2.7 33% Orange carrot 3.0 0% Orange 2.3 33% Soy shake 2.0 33% People ranked each from 1 (worse) to 4 (best). How could you explain this paradox? Fresh-Samantha Paradox Group 1st 2nd Orange Group A Carrot carrot Orange Group B Orange carrot Orange Group C Soy shake carrot 3rd Orange Carrot Carrot Ranking (or ratings) do not always translate to behavior? The 2nd best problem 4th Soy shake Soy shake Orange The 2nd best problem It is hard to infer from ratings or ranking choice At the same time it is hard to infer relative location from choice B1 A1 B2 A2 Measurements Objective measures are usually easier to make inferences from But, even with objective measures it is important to think what is measured and how it relates to the question of interest Descriptive stats A compact way to represent the main essence about the data What is this main essence depends on the measurements and on the data Each method has +s and -s There are no rules for what method to pick -- do your best! it is your responsibility