Variables, distributions, and samples. Phil 12: Logic and Decision Making Spring 2011 UC San Diego 4/21/2011

Transcription

1 Variables, distributions, and samples Phil 12: Logic and Decision Making Spring 2011 UC San Diego 4/21/2011

2 Midterm this Tuesday! Don t need a blue book or scantron Just bring something to write with Sample midterm Not posting an answer key Check answers by checking text, notes, in section, office hours, If asking me or TAs, must talk through what you think the answer might be, talk through options, reasoning

3 Anonymous clicker question Do you want me to hold office hours Monday afternoon or evening? A. Yes, Monday 2-4pm B. Yes, Monday 3-5pm C. No, I m good 3

4 Review Observational research involves careful recording and analysis of what is observed - Without an attempt to manipulate what happens Naturalistic vs. participant observation Risks that must be minimized: - Observer bias - Reactivity - Anthropomorphizing

5 Coding Schemes A coding scheme is a set of categories used to classify observed phenomena - extract data so as to learn from the observations How can a coding scheme be poorly designed? - fail to have a category for some phenomena you care about recording and analyzing - use one category for phenomena you would like to distinguish

6 Recording continuously vs. selectively Continuous observation: record what is happening at every moment of time Time sampling: recording what is happening at predetermined intervals Event sampling: recording whenever an event of a specified kind occurs Situation sampling: recording what happens in a variety of different situations (locations) 6

7 Clicker question To determine how many students carry backpacks, a researcher sits outside the library and records, for every fifth students who exits, whether they have a backpack. The researcher is performing A. Continuous observation B. Time sampling C. Event sampling D.Situation sampling 7

8 Variables The data from observational research is analyzed in terms of variables A variable is a characteristic or feature of an event that varies(i.e., takes on different values) - Variables of a thrown ball: velocity, momentum, direction, spin,... - Variables of human hair: color, length, texture,... - Variables of human cognition: memory span, speed of reasoning, emotional state,...

9 Types of variables Variables differ in the type of measurement of the values of the variable that is possible. Sometimes one refers to types of scales rather than types of variables. 1. Categorical or nominal variables 2. Ordinal or rank variables 3. Interval variables 4. Ratio variables

10 Types of variables - 1 Categorical or nominal variables: items can be assigned to a category (whose members can then be counted, or compared on another variable) - Examples: Gender: male/female Major: psychology, political science, economics,... Organisms: Plant, Animal, Bacteria, Virus,...

11 Types of variables - 2 Ordinal or rank variables: There is a rank-order to the values the variable may take - Numbers might be assigned to the items, but since there is no metric one cannot compare how much higher or lower one item on the scale is than another - Examples: Movies; *, **, ***, **** Class rank: top 10, next 10, etc. Patient condition: resting and comfortable, stable, guarded, and critical Socioeconomic class: low, middle, high

12 Types of variables - 3 Interval variables: equal differences between numbers assigned to items reflect equal differences between the values being measured. - Allows additive comparison (e.g., x is three more than y) - But lacking a natural zero-point, does not permit multiplicative comparison (e.g., x is three times y) - Examples: Intelligence: IQ score Temperature: in degrees Celsius or Fahrenheit Personality: degree of extroversion

13 Types of variables - 4 Ratio variables: items are rated on a scale with equal intervals and a natural 0-point. - Allows for both additive and multiplicative comparison - Examples: Age: in year, months, days,... Temperature: in degrees Kelvin Time: in milliseconds, seconds, years,... Velocity, acceleration, etc. - Interval and ratio data often treated similarly and counted as score data

14 Summary: Types of Variables Type of variable Example Categorical or nominal college major Score variables Ordinal or rank Interval Ratio patient condition temperature in degrees Fahrenheit age

15 Clicker question The variable number of clicker responses is A. A categorical or nominal variable B. An ordinal or rank variable C. An interval variable D. A ratio variable

16 Clicker question On the CAPE evaluations, you respond to questions such as Exams are representative of the course material (the variables being measured) using the following answer choices (values): 1 = strongly disagree 2 = disagree 3 = neither 4 = agree 5 = strongly agree What type of variable are these questions? A. A categorical or nominal variable B. An ordinal or rank variable C. An interval variable D. A ratio variable

17 Visual representations of data

18 Nominal & ordinal variables: Bar graphs & Pie Charts Example: Profile of pet ownership in San Diego County

19 Score variables: Histograms Histograms rather than bar graphs used because score variables are continuous This is done by creating bins and tabulating the number of items in each bin The size of bins can create radically different pictures of the distribution! bin size: 0.25 bin size: 1

20 Daily Life Activities Bin size: 1 hr 25 Studying (online + offline) 20 No. of people Hours

21 Daily Life Activities Bin size: 0.5 hr 25 Studying (online + offline) 20 No. of people Hours

22 Daily Life Activities Bin size: 0.25 hr 25 Studying (online + offline) 20 No. of people Hours

23 Normal and non-normal distributions Normal distributions - Have a single peak - Scores equally distributed around the peak - Fewer scores further from the peak Non-normal distributions Skewed Bimodal

24 Daily Life Activities N = Studying (online + offline) 20 No. of people Hours

25 Daily Life Activities N = In class 20 No. of people Hours

26 Clicker question The distribution below is < > A. Normal since it has one peak B. Normal since scores are equally distributed around the peak C. Not normal since because there are not fewer scores further from the peak D. Not normal because scores are not equally distributed around the peak

27 Describing distributions Two principal measures: 1. Central the standard tendency deviation Two comparable distributions differing in central tendency 2. Variability Two distributions with same central tendency but differing in variability

28 Three measures of central tendency Mean: the arithmetic average--sum of all the scores divided by the number of instances Median: the score of which half are higher and half are lower Mode: the most frequent score Consider this distribution of values: 2, 6, 9, 7, 9, 9, 10, 8, 6, 7 mean = 73 / 10 = 7.3 median = mode = 7.5 9

29 Which measure to use? If the distribution is normal, all three measures of central tendency give the same result - The mean is the easiest to calculate and the most frequently reported If there are extreme outliers in one direction, the mean may be distorted - Exam scores: 21, 72, 76, 79, 82, 84, 87, 88, 90, 91, 95 Mean: 78.6 Median: 84 - In such a case, the median gives a better picture of the central tendency of the class

30 Measures of variability Variability concerns: How much do the scores vary? Range: the lowest value to the highest value

31 Measures of variability Variability concerns: How much do the scores vary? Range: the lowest value to the highest value Variance: (X-mean) 2 N Standard deviation: Variance Mean = 5.0 SD = 0 Mean = SD = 1.04

32 Measures of variability Variability concerns: How much do the scores vary? Range: the lowest value to the highest value Variance: (X-mean) 2 N Standard deviation: Variance - Intuitive interpretation: 1 SD: the part of the range in which 68% of the scores fall 2 SD: the part of the range in which 95% of the scores fall 3 SD: the part of the range in which 99% of the scores fall

33 Variance Consider a distribution: Mean = 6 X - mean (X-mean) 2 (X-mean) 2 Variance = = N 12 9 = 1.33 SD = variance = 1.33 = 1.15 Range of 1 SD Range of 2 SD = 6 ± 1.15 = 4.85 to 7.15 = 6 ± 2.30 = 3.70 to 8.30

34 Range and Standard Deviation range 68% of scores 95% of scores

35 Clicker question On an exam on which scores were distributed normally and the mean was 86 and the SD was 4, A. 68% of the scores were between 78 and 94 B. 68% of the scores were between 82 and 90 C. 95% of the scores were between 78 and 94 D. 95% of the scores were between 82 and 90 E. None of the above

36 Populations The phenomena about which we seek to draw conclusions in a study are known as the population. Sometimes one can study each member of the population of interest But if the population is large: - - it may be impossible to study the whole population there may be no need to study the whole population

37 Samples A sample is a subset of the population chosen for study. From studying the distribution of a variable in a sample, one makes an estimate of the distribution in the actual population Sometimes the estimate from a sample may be more accurate than trying to study the population itself - U.S. Census

38 Is the sample biased? If information about the sample is to be informative about the actual population, the sample must be representative - Randomization: attempt to insure that the sample is representative by avoiding bias in selecting the sample Risk: inadvertently developing a misrepresentative sample - E.g., using telephone numbers in the phonebook to sample electorate

39 Does the sample reflect the population? Does the mean of the sample reflect the mean of the actual population? Sampling distribution simulation Very unlikely that the mean of the sample will exactly equal the mean of the population Key question: how much does the mean of the sample vary from the mean of the actual population? Given the mean of a sample, what is the range within which the mean of the actual population lies? - To determine this, the standard deviation measure is very useful

40 Standard deviation and mean In 68% of samples, the mean of the population will fall within 1 standard deviation of the mean of the sample Sample mean In 95% of samples, the mean of the population will fall within 2 standard deviations from the mean of the sample

41 What happens as sample size gets larger? As sample size grows, the SD of the sample shrinks So with larger samples, the range of 2 standard deviations shrinks Assume sample mean is 50: Sample size Range of 2 SD (95% confidence interval) Range of 3 SD (99% confidence interval)

42 Example of estimating population mean from sample mean Example: age of people eating at the Food Court - Draw a sample to make inference of average age of people eating at the Food Court < >25 Population Sample

43 Estimating real distribution < >25 Population Sample 1 (n = 10) Sample 2 (n=20) Mean of the actual population: Sample 1 Sample 2 Mean of the sample: SD of the sample: Range of 1 SD: Range of 2 SD: Want to predict more accurately? Use a larger sample size

44 Review Four types of variables: - Nominal ordinal interval ratio Values of variables are distributed - Important goal: characterizing the distribution Graphs - Bar graphs for nominal and ordinal variables - Histograms for score variables Normal versus non-normal distributions - Skewed, bimodal, etc

45 Review Two principal measures of distributions - Central tendency Mean, median, mode - Variability Range, variance, SD - 1 SD includes approx. 68% of scores - 2 SD includes approx. 95% of scores - 3 SD includes approx. 99% of scores

46 Review Population and samples - From studying the distribution in sample, estimate the distribution in the actual population - Mean of actual population will Fall within one SD of mean of sample for 68% of samples Fall within two SD of mean of sample for 95% of samples Fall within three SD of mean of sample for 99% of samples - Larger sample yields smaller SD and hence more precise estimate - Hence, to improve the precision of an estimate, use a larger sample