1. N318b Winter 2002 Nursing Statistics Lecture 2: Measures of Central Tendency and Variability

2. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 2 Today’s Class(es)  mean, median, mode  range, standard deviation, variance >  Some examples  Applying knowledge to assigned readings (Arathuzik; Hayman et al.) focuses on determining and interpreting measures of central tendency and dispersion Followed by small groups from 12-2 PM

3. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 3 A Quick Review from Last Week - 1 Measurement Scales Nominal data Ordinal data Interval data Ratio data Variable Types Dependent Independent

4. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 4  mean  median  mode A basic cornerstone of most research statistics is that numeric data points tend to group together, usually in identifiable (predictable) ways – i.e. they tend to congregate around a common value Measures of Central Tendency You should know what these three things are and how they differ from each other

5. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 5 Mean Most appropriate for ratio or interval data (i.e. continuous numeric data) but not if strongly skewed  = (x 1 + x 2 + x 3 + x n ) / N Where x 1 + x 2 + x 3 + x n are independent data points and N is the total number of data points Note: x 1 + x 2 + x 3 + x n also written as “  X”

6. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 6 Some Properties of the Mean  All data points contribute to its value  Sensitive to extreme values  Sum of deviations always zero i.e.  (x-  )=0  Sum of squared deviations at a minimum - i.e.  (x-  ) 2 lower for mean than other terms  Mean is algebraic thus it can be manipulated making it more useful statistically  When sample large enough (e.g. >25) it does a good job estimating true population mean

7. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 7 Median Most appropriate for ratio data (i.e. continuously scaled) even if skewed median = mid-point of distribution (i.e. the 50 th percentile) Divides the data into two equally sized groups (i.e. same frequency or count in each)

8. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 8 Some Properties of the Median  Typically not calculated as it is simply the mid- point (but data must be sorted/ordered)  Median not sensitive to extreme values thus useful if data skewed  Not used with nominal data since it requires data to have an order  Does not have to actually exist as a data point (e.g. mid-point between adjacent data points)

9. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 9 Mode Typically more useful for grouped data (i.e. ordinal or re-scaled continuous data) mode = most common value Has descriptive value but it is not a widely used statistic

10. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 10 Some Properties of the Mode  Not calculated (but observed)  If all values unique then no mode  May be more than one mode (e.g. bimodal, trimodal, etc.)  Only measure of central tendency for strictly nominal data

11. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 11 Mean, Median and Mode When distribution of data points is very even (i.e. normally distributed), then the three converge centrally Mean, median, mode all in same position in a perfect distribution

12. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 12 Mean, Median and Mode “real” data points rarely (never!) perfectly normally distributed thus typically some differences do exist Mean Median Mode Sample “left” skewed as mean is less than median

13. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 13 Mean, Median and Mode Age groups Group 1 = (11, 12, 13, 13, 14, 15) Mean affected by extreme value  1 = 13 Group 2 = (11, 12, 13, 13, 14, 25)  2 = 17 Median is 13 – divides data in half Mode is 13 – most common value

14. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 14  Standard deviation  Variance  Percentiles  Range Viewed another way, most research statistics that are numeric data points also tend to vary from each other, usually in identifiable (predictable) ways – i.e. they tend to be spread out Measures of Dispersion You should know what these four things are and how they differ from each other The “Flip-side”:

15. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 15 Dispersion (or spread) Two samples with the same mean can have very different dispersion Sample B: More dispersed Sample A: Less spread,  SD of A < SD of B Sample A measured more precisely? Mean

16. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 16 Standard deviation 2 SD includes about 95% of sample -2 SD+2 SD 1 SD either side of mean includes about 68% of sample -1 SD +1 SD Mean

17. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 17 SD =  (x-  ) 2 / N-1  Standard deviation Key indicator of the average point deviation from the sample mean If SD is low relative to the mean then measure is more precise (see coefficient of variation in textbook) SD - most important dispersion measure

18. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 18 Variance: squared deviations from mean; important for later methods Other measures of deviation Range: maximum value - minimum value; useful for describing sample Percentiles: Value above which and below which a certain proportion of the sample falls

19. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 19 10 minute break !

20. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 20 Assignment #1 Marks Example 1

21. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 21 What happens if we remove the zeros – i.e. the most influential (outlying) observations?

22. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 22 Assign #1 – Zeros dropped Example 2

23. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 23 Part 2: Application to the Assigned Readings

24. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 24 Arathuzik (1994) Quick summary of the paper: – a pilot study examining the effects of a combination of interventions on pain perception, pain control and mood in metastatic breast cancer patients – pre-test / post-test experimental design – 3 groups enrolled with 24 (convenience sample) subjects randomly allocated to the intervention groups

25. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 25 Q1. What do you think of the sample size Only 8 per group gives little chance to accurately address hypotheses What happens if you change age categories of only 2 subjects in Table 1? What about education level? Small samples are unstable ! A few questions …

26. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 26 Table 1 – Descriptive data

27. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 27 Q2. How are the pain scales expressed? Visual analogue scales with 0 being no pain and 10 being extreme pain How are they treated in the analysis? Table 2 - Continuous data A few questions … no pain extreme pain 010 - this may make it even harder to see an effect since they are not very precise

28. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 28 Table 2 – Descriptive data

29. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 29 Hayman et al. (1995) Quick summary of the paper: – matched pair analysis of twins to examine nongenetic influences of obesity on lipid profile and blood pressure both cross-sectionally (Phase 1, N=73 pairs) and longitudinally (Phase 2, N=56 pairs)

30. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 30 Q1. Describe the sample population in terms of race, age and sex? Did it change much over time? Age: at Phase 1: Sex: at Phase 1: A few questions … at Phase 2: Race – at Phase 2: all white, both Phases M=8.5 yrs, SD = 1.8 yrs M=12.5 yrs, SD = 1.8 yrs 43.8% male, 56.2% female 44.6% male, 55.4% female

31. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 31 Q2. how long was follow-up period p278 - “median interval between measurements was 40 months What does this mean? Roughly half the time periods were longer than 40 months and half were less than 40 months (i.e. it was the “dividing line”) A few questions …

32. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 32 Table 1 – Descriptive data

33. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 33 Table 2 – Contrasting twins

34. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 34 Table 3 – Descriptive data for the follow-up study

35. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 35 Next Week - Lecture 3: Graphs, Normal Curve and Central Limit Theorem For next week’s class please review: 1.Page 13 in syllabus 2.Textbook Chapter 2, pages 46-57 3.Textbook Chapter 3, pages 65-70 4.Syllabus papers: i) Kilpack (1991) ii) Paulson & Altmaier (1995)

36. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 36 Workshop Rooms: H018, H19 and H9 MS016, MS017, MS018, MS022 MS023, MS027, MS028, and MS029 All rooms are now confirmed for rest of the year so please go to the same room with your group as last time

37. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 37 ValidCumulative ValueFrequencyPercent 20.0014.0 22.0028.0 12.0 23.00312.0 24.0 23.5014.0 28.0 24.00312.0 40.0 24.5014.0 44.0 25.00416.0 60.0 25.5028.0 68.0 26.00520.0 88.0 26.5014.0 92.0 27.5014.0 96.0 28.0014.0 100.0 Total25100.0 “In Group”Session – Q#1: 3 rd column not necessary – i.e. no missing data !

38. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 38

39. School of Nursing Institute for Work & Health Nur 318b 2002 Lecture 2: page 39 A Quick Review from Last Week - 2 Summarizing Hypotheses  Null or Research?  Directional or Non-directional?  Causal or Associative?  Simple or Complex?