Descriptive Statistics

1. Descriptive Statistics Describe the basic features of the data that were gathered in a study. Measurements or observations of a variable

2. Example study spreadsheet 95 cases 28 variables →

3. DSs provide summaries about various aspects of a sample or a population • Distinguished from inferential statistics • DSs Include measures of - • Distribution • Frequencies of the values of observations • Central tendency • Define the middle of a distribution of values • Dispersion • The spread of values around the central region of a distribution

4. Example Data Series • Number of visits provided to whiplash patients - Case # Visits 1 7 2 2 3 2 4 3 5 4 6 3 7 5 8 3 9 4 10 6 11 2 12 3 13 7 14 4

5. Distribution • Provides a summary of: • Frequencies of each of the values • 2 – 3 • 3 – 4 • 4 – 3 • 5 – 1 • 6 – 1 • 7 – 2 • Ranges of values • Lowest = 2 • Highest = 7 Case # Visits 1 7 2 2 3 2 4 3 5 4 6 3 7 5 8 3 9 4 10 6 11 2 12 3 13 7 14 4

6. Frequency distribution table FrequencyPercentCumulative % • 2 3 21.4 21.4 • 3 4 28.6 50.0 • 4 3 21.4 71.4 • 5 1 7.1 78.5 • 6 1 7.1 85.6 • 7 2 14.3 100.0

7. Graphing Frequencies • Histogram – commonly used for frequencies

8. Bar charts for categorical or dichotomous data

9. Case # Visits 1 7 2 2 3 2 4 3 5 4 6 3 7 5 8 3 9 4 10 6 11 2 12 3 13 7 14 4 Excel example

10. Measures of central tendency • Mean • Most commonly used • AKA as average • All values of a series of numbers are added, then divided by the number of elements • Mode • Most frequently occurring value • Median • The number that divides a series in half when they are all listed in order

11. Mean Visits 7 2 2 3 4 3 5 3 4 6 2 3 7 4 Sample Population __ 55 ∕ 14 = 3.93

12. Mode • Most frequent value • 2 – 3 • 3 – 4 • 4 – 3 • 5 – 1 • 6 – 1 • 7 – 2 The number 3 occurs most frequently, so it is the modal value

13. Median Case # Visits 1 7 2 2 3 2 4 3 5 4 6 3 7 5 8 3 9 4 10 6 11 2 12 3 13 7 14 4 List scores in order from least to most, then find the value in the middle 2 2 2 3 3 3 3 4 4 4 5 6 7 7 Average = (3 + 4)/2 = 7/2 = 3.5 • Odd number of values, it’s the middle value • Even number of values, count from each end ofthe series toward the middle and average the2 middle values

14. Most biological data are symmetrically distributed Curve (line) depicting shape of the data Normal (bell-shaped) curve

15. These data are not symmetric Mode=3 Median=3.5 X=3.93

16. Fitting a curve to a graph This curve is the best fitof a line that depicts the shape of the data This distribution is skewed to the right

17. The direction of skew depends on which way the longer tail is pointing

18. Recall calculating SD Wider spread = higher SD Narrow spread = lower SD

19. Spread influences SD

20. Normal curve X = Median = Mode Symmetric

21. All “Normal” distributions • Are symmetric • Have a single peak at their mean (which is also the median) • Are bell-shaped curves • Extend ≈ 3 standard deviations above and below the mean • Have a total area under the curve of 100%

22. Properties of Normal Distribution 95% Actually, it’s 68.26%, 95.44%, and 99.72

23. Normal Distribution Cont. • The proportion under the normal distribution is constant for a given number of SDs • By knowing how many SDs above or below the mean a given value is, we can determine the associated probability • Rather than calculating values, there are tables available that use these properties

24. Normal curve calculated from diastolic blood pressures of 500 men % above 2 SDs? Mean 82 mmHg, standard deviation 10 mmHg.

26. Comparing Means

27. Four Levels of Measurement • Each level has specific rules for the operation and interpretation of associated data • Understanding these rules is necessary in order to determine which statistical test should be used • Not uncommon to see violations in the biomedical literature

28. Nominal Measurement • AKA classificatory scale because individuals or objects are classified into categories • A code in the form of a number, name, or letter is assigned to each category • Examples • Agree (0) disagree (1) • Hair color – blonde (0), brunette (1), red (2), black (3) • Counting the number of elements in the categories is the only permissible mathematical operation (i.e., 25 males and 30 females)

29. Ordinal Measurement • Includes categories, but they are rank-ordered • Examples • Lieutenant, captain, major, colonel, general • Normal, minimal, moderate, severe • Ordinal value only represents position of order, not quantity • Counting the number of elements in the categories is the only mathematical operation allowed

30. Interval Measurement • Exhibit equal intervals between ordered measurements • Does not have a true zero • Examples • Fahrenheit scale – has a zero, but does not signify the absence of heat (not a true zero) • Height, weight, and age qualify as interval, but also qualify at a higher level of measurement . . . ratio

31. Ratio Measurement • There are equal intervals and a true zero • The most advanced level of measurement • Examples • Cervical range of motion • A patient’s capacity for lifting • Diastolic blood pressure

32. Ratio Measurement Cont. • To determine if a measurement qualifies as being ratio, ask . . . • Does a measurement of zero represent an absence of the characteristic being tested? • Do differences between consecutive measured numbers represent equal amounts of a characteristic?

33. NOIR • A helpful mnemonic • NominalOrdinalIntervalRatio

34. The Importance of NOIR • The first question to ask when critiquing the data analysis presented in a scientific article is, “what level of measurement was involved?” • As stated before, it is not uncommon to see the wrong statistical test performed because of the improper identification of level of measurement

35. When to use mean, median, or mode

36. Qualitative vs. Quantitative