Lecture Week 4 Inspecting Data: Distributions

Transcription

1 Lecture Week 4 Inspecting Data: Distributions Introduction to Research Methods & Statistics Hemmo Smit

2 So next week No lecture & workgroups But Practice Test on-line (BB) Enter data for your own research Practice SPSS skills with own data

3 Overview Descriptive research Describing and presenting data Frequency distributions Graphical displays (1) Measures of Central Tendency and Variability Graphical displays (2): Boxplots Read: Leary: Chapter 6 Howell: Chapter 2

4 Types of descriptive research Survey research Demographic research Attitudes, lifestyles, behaviors, problems Patterns of basic life events: birth, marriage, migration, death. Epidemiological research Occurrence of disease and death

5 3 types of surveys Cross-sectional Successive independent samples Longitudinal (panel survey design) One-shot cross-section of the population Changes over time Different respondents each time! Are samples comparable? Changes over time Same respondents more than once! Drop out

6 Describing and presenting data 3 criteria for a good description: 1) Accurate 2) concise Trade-off 3) comprehensible - Loss of information - Possible distortion Data can be presented in numerical and graphical format TIP: Always start with graphs Beware: Scale of measurement?!?

7 ( y y) ) How to describe a distribution? A) Overall pattern 1) Shape - number of peaks (uni-, bi- of multi-modal)? - symmetrical or skewed? 2) Central tendency / Location: midpoint 3) Spread: a little or a lot? B) Deviations from the pattern - Outliers: observations that lie far from the majority - Tails: thick or thin?

8 Frequency distributions: Example How do children recall stories? Respondents: 25 children Task: Tell researcher about a movie Dependent variable: number of and then statements (see Howell, Exercise 2.1, p.55)

9 Raw data and frequency distributions Table 1. # and then statements Table 2. # and then statements Score f P Total

10 Absolute and relative frequencies Absolute frequency (f) = Number of respondents with a given score Disadvantage: hard to interpret / compare Relative frequency (P) = Proportion of the total with a given score (P = f / n) Advantage: easy to interpret Note: 0 < P < 1 P x 100 = %

11 SPSS: Frequencies - Menu Analyze > Desciptive Statistics > Frequencies

12 SPSS: Frequencies Dialog box

13 SPSS: Frequencies - Output

14 Grouped frequency distribution (1) Simple frequency distributions unclear in case of: - small number of participants in each category and/or - variables with many categories Solution: grouped frequency table Distribute the raw data over K class intervals and make a new frequency distribution Make sure all intervals are: - exhaustive and mutually exclusive - of equal width

15 Grouped frequency distribution (2) Rule 1: number of classes (K) = n Rule 2: class interval width (I) = range / number of classes (Range (R) = highest score lowest score) In our example Number of intervals = 25 = 5 Range = = 11 Interval width = 11 / 5 2 or 3 Score f P total

16 SPSS: Grouped frequency distribution (1)

17 SPSS: Grouped frequency distribution (2) 1 2

18 SPSS: Grouped frequency distribution (3) 2 1 3

19 SPSS: Grouped frequency distribution (4) 1 2

20 SPSS: Grouped frequency distribution (5)

21 Cumulative frequency distributions (1) Class interval Real lower limit Real upper limit Total Real lower limit = lower limit 0.5 Real upper limit = upper limit Midpoint = upper limit + lower limit / 2 Midpoint f P F

22 Cumulative frequency distributions (2) Class interval Real lower limit Real upper limit Midpoint f P F Total F = Cumulative Relative Frequency (CRF): add all previous proportions.

23 Cumulative frequency distributions (3) Class interval Real lower limit Real upper limit Midpoint f P F Total NB. Also possible: cumulative absolute frequency

24 ( y y) ) Cumulative frequency distributions (4) The cumulative relative frequency polygon graphs the possibility that someone has a score of X or lower.

25 Count Count Graphical displays: Nominal / Ordinal Raw data Grouped Bar score score Pie

26 Graphical displays: Interval Histograms Stem & Leaf Display Freq. Stem & Leaf 1,00 Extremes (=<12,0) 2, , , , , , , , , Stem width: 1 Each leaf: 1 case(s)

27 Histogram symmetrical or skewed? Symmetrical Negatively skewed Positively skewed

28 SPSS: Graphs Chart Builder / Legacy Dialogs

29 SPSS: Graphs > Legacy Dialogs

30 SPSS - Graphs > Chart builder 3 1 2

31 Measures of central tendency 1. Mode (Mo) = most common score 2. Median (Mdn) = middle score (50 th percentile) Median location 1 N 2 3. Mean (M) = average x x 1 x 2... n x n or x 1 n x i

32 s2 sxx Central tendency and skewness Shape Mode Median Mean positive skew symmetrical negative skew A B C A A A C B A

33 Measures of variability 1. Range (R) = Highest score Lowest score 2. Interquartile range (IQR) = Q3 Q1 3. Standard deviation (s or σ) = spread around the mean 4. Variance (s² or σ²) = spread around the mean

34 Variance and standard deviation Score Deviation Squared x 1 x 2 x 3 x x 1 x n i x x 2 x x 3 x n x ( x1 x) ( x2 x) ( x x Sum x ) 2 ( x n x) Variance s Standard deviation s 2 x x ( x n i x) 1 ( xi x) n The standard deviation and variance are: only suitable as measures of spread around the mean Not robust against outliers

35 Five-number summary and boxplot Five-number summary consists of: Minimum = Lowest (non-outlying) score Q1 = 25 th percentile (25% lower, 75% higher) Median (=Q2) = 50 th percentile Q3 = 75 th percentile Maximum = Highest (non-outlying) score Graphical display: Boxplot

36 Boxplot - Example Data: Nummerical (five-number summary) Max = 93 Q3 = 45 M = 28 Q1 = 19 Rule of thumb Outlier = observation that lies 1.5 x IQR above Q3 or below Q1. Min = 3 IQR = = 26 Graphical (boxplot) Q1 1.5*IQR = -20 Q *IQR = 84

37 Overview Scale of Measurement Nominal Graphical CT Spread Bar chart Mode --- (Pie chart) Ordinal Boxplot Median Range IQR Interval Histogram Mean - Standard dev. (and higher) (Stem&Leaf display) - Variance

38 What have you learned today? What are the various ways to represent distributions numerically? What are the various ways to represent distributions graphically? How to describe a distribution How to create and evaluate various numerical and graphical representations of distributions How to determine what numerical and graphical representation is suitable for a variable.

39 Next week No lecture and workgroups Practice test on Blackboard Enter your own data In two weeks Normal distribution and standard scores Read: Howell: Chapter 3