Slides to accompany Weathington, Cunningham & Pittenger (2010), Statistics Review (Appendix A) Bring all three text books Bring index cards Chalk? White-board pen?
Objectives Variables Parameters vs. estimates Measures of central tendency Measures of variability Standardized scores
Variables X and Y represent the variables and/or sets of data N and n to indicate the number of observations N for the total number of observations n for the observations in a subset Subscripts (e.g., X1 , Y1) represent an individual score within a specific group
Parameters vs. Estimates Population parameters describe a population Estimates pertain to sample characteristics Examples: µ = M σ = SD ρ = r
Measures of Central Tendency Descriptive statistics Indication of the “typical” score Three general types: Mode Median Arithmetic Mean For summarization and interpretation Represent the “typical score”
Mode Mo Most frequently occurring score in a set of data What’s the Mode of this set? X {9 1 3 1 3 9 7 4 7 8 9 8 9} X {1 1 3 3 4 7 7 8 8 9 9 9 9} 9 Easiest way to find it is to rank them all in order Good for finding a cluster of common scores Great when there are multiple “peaks” in a frequency plot (multimodal) or when data are discrete (e.g., number of children, hours worked)
Median Mdn, Q2 The score that divides ranked data in half X {1 1 3 3 4 7 7 8 8 9 9 9 9} (13 + 1)/2 = 7 Count 7 up from lowest score The middle-most score Good for skewed data b/c not affected by outlying scores (give a demonstration)
Arithmetic Mean M Sum of observed scores divided by n X {1 1 3 3 4 7 7 8 8 9 9 9 9} ΣX = 78 M = 78/13 = 6 There is more than one type of Mean – we are thinking of the arithmetic one here usually See the evidence of skew, given all the 9’s? Sensitive to outliers and skewed distributions But, M has the smallest total difference between itself and each observation Σ(X – M) = 0 This is important for many statistical analyses
Measures of Variability Quantify spread of data around a set’s central tendency Proper measure depends on scale type, distribution symmetry, and desired inferences Simple Range Semi-Interquartile Range SD and Var
Discuss this figure, highlighting the use of central tendency and variability statistics to describe the data (they are descriptive statistics) Figure A.1. Four distributions of data, each with a mean of 50 and standard deviation of 5, 10, 15, or 20. As the standard deviation increases, the spread of the scores around the mean becomes much wider.
Simple Range Simplest measure of dispersion Range = Highest score – Lowest score X {1 1 3 3 4 7 7 8 8 9 9 9 9} Range = 9 – 1 = 8 Greatly affected by outliers Only good for general description of a data set
Semi-Interquartile Range SIR Difference between 75th and 25th percentile scores, divided by 2 (Q3 – Q1) / 2 X {1 1 3 3 4 7 7 8 8 9 9 9 9} Find Median (Q2) (13 + 1)/2 = 7 Determine location of Q1, Q3 (7 + 1)/2 = 4 Locate Q1, Q3 75% of the scores in the set are at or below Q3 and 25% at or below Q1 Not affected by extreme scores, so this is a good statistic for describing skewed data
SD and Var s and s2 if parameters, both indicate dispersion Calculation depends on whether working with parameters or estimates SD2 = Var X {1 1 3 3 4 7 7 8 8 9 9 9 9} Var = Σ(X – M)2/(n-1) = ((ΣX2) – ((ΣX)2/n))/ (n-1) = ((12+12...92) – (78)2/13)/(13-1) = (586 – 468)/12 = 9.83 SD = √VAR = √9.83 = 3.14 Usually we are working with estimates (n – 1) Conceptually, sd = typical distance of scores from the M
Standardized Scores z-score “Equalizes” scales from scores in different groups makes comparison possible Can help with interpretation of data X {1 1 3 3 4 7 7 8 8 9 9 9 9} z = (X – M)/SD e.g., (1-6)/3.14 = -1.59 Perform calculation for each score in X Xz {-1.59 ... .96} M = 0, SD = 1 Walk through the calculations with them and make sure they understand what’s going on. Note that larger z-scores denote further distance from the mean Among other uses, z-scores are useful when we want to compare a person’s scores on several measures, each of which may have a different M and SD