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Chapter 21 Describing Data Through Statistics Copyright © 2008 Wolters Kluwer Health | Lippincott Williams & Wilkins Types of Statistics Descriptive statistics Used to describe and synthesize data Inferential statistics Used to make inferences about the population based on sample data Copyright © 2008 Wolters Kluwer Health | Lippincott Williams & Wilkins Frequency Distributions • A systematic arrangement of numeric values on a variable from lowest to highest, and a count of the number of times each value was obtained • Frequency distributions can be described in terms of: – Shape – Central tendency – Variability Copyright © 2008 Wolters Kluwer Health | Lippincott Williams & Wilkins Construction of Frequency Distributions • Can be presented in tabular form (counts and percentages) • Can be presented graphically – Histograms – Frequency polygons Copyright © 2008 Wolters Kluwer Health | Lippincott Williams & Wilkins Shapes of Distributions 1. Symmetry • Symmetric • Skewed (asymmetric) Positive skew (long tail points to the right) Negative skew (long tail points to the left) Copyright © 2008 Wolters Kluwer Health | Lippincott Williams & Wilkins Examples of Symmetric Distributions Copyright © 2008 Wolters Kluwer Health | Lippincott Williams & Wilkins Examples of Skewed Distributions Copyright © 2008 Wolters Kluwer Health | Lippincott Williams & Wilkins Shapes of Distributions (cont.) 2. Peakedness (how sharp the peak is) 3. Modality (number of peaks) Unimodal (1 peak) Bimodal (2 peaks) Multimodal (2+ peaks) Copyright © 2008 Wolters Kluwer Health | Lippincott Williams & Wilkins Modality • Symmetric • Unimodal • Not too peaked, not too flat Copyright © 2008 Wolters Kluwer Health | Lippincott Williams & Wilkins Distribution of Values: Central Tendency Index of “typicalness” of set of scores that comes from center of the distribution Mode—the most frequently occurring score in a distribution 2 3 3 3 4 5 6 7 8 9 Mode = 3 Copyright © 2008 Wolters Kluwer Health | Lippincott Williams & Wilkins Distribution of Values: Central Tendency (cont.) • Median—the point in a distribution above which and below which 50% of cases fall 2 3 3 3 4 5 6 7 8 9 Median = 4.5 • Mean—equals the sum of all scores divided by the total number of scores 2 3 3 3 4 5 6 7 8 9 Mean = 5.0 Copyright © 2008 Wolters Kluwer Health | Lippincott Williams & Wilkins Indexes of Central Tendency • Mode, useful mainly as gross descriptor, especially of nominal measures • Median, useful mainly as descriptor of typical value when distribution is skewed • Mean, most stable and widely used indicator of central tendency Copyright © 2008 Wolters Kluwer Health | Lippincott Williams & Wilkins Variability of Distributions The degree to which scores in a distribution are spread out or dispersed • Homogeneity—little variability • Heterogeneity—great variability Copyright © 2008 Wolters Kluwer Health | Lippincott Williams & Wilkins Two Distributions of Different Variability Copyright © 2008 Wolters Kluwer Health | Lippincott Williams & Wilkins Indexes of Variability • Range: highest value minus lowest value • Standard deviation (SD): average deviation of scores in a distribution • Variance: a standard deviation, squared Copyright © 2008 Wolters Kluwer Health | Lippincott Williams & Wilkins Standard Deviations in a Normal Distribution Copyright © 2008 Wolters Kluwer Health | Lippincott Williams & Wilkins Contingency Table (or Crosstab) • A two-dimensional frequency distribution; frequencies of two variables are crosstabulated • “Cells” at intersection of rows and columns display counts and percentages • Variables must be nominal or ordinal Copyright © 2008 Wolters Kluwer Health | Lippincott Williams & Wilkins Contingency Table for Gender and Smoking Status Relationship Copyright © 2008 Wolters Kluwer Health | Lippincott Williams & Wilkins Correlation Procedures • Indicate direction and magnitude of relationship between two variables • Used with ordinal, interval, or ratio measures • Can be shown graphically (scatter plot) • Correlation coefficient (usually Pearson’s r) can be computed • With multiple variables, a correlation matrix can be displayed Copyright © 2008 Wolters Kluwer Health | Lippincott Williams & Wilkins Various Relationships Graphed on Scatter Plots Copyright © 2008 Wolters Kluwer Health | Lippincott Williams & Wilkins