Descriptive Statistics. Applied Statistics Advanced Data Analysis. Variable Types. Descriptive Statistics DS - 1

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1 Applied Statistics Advanced Data Analysis Andy Chang Youngstown State University Statistics in Broader Sense Statistics is a field of study concerned with the 1) data collection, [Producing data] 2) organization, summarization, examination and providing an overview of the general features of data, [Exploring Data] 3) and the drawing of inferences about a body of data (population) based on the properties of a part of the data (sample) observed. [Statistical Inference] 1 Producing Data Exploring Data Statistical Inference 2 Areas of Statistics 1) Sampling [Representative sample] 2) Descriptive statistics 3) Inferential statistics Estimation Hypothesis Testing Variable Types Quantitative Variables (numeric) [height, number of subscriptions,...] Continuous: a variable that has an uncountable number of possible values. (measurements) Discrete: a variable that has a countable number of possible values.(counts) Qualitative (Categorical) Variables [hair color, gender,...] 3 4 Measurement Scales Nominal: consists of labels, names or categories. Ordinal: data that the order or rank is meaningful. Interval: numerical data that arithmetic operations are meaningful. Ratio: data that the ratio of two data is meaningful. Descriptive Statistics Data Presentation Grouping tables (Frequency Distribution Tables) Graphical summary Numerical Summary Center Dispersion (Spread) Exploratory Data Analysis 5 6 DS - 1

2 Data Presentation What type of statistical technique is appropriate for Data Presentation? Categorical variable? Quantitative variable? 7 (A complete list) ID Height Weight BirthMonth Exp. Gender H F H F T M H F T F H M H F H M T M H M T M T F H M T F H M H F T F T M H M T M H M H M 8 Univariate Analysis Analyze observations on a single variable. Grouping and Displaying Categorical Data Frequency Table and Charts (One Categorical Variable) Class Frequency Relative Frequency Female 9 9/22 =.409 = 40.9% Male 13 13/22 =.591 = 59.1% Total 22 % Count Pareto chart Percent 6585 Female 40.9% Male 59.1% White Asian N. Hawaiin/Other Pac 20 Pe rcent 10 0 Female Male Black or African Ame Hispanic or Latino Multiple - Non-hispa Multiple - Hispanic How do you describe yourself Am. Indian or Alaska sex 11 * Bars arranged according to their frequencies. 12 DS - 2

3 Frequency Distribution Table Data: 135, 119, 175, 106, 135, 170, 180, 205, 195, Grouping and Displaying Quantitative Data Class Tally Frequency - < < <1 1 - < < < < <2 2 - < <300 Total to less than Frequency Distribution Table Frequency Distribution Table Data: 135, 119, 175, 106, 135, 170, 180, 205, 195, Class Tally Frequency - < < < < < < < < < <300 1 Total 22 (From data sheet) Class Frequency Relative Freq. Cumulative R.F. - < /22 =.136 3/ < /22 =.136 6/ <1 2 2/22 =.091 8/ < /22 = / < /22 = / < /22 = / < /22 = / <2 0 0/22 = / < /22 = / < /22 = /22 Total Classes: Categories for grouping data. Frequency (class frequency): The number of data values in a class. Relative frequency: The ratio of the frequency of a class to the total number of pieces of data. Frequency distribution: A listing of classes and their frequencies. Relative Frequency distribution: A listing of classes and their relative frequencies. Upper class limit: The largest value that can go in a class. Lower class limit: The smallest value that can go in a class. Class width: The difference between the lower class limit of the given class and the lower class limit of the next higher class. Class midpoint (class mark): The midpoint of a class. Guidelines for grouping data: (for quantitative variable) There should be between five and twenty classes. Each piece of data must belong to one, and only one, class.(mutually Exclusive) Whenever feasible, all classes should have the same width DS - 3

4 To build a Frequency Table: Find the range of the data: Range = Largest value smallest value Use the range and try different class width to determine how many classes you need to make frequency table or histogram. Student data example: Range = = 179/20 9 If using a class width of 20, there ll be about 9 classes which is good. Frequency Distribution Table (From data sheet with different boundaries) Class Frequency Relative Freq. Cumulative R.F. - < /22 =.136 3/ < /22 =.136 6/ <1 2 2/22 =.091 8/ < /22 = / < /22 = / < /22 = / < /22 = / <2 0 0/22 = / < /22 = / < /22 = /22 Total Histogram (SPSS) Polygon (SPSS) Polygon (SPSS) Cumulative R. F. Histogram % % DS - 4

5 Cumulative R. F. Polygon (Ogive) What to observe in Histograms? % % Outliers: observations that stand out from the rest for some reason. Center: the middle of the data. Spread: the range; the extent of the data; how far the values are from each other. Shape: distribution pattern. [Skewness, symmetry, uniform, Normal,...] Test of Normality, Q-Q Plot Symmetric (Bell) shape Skewed to the right, or positively skewed Stemplots (or Stem-and-leaf plots) Bimodal -- leading digits are called stems -- final digits are called leaves Uniform Skewed to the left, or negatively skewed Example: (number of hysterectomies performed by 15 male doctors) 27,, 33, 25, 86, 25, 85, 31, 37, 44, 20, 36, 59, 34, 28 Example: (number of hysterectomies performed by 15 male doctors) 27,, 33, 25, 86, 25, 85, 31, 37, 44, 20, 36, 59, 34, Stemplot 29 Ordered Stemplot 30 DS - 5

6 Example: Number of hysterectomies performed by 15 male doctors: 27,, 33, 25, 86, 25, 85, 31, 37, 44, 20, 36, 59, 34, 28 Back-to-back stem-plot by 10 female doctors, the numbers are: 5, 7, 10, 14, 18, 19, 25, 29, 31, 33 (Male) (Female) (Female) (Male) Box Plot Examine Bivariate Data (Bivariate Analysis) Examine the relation between two variables N = 21 HEIGHT Two Categorical Variables Contingency Table Two Categorical Variables Cluster bar chart Smoker Non- Smoker Column Total Cancer No cancer Row Total 20 (20%) 5 (5%) 25 (25%) 30 (30%) 45 (45%) 75 (75%) (%) (%) Odds of smoker to have cancer: 20/30 = 6/9 Odds of nonsmoker to have cancer: 5/45 = 1/9 Odds Ratio = (6/9)/(1/9) = DS - 6

7 Two Quantitative variables Two Quantitative Variables Data: Temperature Mortality Index Average annual temperature and the mortality index for a type of breast cancer in women in certain region of Europe. Mortality Index Average Temperature 38 Response/Explanatory Variables Response (Dependent, Outcome) Variable Lung Cancer, Mortality Index Explanatory (Independent, Predictor) Variable Smoking, Average Temperature Time Plot Rate Time Rate Time A Categorical & A Quantitative Variables Side-by-side Boxplot Multivariate Analysis Analyze observations on two or more than two variables HEIGHT N = 8 13 Female Male sex DS - 7

8 Type of Statistical Studies Observational Study: conditions to which subjects are exposed are not controlled by the investigator. (no attempt is made to control or influence the variables of interest) Results from observing behavior and outcomes from the use of medicine for 200 randomly selected patients. (Patients chose their medicine) Treatment Drug A Hypertension Yes 44 No 56 Total Experimental (Controlled) Study: conditions to which subjects are exposed to are controlled by the investigator. (treatments are used in order to observe the response) (Randomization, Replications) Drug B Total Drug A: 44/ = 44% Drug B: 29/ = 29% Simpson s Paradox Hypertension Treatment Below 65 Yes No Total Yes 65+ No Total Treatment 1 & Treatment 2 Cause? Patient s Survival Drug A Drug B * Older patients prefer Drug A OR <65: Drug A: 5/23 = 22% Drug B: 17/77 = 22% OR 65+: Drug A: 39/77 = 51% Drug B: 12/23 = 52% 45 Confounding variables Patient s Age & Health Condition 46 Confounding Effect What can we do? Variables, whether part of a study or not, are said to be confounded when their effects on the outcome cannot be distinguished from each other Age may affect the reaction to drug and may also affect drug choosing decision. Completely Randomized Design Control Group, Placebo Group Treatment Group Multiple Regression DS - 8