Fundamental Elements of Statistics

Transcription

1 Fundamental Elements of Statistics Slide Statistics the science of data Collection Evaluation (classification, summary, organization and analysis) Interpretation Slide

2 Population Sample Sample: A subset of the population Slide The objective of inferential statistics is to make inferences (prediction, decision) about a population based on the information contained in a sample. Example. To estimate the average height of all eighteenyear-old boys in USA. Example. To estimate the proportion of cola drinkers in the world who prefer Pepsi. Slide

3 Population the set representing all measurements of interest to the investigator Example. the set of the heights of all eighteen-year-old boys in USA Example. the set of cola preferences of all cola drinkers in the world Slide Sample a subset of population. Example. the set of the heights of, randomly selected eighteen-year-old boys in USA Example. the set of cola preferences of, randomly selected cola drinkers in the world Slide

4 Statistical Inference generalization about a population based on sample data Example. concluding that the average heights of eighteen-year-old boys in USA is Example. concluding that the proportion of cola drinkers in the world who prefer Pepsi is Slide 7 Measure of Reliability statement about the uncertainty associated with an inference How good is the result of the statistical inference? How precise are the estimations? These questions will be answered in the statistical inference part. Slide 8

5 Describe Data using Graphs (Graphical Method) Describe Data using Numerical Measures (Numerical Method) Slide 9 Variable a characteristic that changes or varies over time, or a characteristic that varies across individuals or objects under the consideration at a particular point in time Example. To estimate the average height of all eighteen-yearold boys in USA. (variable: height) Example. To estimate the proportion of cola drinkers in the world who prefer Pepsi. (variable: cola preference) Slide

6 Experimental Unit individual or object on which a variable is measured Example. To estimate the average height of all eighteen-yearold boys in USA. (experimental units: selected eighteen-year-old boys) Example. To estimate the proportion of cola drinkers in the world who prefer Pepsi. (experimental units: selected cola drinkers) Slide Variables Quantitative (Numerical) Qualitative (Categorical) Continuous Variables Discrete Variables A continuous variable is one that can assume all of the infinitely many values corresponding to a line Interval. A discrete variable can assume only a countable number of values. Slide

7 Example Identify each of the following variables as qualitative or quantitative. If a variable is quantitative, is it a continuous variable or discrete variable? a. The most frequent use of your microwave (reheating, defrosting, warming, other) during December. qualitative b. The number of consumers refusing to answer a telephone survey. quantitative, discrete Slide c. The type of cable service delivered to residences (standard cable, premium cable or antenna only) in Atlanta. qualitative d. The completion time for a particular task performed by a computer software program. quantitative, continuous e. The number of stocks on the New York Stock Exchange showing a gain from March,, to July, quantitative, discrete Slide 7

8 Graphical Method: Pareto Chart Slide Carbon Monoxide and Ozone Sources (million metric tons / year) Carbon Monoxide Transportation Fuel combustion Industrial processes Solid waste Miscellaneous Total Volatile Organic Compounds Transportation Fuel combustion Industrial processes Solid waste Miscellaneous Total Slide 8

9 Carbon Monoxide Transportation Fuel combustion Industrial processes Solid waste Miscellaneous Total Slide 7 Carbon Monoxide Transportation Fuel combustion Industrial processes Solid waste Miscellaneous Total Slide 8 9

10 Graphical Method: Histogram Slide 9 Largest Smallest Slide

11 Histogram Slide Step. Determine the class width. class width largest number smallest number = number of classes EPA Mileage example: smallest number =. largest number =.9 class width Steps for Constructing Histogram Step. Determine the number of classes (usually - classes). EPA Mileage example: Ten classes are used..9. = =.9. Slide

12 Step. Locate the class boundaries. The lowest class must cover the smallest number and the highest class must cover the largest number. EPA Mileage example: Classes: [.,.), [.,.), [.,.),, [.,.) Step. Find the frequencies and relative-frequencies for all the classes. The frequency of a class is the number of items in the class. frequency relative - frequency =. number of items in the entire data set Step. Construct frequency and relative-frequency table. Slide Classes [.,.) [.,.) [.,.) [.,.) [., 7.) [7., 9.) [9.,.) [.,.) [.,.) [.,.) Total Frequency 9 8 Slide

13 Classes [.,.) [.,.) [.,.) [.,.) [., 7.) [7., 9.) [9.,.) [.,.) [.,.) [.,.) Total Frequency 9 8 Relative Frequency Slide Step. Draw frequency histogram and relative-frequency histogram. Frequency Histogram Slide

14 . Relative-frequency Histogram.... Slide 7 Graphical Method: Stem-and-leaf Display (Stemplot) Slide 8

15 Number of PSI Days Greater Then at All Ozone Trend Sites O Trend Sites Atlanta Boston Chicago 9 Dallas 7 Denver Detroit 8 7 Houston 9 9 Kansas City Los Angeles 9 7 New York 8 Philadelphia 9 7 Pittsburgh 9 7 San Francisco Seattle Washington Total Slide 9 O Trend Sites Atlanta Chicago 9 Slide

16 O Trend Sites Houston 9 9 New York 8 Slide

17 Stemplot vs. Histogram Slide Graphical Method: Boxplot Slide 7

18 Basic Steps for Constructing Modified Boxplot:. Find the median (Q M ), lower quartile (Q L ) and upper quartile (Q U ) of the data.. Calculate the interquartile range (IQR), lower fence and upper fence. Example n = Ordered data:,,,,,, 9, 9,,,,,,, Q M = 9, Q L =, Q U = IQR = lower fence = Q L. IQR = upper fence = Q U +. IQR = outlier:. Draw a rectangular box whose left edge is at the lower quartile and whose right edge is at the upper quartile.. Draw a vertical line segment inside the box at the median. 9 8

19 . Place marks at distances. IQR from either end of the box - these are the fences.. Extend horizontal line segments from each end of the box out to the most extreme observations that are still within the fences. 7. Outliers are observations beyond the fences. Show each outlier using an asterisk. * 9 Lower Fence Upper Fence 9

20 Graphical Method: Time Series Plot Slide 9 Carbon Monoxide Total Volatile Organic Compounds Total Slide

21 O Trend Sites Total Slide Graphical Method: Scatter Plot Slide

22 Midterm Final Midterm Final This data set contains midterm exam scores and final exams score of students. The goal is to find the relationship between the midterm exam score (x) and the final exam score (y). Slide Scatter Plot of Final vs. Midterm Slide