Statistics Fundamentals: Data Analysis and Display Techniques

1. Lecture 1 STAT 211 – 019 Dan Piett West Virginia University

2. Overview • 1.1 Science of Statistics • 1.2 Displaying Small Sets of Numbers • 1.3 Graphing Categorical Data • BREAK - Problems • 1.4 Frequency Histograms • 1.5 Density Histograms • 1.6 Centers of Data • 1.7 Mean vs Median

3. Section 1.1 Science of Statistics

4. Definitions • Statistics • The science of collecting, organizing, and analyzing data for the purpose of estimation and making inferences. • Data • Values which arise from observing characteristics on a selected group (sample) of individuals. The characteristic(s) which are observed are called variables. • Population • The entire group of “things” from which data may be collected. • Sample • The selected group of the population that is studied.

5. Population vs. Sample • Population • All undergraduate students at WVU • Sample • A group of 100 undergraduate students at WVU All WVU Students (population) Group of 100 Students (sample)

6. Types of Data • Variables • The characteristics which we look at when observing data. These can be classified into four groups. • Numeric • Discrete • Continuous • Categorical • Ranked (Ordinal) • Unranked (Nominal)

7. Numeric Variables • Numeric Variables • Variables whose values represent quantities. This can be further broken down into discrete and continuous. • Discrete Numeric • Variables which usually arise by counting • Examples • Number of cars in a parking lot • Number of courses taken in a semester • Continuous Numeric • Variables which arise by measuring • Examples • Time spent running a marathon • Weight

8. Categorical Variables • Categorical Variables • Variables that are not numeric. These can be further classified into ranked and unranked. • Ranked Categorical Variables • If the possible values of a categorical value follow a “natural” ordering • Example • Olympic Medal • Unranked Categorical Variables • Anything that is not ranked • Example • Blood Type

9. Two Major Branches of Statistics Descriptive Statistics Inferential Statistics Use graphical displays and numeric summarizations to represent data. First half of this semester Use analytic methods and theory of probability to draw conclusions or make decisions. Second half of this semester

10. Section 1.2 Displaying Small Sets of Numbers

11. Displaying Small Sets of Data • We will be looking at three different ways to display small sets of data. • Dot Plots • Stem and Leaf Display • Histograms

12. Dot Plot • Each Dot represents one data value • Example – Height of Students 62, 71, 65, 68, 64, 72, 66, 68, 70, 67, 67, 68, 64, 65, 68

13. Stem and Leaf Display • Represents the data using the actual digits that make up the data. • Leading digit becomes the stem • Trailing digit becomes the leaf

14. Stem and Leaf Examples • Example – Exam Grades 76, 74, 82, 96, 66, 76, 93, 86, 84, 62, 82, 75, 58, 71, 73, 79, 65, 80 Example - Decimals 1.3, 2.4, 1.7, 3.2, 5.6

15. Outliers • Outlier • An observation whose value is unusual or extreme. • Example 15, 21, 13, 18, 23, 19, 26, 16, 71, 22, 14, 21 71 is an outlier

16. Section 1.3 Graphing Categorical Data

17. Grouped Frequency Table • Frequencies are tabulated for each value of the categorical variable. • Example – STAT 101 Grade Distribution (2000)

18. Bar Graph • Represents categorical data by showing the amount of data that belong to each category as proportionally sized rectangles. • Example: US Coal Production (in millions) WV – 172.0 PA – 189.2 KY – 154.8 WY – 233.6 Other – 120.4

19. Pie Chart (Circle Graph) • A pie chart shows the amount of data that belongs to each category as a proportional part of a circle. • Example: US Coal Production (in millions) WV – 172.0 PA – 189.2 KY – 154.8 WY – 233.6 Other – 120.4

20. Section 1.4 Frequency Histograms

21. Grouped Frequency Distributions • Grouped Frequency Distributions • A list (or table) which pairs ranges for values of a variable with their frequencies (counts). Each range is called a class. • Rules for constructing a grouped frequency distribution • Each class should be the same width. • Classes should not overlap. • Each observation falls into one and only one class. • Use between 3 and 15 classes.

22. Frequency Histogram • Frequency Histogram • A graphical representation of a frequency distribution • Example – Math3 Exam Scores

23. Shape of a Distribution • Symmetric Shaped • Mound Shaped • U-Shaped • Uniform

24. Shape of a Distribution • Asymmetric Shaped • Skewed Right (Positive) • Skewed Left (Negative)

25. Break

26. Section 1.5 Density Histograms

27. Density Histogram • Relative Frequency Histogram • A histogram in which the vertical axis represents percentages or proportions, rather than counts. • Example – Math3 Exam Scores

28. Constructing Density Histograms • For each class, compute the percent of observations in each class. • Divide the percent associated with each class by the width of that class. This yields the percent of observations associated with each unit of the measurement scale. • Draw the histogram using the values computed in step 2.

29. More on Density Histograms • If the bars of a histogram are arranged from largest to smallest, it is called a Pareto Chart. • A density histogram is a histogram whose vertical axis is scaled so that the sum of the areas of it’s rectangles is 1 square unit. • If the sample size, n, is very large, a density histogram can be used to estimate the distribution of the population from which the data was obtained.

30. Section 1.6 Centers of Data

31. Averages • Average • An average is a single value which represents all of the data. • Types of Averages • Sample Mean • Sample Median • Sample Mode

32. Sample Mean (x) • Sample Mean • Formula • Example x: 14, 23, 8, 19, 41 Sample Mean = (14 + 23 + 8 + 19 + 41)/5 = 105/5 = 21

33. More on Sample Means • If we acquire data from all members of a population, we can compute the population mean ( ) • Because of this, we can use the sample mean to estimate the population mean

34. Sample Median ( ) • Sample Median • The middle value of the observed data values ranked from lowest to highest • If the sample contains n observations, the median is the ½(n+1) ranked value • Example: x: 14, 23, 8, 19, 41 Ranked: 8, 14, 19, 23, 41 n = 5 Position of the median: ½(5+1) observation = 3rd Median = 19

35. Sample Median Even Example • In the previous example, n was odd • Example with an even n: x: 43, 26, 37, 19, 52, 80 Ranked: 19, 26, 37, 43, 52, 80 n=6 Position of the Median: ½ (6+1) observation =3½ position Median is the mean of the 3rd and 4th ranked observation Median = (37+43)/2 = 40

36. Sample Mode • Sample Mode • The observed value which occurs with the greatest frequency • Example: x: 4, 3, 1, 4, 0, 3, 3, 1, 4, 0, 1, 1, 2, 2 Mode = 1

37. More Mode Examples • A set of x might have no mode (every value occurs once) • Mode = None • NOT Mode = 0 • A set of x might have more than one mode • Mode = 5 and 6

38. Section 1.7 Mean vs. Median

39. Best Measure of the Center Mean Median Use for approximately symmetric distributions Greatly influenced by outliers Use for significantly skewed distributions Not greatly influenced by outliers

40. How to Handle Outliers • If the outlier is a “mistaken” data value, eliminate it from the analysis • If the outlier is an actual data value, do not eliminate it

41. End of Class