Chapter 1 Section 1.1

1. Chapter 1Section 1.1 Displaying Distributions with Graphs

2. Statistics • the science of collecting, analyzing, and drawing conclusions from data

3. Main Types of Statistics • Descriptive • Inferential

4. Descriptive Statistics • the methods of organizing & summarizing data

5. Inferential Statistics • involves making generalizations from a sample to a population

6. Population • the entire collection of individuals or objects about which information is desired

7. Sample • a subset of the population, selected for study in some prescribed manner

8. Variable • any characteristic whose value may change from one individual to another

9. Data • observations on single variable or simultaneously on two or more variables

10. When you see a set of data, ask: • Who? • What? • Why?

11. Who? • What individuals do the data describe? • How many individuals appear in the data?

12. What? • How many variables? • What are the definitions of these variables? • What units?

13. Why? • What is the reason data were gathered?

14. Types of Variables

15. Categorical Variables • Also called “qualitative” • Identify basic differentiating characteristics of the population

16. Quantitative Variables • Also called “numerical” • Observations or measurements that take on numerical values • Makes sense to average these values • Two types-discrete & continuous

17. Discrete (numerical) • Listable set of values • Usually counts of items

18. Continuous (numerical) • Data can take on any value in the domain of the variable • Usually measurements of something

19. Classifications by the number of variables • Univariate-data that describes a single characteristic of the population • Bivariate-data that describes two characteristics of the population • Multivariate-data that describes more than two characteristics

20. Categorical or Quantitative • Which category of variables would the following be? • Gender • Age • Hair color • Smoker • Systolic blood pressure • Number of girls in class

21. Types of Distributions 4 Common Types

22. Distribution • Tells us what values the variable takes and how often it takes these values • One variable may take values that are very close together while others might be spread out

23. Symmetrical • Refers to data in which both sides are (more or less) the same when the graph is folded vertically down the middle • Bell-shaped is a special type • Has a center mound with sloping tails

24. Uniform • Refers to data in which every class has equal or approximately equal frequency

25. Skewed (left or right) • Refers to data in which one side (tail) is longer than the other side • The direction of the skewness is on the side of the longer tail Skewed right Skewed left

26. Bimodal (multi-modal) • Refers to data in which two or more classes have the largest frequency and are separated by at least one other class

27. How to describe a graph

28. Shape of Graphs • Normal, Symmetrical • Skewed • Uniform • Bimodal

29. Center of Graphs • Where the middle of the data falls • 3 types of central tendency • Mean, median, mode

30. Spread • Shows how spread out the data is • Refers to the variability of the data • Range, standard deviation, IQR

31. Unusual Occurrences • Outliers-value that lies away from the rest of the data • Clusters • Gaps

32. Types of Graphs • Bar • Pie Chart • Dotplot • Stem-and-Leaf • Histogram • Relative cumulative frequency graph • Time Plot • Box & Whisker • Scatter

33. 1. Bar Graphs • Must label the axes and title the graph • Scale your axes • Draw vertical bar above each category name to a height that corresponds to the count in that category

34. Bar Graph (example) The graph below shows the proportion of the female labor force aged 25 and older in the United States that falls into various educational categories. The coding used in the plot is as follows: 1. none-8th Grade 6. bachelor’s degree 2. 9th grade-11th grade 7. master’s degree 3. high school graduated 8. professional degree 4. some college, no degree 9. doctorate degree 5. associate degree Proportion Educational Attainment (women)

35. 2. Pie Chart • Must include all categories that make up a whole.

36. 3. Dotplot Tally marks are made for each set of data. The data below is graphed on the dotplot on the right. 195 194 204 199 204 204 192 204 192 192 193 214 209 222 209

37. 4. Stem and Leaf Plot A stem and leaf plot displays data like a bar graph, but we break the data apart into stem and leaf plots. The data point 30 is broken up into 3 0 stem leaf The data point for 304 is broken up into 30 4 stem leaf

38. Stem and Leaf Plot The data set {30, 27, 34, 28, 45, 31, 34, 40, 29} is represented by: 2: 7 8 9 3: 0 1 4 4 4: 0 5 The 2 and 9 represents the data point 29. The stems are the tens place and the leaves are the ones digits. Notice numbers are listed in order from smallest to largest.

39. 5. Histogram • Draw a bar graph that represents the count in each class. Leave no horizontal space (unlike a bar graph). • The data below is graphed on the dotplot on the next slide. 195 194 204 199 204 204 192 204 192 192 193 214 209 222 209

40. Histogram (example)

41. 6. Relative Cumulative Frequency Graph • Also called an Ogive. • A relative frequency histogram has the same shape and the same horizontal scale as the corresponding frequency histogram. The difference is that the vertical scale measures the relative frequencies (measured as a percentage), not frequencies.

42. Relative Cumulative Frequency Graph The frequency needs to be changed to percentages.

43. 7. Time Plot • Place the time on the x axis. Time is the explanatory variable. • Place the observations on the y axis. The observations represent the response variable.

44. Time Plot (example)

45. 8. Box & Whisker/9. Scatter Plot • These will be discussed later. • Box and Whisker-Section 1.2 • Scatter Plot-Section 3.1

46. Summary • The distribution of a variable tells us what values it takes and how often it takes these values. • To describe a distribution, begin with a graph. • Bar graphs and pie charts display the distributions of categorical variables. These graphs use the counts or percents of the categories. • Stem & Leaf plots and histograms display the distributions of qualitative variables. Stem & Leaf Plots separate each observation into a stem and a one-digit leaf. Histograms plot the frequencies (counts) or percents of equal-width classes of values.

47. Summary continued… • When examining a distribution, look for the shape, center and spread, and for clear deviations from the overall shape. Some distributions have simple shapes, such as symmetric or skewed. Others may be bimodal (more than one major peak). • Outliers are observations that lie outside the overall pattern of a distribution. Always look for outliers and try to explain them.

48. Summary continued • A relative frequency graph (ogive) is a good way to see the relative standing of an observation. • When observations on a variable are taken over time, make a time plot that graphs horizontally and the values of the variable vertically. A time plot can reveal trends (patterns) or other changes over time.

49. Assignment • On page 64 in your textbook, complete exercises 1.13 and 1.16.