This document provides an overview of key concepts in descriptive statistics, including frequency tables, histograms, and distributions. It explains that frequency tables and histograms summarize and describe patterns in a set of scores or values. A frequency table shows the counts of observations at each value, while a histogram uses bar heights to represent frequencies, making distributions visually apparent. An example shows how to construct a frequency table and histogram from a set of job happiness ratings ranging from 0 to 10.

1. Statistics for the Behavioral and Social Sciences: A Brief Course Fifth Edition Arthur Aron, Elaine N. Aron, Elliot Coups Copyright © 2011 by Pearson Education, Inc. All rights reserved . This multimedia product and its contents are protected under copyright law. The following are prohibited by law: -any public performance or display, including transmission of any image over a network; -preparation of any derivative work, including the extraction, in whole or in part, of any images; -any rental, lease, or lending of the program. Prepared by: Genna Hymowitz Stony Brook University

2. Displaying the Order in a Group of Numbers Using Tables and Graphs Chapter 1 Copyright © 2011 by Pearson Education, Inc. All rights reserved .

3. Chapter Outline The Two Branches of Statistical Methods Some Basic Concepts Kinds of Variables Frequency Tables Histograms Shapes of Frequency Distributions Frequency Tables and Histograms in Research Articles Copyright © 2011 by Pearson Education, Inc. All rights reserved.

4. Why Learn Statistics? Increase your understanding of research articles Help you learn how to do your own research Can improve your reasoning and intuition Help you make sound decisions Copyright © 2011 by Pearson Education, Inc. All rights reserved .

5. What is Statistics? A branch of mathematics that focuses on the organization, analysis, and interpretation of a group of numbers Two Main Branches of Statistics descriptive statistics: used to summarize and describe a group of numbers from a research study inferential statistics: procedures for drawing conclusions based on the scores collected in a research study but going beyond them Copyright © 2011 by Pearson Education, Inc. All rights reserved .

6. Basic Concepts Variable characteristic or condition that can have different values e.g., level of stress age Gender Value possible number or category a score can have e.g., 0–10 35 Male Score particular person’s value e.g., a study participant rates her current level of stress as a 5 on a scale of 0–10 Copyright © 2011 by Pearson Education, Inc. All rights reserved .

7. Kinds of Variables Numeric (Quantitative)Variable variable that has values that are numbers Nominal (Categorical)Variable variable that has values that are names or categories e.g., gender, religion, ethnicity Copyright © 2011 by Pearson Education, Inc. All rights reserved .

8. Level of Measurement Type of underlying numerical information provided by a measure equal-interval numeric variable in which differences between values correspond to differences in the underlying thing being measured interval – a scale in which the units of measurement (intervals) between the numbers are all equal in size but there is no absolute zero. e.g. , intelligence, temperature ratio – in addition to order and equal units of measurement, there is an absolute zero that indicates an absence of the variable being measured. e.g. , height, weight Copyright © 2011 by Pearson Education, Inc. All rights reserved . Not in text

9. Level of Measurement Type of underlying numerical information provided by a measure rank-order (ordinal) numeric variable in which values correspond to the relative position of things measured e.g., class standing, birth order, position in a race Copyright © 2011 by Pearson Education, Inc. All rights reserved .

10. Level of Measurement Type of underlying numerical information provided by a measure nominal variable in which values are categories e.g., gender, religion, ethnicity Copyright © 2011 by Pearson Education, Inc. All rights reserved .

11. How Are You Doing? You are conducting a study to evaluate how happy people are in their job. For this study, you ask people to indicate their job title. What is your variable of interest? Is your variable numeric nominal What level of measurement are you using? ratio interval rank-order variable nominal Copyright © 2011 by Pearson Education, Inc. All rights reserved .

12. How Are You Doing? You are conducting a study to evaluate how happy people are in their job. . For this study, you ask people to rate their level of happiness on a scale of 0–10. What is your variable of interest? Is your variable numeric nominal What level of measurement are you using? ratio interval rank-ordered nominal Copyright © 2011 by Pearson Education, Inc. All rights reserved .

13. How Are You Doing? You are conducting a study to evaluate how happy people are in their job. For this study, you ask people to rate their level of happiness as “very happy”, “happy”, “unhappy”, “very unhappy”. What is your variable of interest? Is your variable numeric nominal What level of measurement are you using? ratio interval rank-ordered nominal Copyright © 2011 by Pearson Education, Inc. All rights reserved .

14. http://www.edugamer.net/app/playGame.aspx?userGameId=4213

15. Frequency Given a set of numbers, how can we make sense of them? Scores on a Job Happiness Survey 8, 2, 3, 1, 2, 9, 1, 5, 6, 9, 4, 4, 2, 3, 3, 5, 4, 7, 5, 3 Copyright © 2011 by Pearson Education, Inc. All rights reserved .

16. Frequency Given a set of numbers, how can we make sense of them? frequency number of scores with a particular value If 5 students reported that their level of happiness on the job was a 2 on a 0–10 scale, the frequency for a rating of 2 would be 5. frequency table a table displaying the pattern of frequencies over different values Copyright © 2011 by Pearson Education, Inc. All rights reserved .

17. Steps for Making a Frequency Table Step 1: Make a list down the page of each possible value, from lowest to highest. Step 2: Go one by one through the scores, making a mark for each next to its value on the list. Step 3: Make a table showing how many times each value on your list was used. Step 4: Figure the percentages of scores for each value. Copyright © 2011 by Pearson Education, Inc. All rights reserved .

18. Frequency Table Step 1 Your research study used a happiness scale that ranges from 0 (not at all happy) to 10 (extremely happy). Copyright © 2011 by Pearson Education, Inc. All rights reserved . Happiness Rating 0 1 2 3 4 5 6 7 8 9 10

19. Frequency Table Step 2 Your study resulted in the following scores: 8 2 3 1 2 9 1 5 6 1 9 4 4 2 3 3 5 4 7 5 3 Copyright © 2011 by Pearson Education, Inc. All rights reserved . Happiness Rating Frequency Tally 0 1 II 2 III 3 IIII 4 III 5 III 6 I 7 I 8 I 9 II 10

20. Frequency Table Step 3 Copyright © 2011 by Pearson Education, Inc. All rights reserved . Happiness Rating Frequency Tally Frequency 0 0 1 II 2 2 III 3 3 IIII 4 4 III 3 5 III 3 6 I 1 7 I 1 8 I 1 9 II 2 10 0

21. Frequency Table Step 4 Figure the percentage of scores for each value. Take the frequency of the value, divide it by the total number of scores, and multiply by 100. Copyright © 2011 by Pearson Education, Inc. All rights reserved .

22. Completed Frequency Table Copyright © 2011 by Pearson Education, Inc. All rights reserved . Happiness Rating Frequency Percent 0 0 0% 1 2 10% 2 3 15% 3 4 20% 4 3 15% 5 3 15% 6 1 5% 7 1 5% 8 1 5% 9 2 10% 10 0 0%

23. Another Example: 4 3 10 5 4 2 9 6 8 3 1 7 5 5 6 2 5 4 6 7 8 7 3 5

24. Frequency Tables for Nominal Variables Follow the same four steps that you would for a numeric variable. Remember that the values in which you are interested are names or categories rather than numbers. Copyright © 2011 by Pearson Education, Inc. All rights reserved . Major Frequency Percent Psychology 5 25 Sociology 8 40 Anthropology 3 15 Political Science 4 20

25. Another Example: Psychology majors? Sociology majors? CJ majors? Other majors?

26. Grouped Frequency Table A frequency table that uses intervals of values Lists the number of participants for each interval of values If the list of possible values ranges from 0–10, a possible set of intervals is: 0–1 2–3 4–5 6–7 8–9 10–11 Copyright © 2011 by Pearson Education, Inc. All rights reserved .

27. Histogram Graph of the information on a frequency table The height of each bar is the frequency of each value in the frequency table. Copyright © 2011 by Pearson Education, Inc. All rights reserved .

28. Histogram Graph of the information on a frequency table The height of each bar is the frequency of each value in the frequency table. Copyright © 2011 by Pearson Education, Inc. All rights reserved . 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10

29. How to Make a Histogram Step 1 Make a frequency table or grouped frequency table. Copyright © 2011 by Pearson Education, Inc. All rights reserved . Scores on a Job Happiness Survey 8, 2, 3, 1, 2, 9, 1, 5, 6, 9, 4, 4, 2, 3, 3, 5, 4, 7, 5, 3 Happiness Rating Frequency 0 0 1 2 2 3 3 4 4 3 5 3 6 1 7 1 8 1 9 2 10 0

30. How to Make a Histogram Step 2 Put the values at the bottom of the page going from left to right, from lowest to highest Step 3 Make a scale of frequencies along the left edge of the page (0 will be at the bottom and the highest value will be at the top). Copyright © 2011 by Pearson Education, Inc. All rights reserved . 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10

31. How to Make a Histogram Step 4 Make a bar for each value. Copyright © 2011 by Pearson Education, Inc. All rights reserved . 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10

32. Frequency Distributions Show the pattern of frequencies over the various values (how the frequencies are spread out). Copyright © 2011 by Pearson Education, Inc. All rights reserved . 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10

33. Frequency Distributions Copyright © 2011 by Pearson Education, Inc. All rights reserved . 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10

34. Frequency Distributions Show the pattern of frequencies over the various values (how the frequencies are spread out). unimodal distribution - a histogram with one very high area Copyright © 2011 by Pearson Education, Inc. All rights reserved . 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10

35. Frequency Distributions bimodal distribution a distribution with two fairly equal high points Copyright © 2011 by Pearson Education, Inc. All rights reserved . 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10

36. Frequency Distributions multimodal distribution a distribution with two or more high points Copyright © 2011 by Pearson Education, Inc. All rights reserved . 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10

37. Frequency Distributions rectangular distribution when all values have approximately the same frequency Copyright © 2011 by Pearson Education, Inc. All rights reserved . 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10

38. Symmetrical and Skewed Distributions In the social and behavioral sciences, most scores are symmetrically distributed. They have approximately the same number of scores on both sides of the distribution. Copyright © 2011 by Pearson Education, Inc. All rights reserved . 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10

39. Symmetrical and Skewed Distributions Skewed distributions are distributions where the scores pile up on one side of the middle. characterized by the side of the distribution where scores are more spread out (tail) negatively skewed distribution tail is to the left Copyright © 2011 by Pearson Education, Inc. All rights reserved . 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10

40. Skewed Distributions positively skewed distribution tail is to the right Copyright © 2011 by Pearson Education, Inc. All rights reserved . 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10

41. Floor and Ceiling Effects Floor Effect Scores pile toward the lower end of the distribution because it is not possible to have a lower score (e.g., number of children). Copyright © 2011 by Pearson Education, Inc. All rights reserved . 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10

42. Floor and Ceiling Effects Ceiling Effect Scores pile toward the upper end of the distribution because it is not possible to have a higher score (e.g., scores on a very easy statistics test). Copyright © 2011 by Pearson Education, Inc. All rights reserved . 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10

43. Normal, Heavy-Tailed, and Light-Tailed Distributions Normal Curve bell-shaped, unimodal, and symmetrical Light-Tailed Distribution There are few scores in the tails (the tails are thin). Heavy-Tailed Distribution There are many scores in the tails (the tails are thick). Copyright © 2011 by Pearson Education, Inc. All rights reserved .

44. Key Points Descriptive statistics are used to describe and summarize a group of numbers from a research study. A value is a number or category; a variable is a characteristic that can have different values; a score is a particular person’s value on the variable. Some numeric variables are rank-ordered and some variables are names or categories and not numbers. A frequency table organizes the scores into a table that lists each possible value from lowest to highest along with the frequency of each value. A grouped frequency table is used when there are many different values. Intervals are given for a range of values. A histogram visually displays the information in a frequency table. The general shape of a histogram can be unimodal, bimodal, multimodal, or rectangular, and the distribution can be symmetrical, skewed to the right, or skewed to the left. Frequency tables, when used in research articles, are used to summarize the characteristics of study participants. Histograms almost never appear in articles, but the shapes of the distribution are sometimes described in words . Copyright © 2011 by Pearson Education, Inc. All rights reserved .