This Document Contains Chapters 1 to 3 Chapter 1 The Role of Statistics in the Social Sciences Learning Objectives: 1. Describe the meaning of the term of “statistics.” 2. Explain why statistics is important in the social sciences. 3. Explain where statistics fit in the research process. 4. Define the difference between the population and the sample. 5. Generally describe the difference between descriptive and inferential statistics. 6. Understand and explain the importance of ethics in statistics. 7. Understand and explain the difference between an independent and dependent variable. 8. Provide examples of independent and dependent variables. 9. Define the four different level of measurement. 10. Identify the level of measurement of different variables. Chapter Summary In this chapter, students were introduced to meaning and importance of statistics. A discussion about where statistics fit within the research process, the difference between a population and a sample, and the difference between descriptive and inferential statistics is provided. Common examples assist students in learning how to identify variables of interest, the difference between independent and dependent variables and the four types of variable measurement. Finally, the importance of conducting ethical research and statistical analysis is discussed. Key Formulas There are no key formulas presented in this chapter. However, for subsequent chapters, we will outline the key formulas students learn throughout their readings. PowerPoint slides are provided for each chapter. In addition to these slides, a PDF file containing only the formulas are also provided. Interactive Figures: The textbook contains interactive figures. You may wish to use these in a lecture. Students also have access to these. For this chapter, there is one interactive figure. 1. “Did You Know?” The Lottery Probability This interactive figure can be found on in the eBook and the Library under Chapter 1 Resources. This Document Contains Chapters 1 to 3 Typical Lecture Material We have provided two sample lectures below. You may wish to add in additional discipline specific information to make these more relevant to your students. Additionally, you may wish to combine this information into one lecture depending on how much time you need to devote to administrative issues pertinent to students in the first lecture. Lecture 1: Objective: Understand the meaning of statistics, its role within the research process, and the difference between populations and samples. Example 1a: Provide the following flow chart to the students in the class Pose the following question to your students: 1. What might be some examples of applied statistics within a “discipline that is the focus of the statistics course” (e.g. developmental psychology, sociology, etc.)? Example 1b: Provide the following flow chart from the text book. Then go through the study planning process with students and identify how statistics should be incorporated through each stage of the research process. An example might be that a researcher is interested in looking at how peer bullying affects students’ grades in school. The theory driving the research question maybe social network theory, which postulates that individuals in a social system (i.e. school) interact with each other and serve as significant reference points in each other’s decision-making and responses to bulling experiences. Given this information, ask students to go through with you each step of the research process (left hand side of Figure 1.3) and the particular statistical considerations that need to be made given this research area and theory (right hand side of Figure 1.3). For example, a potential hypothesis is that individuals who are-bullied will surround themselves with other individuals that are bullied; and therefore, may have similar academic achievement. Example 2: Provide the following Figure to your students from the textbook. Remind students of our previous researcher who wishes to look at the association between peer bullying and academic achievement. Pose the following questions to your students: a) What is the population that the researcher is interested in studying? b) How would the researcher obtain a sample to answer their research question? c.) What would ensure that our sample is representative of the population the researcher is interested in? Lecture 2: Objective: Understand the meaning of variables, the different levels of variable measurement, and the difference between descriptive and inferential statistics. Example 1: Remind students that variables are a phenomenon of interest that can take on different values and can be measured. Provide the following figure to students from the textbook: Draw the following table on the board (include the first bolded row as an example) and then ask students to identify the three other types of variable measurement, their properties and some examples: Measurement Type Properties Examples Nominal Qualitative categories Gender (male vs. female) Partnership Status (Single vs. Married) Ordinal Qualitative categories with an order or ranking Rank favourite TV shows. Activity rating (not-active, somewhat active, highly active) Interval Equal and meaningful difference between levels Zero (0) does not mean the absence of the variable/phenomenon IQ Score Anxiety score (where 0 does not mean total absence of anxiety, but does mean there is not enough anxiety to warrant intervention). Ratio Equal and meaningful difference between levels. Zero is meaningful (i.e. denotes absence of phenomenon). Indicating the number of times they have been bullied in the last school year. Percent of immigrants relocating to Canada over the last year. Example 2: Post the following research questions on the blackboard and ask students to identify the variables included in the question, their likely measurement type, and whether the question is suited for descriptive or inferential statistics. 1. How many 15-18 year old Canadian’s smoke cigarettes? 2. Do individuals who report a history of childhood abuse have a greater likelihood of having failed relationships in adulthood? 3. What is the relationship between age and depression among ethnic minorities in Canada? Solutions to Interactive Exercises Question 1-1 A researcher wanted to study why most of the office bearers of the community leagues in one area of a city leave their position even before completing one year of their assignment. The researcher wants to identify the factors that predict commitment as an office bearer of a community league. The researcher decides to assess the knowledge of the position, attitude towards the policies, and ability to handle conflicts as predictors for commitment to the position. What is the independent and dependent variable(s) of the study? Feedback: Independent Variable: knowledge of position, attitude toward policies, and ability to handle conflicts Dependent Variable: commitment to position (continuing in the position). Question 1-2 A researcher wanted to see whether playing classical music during math test will reduce the number of calculation errors of primary students. A researcher randomly selected 2 Grade 3 classes (30 students in each class) taught by the same teacher in a particular school board. One class was given the test without any music being played during the test, while music was playing during the test in another class. At the end of the study, from the data she collected, the researcher concluded that that the students who wrote the test with classical music will make lesser calculation errors than the students who wrote the test without classical music. a) What is the population of the study? b) What is the sample? c) What branch of Statistics will the conclusion of the study fall? Why? d) What variable is the score of a student in this test? e) What variable is the gender of the student? Feedback: a) Primary students. b) 60 students. c) Inferential statistics as she is making a conclusion about the population. d) Quantitative variable (ratio) e) Nominal Question 1-3 A local super market employee asks 50 customers whether they like the new brand peanut butter. What is the sample and population of this study? Feedback: Sample = 50 customers. Population = All customers. Chapter 2 Describing Your Data: Frequencies, Cross Tabulations, and Graphs Learning Objectives: 1. Define and describe the terms frequency and frequency distribution. 2. Define and describe the terms relative frequency, percentage frequency, and cumulative percentage frequency. 3. Construct frequency tables for nominal and ordinal data. 4. Construct class intervals for interval and ratio data. 5. Construct frequency tables for interval and ratio data. 6. Create cross tabulations. 7. Calculate percentage change, ratios, and rates. 8. Create and interpret pie and bar charts. 9. Create and interpret frequency polygons and cumulative percentage frequency polygons. 10. Create and interpret histograms, stem-and-leaf plots, and boxplots. Chapter Summary In this chapter, students were introduced to various methods for summarizing data. Frequency tables, pie charts and bar charts were shown as methods for describing nominal and ordinal data. Frequency tables, histograms and cumulative frequency polygons were shown as methods for describing interval and ratio data. The terms and equations for calculating percentage, percentage change, ratios and rates, relative frequency and cumulative percentage frequency were also introduced. Finally, students were shown how box-plots and stem and leaf plots can be used to illustrate their data. Key Formulas The following represent the key formulas for this chapter. PowerPoint slides are provided for each chapter. In addition to these slides, a PDF file containing only the formulas is also provided. Relative Frequency (f/n) Percentage Frequency (f/n × 100%) Percentage Change Ratio Rate (per 10,000) Interactive Figures: The textbook contains interactive figures. You may wish to use these in a lecture. Students also have access to these. For this chapter, there are two interactive figures: 1. Figure 2.9 illustrates the probability of matching birthdays. 2. Figure 2.22 provides an interactive example of two boxplots. These interactive figures can be found on in the eBook and the Library under Chapter 2 Resources. Typical Lecture Material We have provided two sample lectures below. You may wish to add in additional discipline specific information to make these more relevant to your students. Lecture 1: Objective: Construct frequencies, relative frequencies, percent relative frequencies for a data set. For an interval or ratio data set choose the end points for classes and then calculate the frequencies in each class. Construct a table containing the frequencies, relative frequencies, cumulative relative frequencies and the percent cumulative relative frequencies. Example 1: Provide students with the following figure (Figure 2.1) from the textbook. Then create a frequency table with the students. A typical frequency table may look like: Life Satisfied Frequency (f) Relative Frequency (f/n) Percentage Frequency (%f) 1 Strongly disagree 1 1/10=0.1 0.1x100%=10% 2 Disagree 2 0.2 20% 3 Neutral 2 0.2 20% 4 Agree 4 0.4 40% 5 Strongly agree 1 0.1 10% Sum n=10 1.0 100% Then pose various questions to the students such as: a) What percentage of respondents stated that they were “neutral”? b) What percentage of the respondents stated that they "agreed" or "strongly agreed"? Example 2: The following scores out of 50 were recorded for 40 individuals. You may wish to tailor this example to your own discipline by stating that the scores represent a specific variable. 25 27 27 27 29 29 29 29 30 30 30 31 31 32 32 32 32 32 32 32 32 33 33 34 34 35 35 35 36 36 36 36 37 37 37 38 38 43 43 49 Draw the following table on the board and then ask students to work through the calculations for each. Class Interval Class Limits Class Boundaries Midpoints Frequency (f) Relative Frequency (f/n) Percentage Frequency (%f) Cumulative Percentage Frequency (c.%f) To create a frequency table with five cells (or bins), students will first have to calculate the range of the data. They should find that the range is 49-25=24. With 5 cells (or bins) the width of our intervals will be 5 (24÷5 =4.9 rounded up). If we start the first interval at 0.5 lower than the lowest value, then we start at 24.5. The first will then have class boundaries from 24.5 to 29.5 with a midpoint of 27. Each subsequent interval is just 5 units higher. The first class interval contains the values 25 to 29. The completed table is as follows: Class Interval Class Limits Class Boundaries Midpoints Frequency (f) Relative Frequency (f/n) Percentage Frequency (%f) Cumulative Percentage Frequency (c.%f) 25 – 29 25, 29 24.5 to < 29.5 27 8 .200 20% 20% 30 – 34 30, 34 29.5 to < 34.5 32 17 .425 42.5% 62.5% 35 – 39 35, 39 34.5 to < 39.5 37 12 .300 30% 92.5% 40 – 44 40, 44 39.5 to < 44.5 42 2 .050 5% 97.5% 45 - 49 45, 49 44.5 to < 49.5 47 1 .025 2.5 100% Total 40 1.000 100% Once students have completed the table, you could then pose various questions to the students such as: a) What percent of the respondents scored less than or equal to 44? b) How many respondents scored between 30 and 34 inclusive? Lecture 2 Objective: Construct charts, histograms and percent cumulative frequency graphs. Figures can be used to lead into numerical measures that describe the data. Discuss the shape of the histograms. Example 1: Histogram and cumulative frequency curve for the 50 scores. Example 1: Cumulative Percent Frequency Plot. The Cumulative Percent Frequency graph plots the percent cumulative frequency versus the upper class boundaries. For the first class we plot 24.5 versus 0% to begin as there were no observations less than 24.5. Also the upper class boundary of the last class is always plotted versus 100% as there are no observations higher than that. Solutions to End-of-Chapter Problems Problem 2-1 a) grade frequency 0≤grade<10 1 10≤grade<20 3 20≤grade<30 2 30≤grade<40 0 40≤grade<50 3 50≤grade<60 5 60≤grade<70 3 70≤grade<80 4 80≤grade<90 4 total 25 b) grade frequency Relative frequency Cumulative frequency 0≤grade<10 1 4% 1 10≤grade<20 3 12% 4 20≤grade<30 2 8% 6 30≤grade<40 0 0% 6 40≤grade<50 3 12% 9 50≤grade<60 5 20% 14 60≤grade<70 3 12% 17 70≤grade<80 4 16% 21 80≤grade<90 4 16% 25 Total 25 100% c) Proportion passed = 44% grade Male Female frequency 0≤grade<10 1 0 1 10≤grade<20 3 0 3 20≤grade<30 2 0 2 30≤grade<40 0 0 0 40≤grade<50 2 1 3 50≤grade<60 2 3 5 60≤grade<70 0 3 3 70≤grade<80 3 1 4 80≤grade<90 2 2 4 Total 15 10 25 d) Percentage of males passed = 20% e) Percentage of females failed = 16% f) Each interval is provided below: grade Male/female 0≤grade<10 100% 10≤grade<20 100% 20≤grade<30 100% 30≤grade<40 0% 40≤grade<50 200% 50≤grade<60 66% 60≤grade<70 0% 70≤grade<80 300% 80≤grade<90 100% Total 150% g) There are 50% more males in the class than females. Solutions to Interactive Exercises Question 2-1 A local restaurant asked 20 randomly selected customers about their wait time between placing the order and serving the food before being the food. The data is given below. 45, 25, 40, 15, 5, 28, 30, 32, 28, 11, 50, 36, 42, 8, 12, 48, 16, 27, 37, 14 Construct a frequency distribution for this data, with a class width of 10 and using 5 as the lower class limit of the first class. Also indicate the class boundaries, frequency and the cumulative frequency in the same table. Feedback: Class Limits Frequency Class Boundaries Cumulative Frequency 5-14 6 4.5-14.5 6 15-24 1 14.5-24.5 6+1=7 25-34 6 24.5-34.5 6+1+6=13 35-44 4 34.5-44.5 6+1+6+4=17 45-54 3 44.5-54.5 6+1+6+4+3=20 Question 2-2 265 people were asked whether they enjoy shopping and their responses are given below: Enjoy Shopping? Male Female YES 32 112 NO 98 23 a) What is the ratio of the respondents who enjoy shopping to those who do not? b) What percentage of females enjoy shopping? c) What percentage of those enjoy shopping are males? Feedback: a) 1.21 b) 82.97% c) 22.22% Question 2-3 The total number of reported crimes in a medium sized city is 22340 in 2009 and 27020 in 2010. What is the percentage change from 2007 to 2008? Feedback: Percentage Change = 20.95% Solutions to SPSS Exercises Question 2-1 The following is the votes provided for the 2010 Democratic nomination in the Washington primary for three of the front runners, along with the age of the voter. Kline = K; Davis = D; Edmunds = E Using the above data, along with SPSS, prepare the table of cross-tabs shown below and answer the following questions. a) At which level is each variable measured? Ans: Name – nominal; Age - interval LO: 1 Page: 39 b) Prepare a frequency table for each of the 2 variables using SPSS placing the age variable into the following categories: 18-29; 30-45; 46-59; 60+. Ans: First label the variables such that k=1; d=2; e=3 and 18-29=1; 30-45=2; 46-59=3; 60+=4. Using analyze – descriptive statistics - frequencies placing the appropriate variable in the right hand side dialog box we find name Frequency Percent Valid Percent Cumulative Percent Valid kline 16 40.0 40.0 40.0 edmunds 17 42.5 42.5 82.5 davis 7 17.5 17.5 100.0 Total 40 100.0 100.0 age Frequency Percent Valid Percent Cumulative Percent Valid 18-29 7 17.5 17.5 17.5 30-45 11 27.5 27.5 45.0 46-59 14 35.0 35.0 80.0 60+ 8 20.0 20.0 100.0 Total 40 100.0 100.0 LO: 1 Page: 39 c) In a similar fashion to that done in 1. b) prepare a cross tabulation of name (in rows) and age (in columns) using the given categories. Ans: Using analyze – descriptive statistics - crosstabs placing the appropriate variable in the right hand side dialog box – name in rows and age in columns we find: name * age Crosstabulation Count age 18-29 30-45 46-59 60+ Total name kline 3 5 5 3 16 edmunds 4 3 6 4 17 davis 0 3 3 1 7 Total 7 11 14 8 40 LO: 6 Page: 49 d) What proportion of voters voted for Edmunds? Ans: Using the frequency table above in b) we read off 42.5% LO: 7 Page: 51 e) Which age group supported each candidate most? Ans: From the table of cross tabulations we look down the columns for each group and find the highest frequency. 18-29: Edmunds 30-45: Kline 46-59: Edmunds 60+: Edmunds LO: 7 Page: 51 f) Using similar intervals as was done in question 1. b), prepare the histogram for each variable. Ans: Follow graphs – legacy dialogs – histogram and put the appropriate variable in the right hand side in variable, yielding: LO: 10 Page: 62 Equations from Chapter 2 Scott R. Colwell and Edward M. Carter c 2012 Equation 2.1: Relative frequency relative frequency = f n Where: f = frequency of specific response n = total number of responses Equation 2.2: Percentage frequency %f = f n × 100 Where: f = frequency of response n = total number of respondents within the variable Equation 2.3: Percentage change between two time periods p = ftime2 − ftime1 ftime1 × 100 Where: ftime1 = frequency of a specific response at time 1 ftime2 = frequency of a specific response at time 2 Equation 2.5: Ratio Ratio = fv1 fv2 Where: fv1 = frequency of the first value to be compared fv2 = frequency of the second value to be compared Equation 2.7: Rates Rate = Number of events for the population of interest Total population of the population of interest ×10, 000 Chapter 3 Describing Your Data: Measures of Central Tendency, Dispersion, and Shape Learning Objectives: 1. Explain why central tendency and dispersion are important. 2. Calculate the mean, median, and mode. 3. Explain for which type of variable the mean, median, and mode are most appropriate 4. Describe what variability is. 5. Calculate range, interquartile range, percentiles, variance and standard deviation. 6. Define and describe skewness and how it affects the mean. 7. Define and describe kurtosis and how it affects the mean. Chapter Summary In this chapter, students were introduced to standard measures of central tendency and dispersion. A description of variable mean, median and mode is given; as well as the appropriate instances where these measures should be used. A discussion of dispersion and variability is provided; including attention paid to the common statistical concepts of dispersion: the percentile, the range, the variance and the standard deviation. The chapter ends with a discussion about describing a distribution, including the definitions of skewness and kurtosis and what these concepts mean in relation to a distribution of data points within a given variable. Key Formulas The following represent the key formulas for this chapter. PowerPoint slides are provided for each chapter. In addition to these slides, a PDF file containing only the formulas are also provided. Mean 𝑥̅= ∑ 𝑥 𝑛 Sample Variance 𝑠2 = ∑(𝑥 − 𝑥̅)2 (𝑛 − 1) Standard Deviation 𝑠 = √ ∑(𝑥 − 𝑥̅)2 (𝑛 − 1) Interactive Figures: The textbook contains interactive figures. You may wish to use these in a lecture. Students also have access to these. For this chapter, there is one interactive figure. 1. Figure 3.4 illustrates the normal distribution, positive skew and negative skew. It also allows you to compare multiple distributions. This interactive figure can be found on in the eBook and the Library under Chapter 3 Resources. Typical Lecture Material We have provided two sample lectures below. You may wish to add in additional discipline specific information to make these more relevant to your students. Lecture 1: Objective: To understand measures of central tendency and when to use these measures to describe data. Remind your students that measures of central tendency refer to those that are used to calculate the middle point of a distribution. Example 1a: Provide the following flow chart to the students in the class. Pose the following question to your students: 1. When using each of these measures of central tendency, what is it that you are doing? Measures of Central Tendency MODE _______________ MEDIAN _________________ MEAN ________________ Example 1b: Draw the following table on the board (include the first bolded row as an example) and then ask students to identify examples of the three other types of variable measurement and their most appropriate measure of central tendency Measurement Type Examples Most Appropriate Measure of Central Tendency Nominal Gender (male vs. female) Partnership status (Single vs. Married) Mode Ordinal Interval Ratio Pose the following question if students did not identify this issue when filling out the table: The textbook argues that you should consider using the median as a measure of central tendency for interval and ratio data; under what circumstances does this apply? Example 2: Provided students with the following data table Participant ID Profession Ranking of Sport Preference Age Hockey Basketball Football 1 Teacher 1 2 3 17 2 Electrician 3 2 1 21 3 Student 1 3 2 28 4 Nurse 2 1 3 32 5 Teacher 1 3 2 45 6 Teacher 1 2 3 55 7 Educational Assistant 2 3 1 61 8 Construction 2 1 3 16 9 Teacher 3 1 2 75 10 Cashier 1 2 3 82 Ask student to identify each variables level of measurement and the appropriate measure of central tendency, and complete is corresponding calculation. Lecture 2: Objective: To understand measures of dispersion and when to use these measures to describe data. Remind your students that measures of dispersion refer to those that describe the variability in the data; and that variability is the extent to which the data varies from its mean. Example 1: Provide the following flow chart to your students: Pose the following question to your students: 2. How do you define each of these different measures of dispersion? Example 2a: The following grade scores out of 100 were recorded for 20 students who completed a first year social statistics course. Measures of Dispersion PERCENTILE ___________ RANGE ___________ VARIANCE ___________ STANDARD DEVIATION ____________ 95 80 85 70 72 70 98 65 60 55 86 90 83 82 76 73 84 69 71 77 Draw the following table on the board and then ask students to work through the calculations for each measure of dispersion. Student ID Class Grade Deviations Squared Deviations 01 95 02 80 03 85 04 70 05 72 06 70 07 98 08 65 09 60 10 55 11 86 12 90 13 83 14 82 15 76 16 73 17 84 18 69 19 71 20 77 Example 2b: Remind your students that skewness is a measure of the amount to which a distribution departs for a symmetrical shape; and that kurtosis is a measure of the peaked ness of the distribution. Post the following diagram on the board for your students and ask the students to match up the diagram with its corresponding kurtosis or skewness description. N = Mean = Sum of Squares = Sample Standard Deviation = Answers: Graph 1 = Positively skewed Graph 2 = Negatively skewed Graph 3 = Platykurtic Graph 4 = Leptokurtic Solutions to End-of-Chapter Problems Problem 3-1 a. the mean = 23.84 (LO2) b. the median = 24 (LO2) c. the mode = 19.7 and 31.9 d. The mean is the best for this data since it is the lowest variance estimator of all linear estimators. (LO3) Problem 3-2 a. Group 1 – the mean = 16.3 (LO2); Group 2 – the mean = 17.6 (LO2) b. Group 1 - the variance = 5.86 (LO4); Group 2 - the variance = 5.92 (LO4) Group 1 - the standard deviation = 2.42 (LO4); Group 2 - the standard deviation = 2.43 (LO4) c. Based on the variance or the standard deviation, group 2 is more variable since the variance and standard deviation are higher. Note that they are quite similar and that with rounding they are almost the same. (LO4) Problem 3-3 a. Histogram provided below but not sure of its content.(LO6) b. Shape of distribution = symmetric(LO6) c. Peakedness = leptokurtic since it appears that the scores are clustered around the centre point. (LO7) Solutions to Interactive Exercises Question 3-1 The average mark obtained by a set of 100 students is 55. The average mark of a set of 200 students is 60. What is the average mark of the 300 students? Feedback: 58.33 0 10 20 30 40 50 90-98 99-107 108-116 117-125 126-134 frequency frequency Question 3-2 The Mayor of Peaceful County is using the number of car vandalism reported in the last two weeks to estimate the average number of car vandalism per week. The data is given below: 7, 5, 7, 11, 4, 8, 9, 15, 12, 13, 9, 5, 7, 8 a) Calculate the Mean, Median and Mode of the data. b) Calculate Standard deviation, Variance, Range of the data. Feedback: a) Mean = 8.57 Median=8 Mode= 7 b) Sample standard deviation = 3.20 Sample variance= 10.24 Range = 11 Question 3-3 President of the faculty association of a small university was compiling data on faculty salary. He realized that the 85th percentile salary is $98,236.00. What does this number tells you? Feedback: Only 15% faculty earn more than $98,236.00 Solutions to SPSS Exercises Question 3-1 The following data was taken from the National Mathlete Competition in problem solving. Each observation represents the time in seconds it took a particular competitor to complete a proof of Pythagoras theorem. 21.0 15.2 12.5 10.3 8.2 19.4 14.7 12.1 10.0 7.8 18.2 14.2 11.7 9.6 7.4 17.3 13.7 11.3 9.3 7.0 16.5 13.3 11.0 8.9 6.5 15.8 12.9 10.6 8.6 5.9 First we need to put the data into the data file. Once done, we can then proceed. a) Given this information, find the sample mean, the median and the mode using. Ans: Using analyze – descriptive statistics - frequencies and uncheck the box for “display frequency tables” on the bottom left of the dialog box. Once done move the seconds variable to the right by highlighting and pressing the arrow to move it over. With the variable selected, click on “statistics” on the right at top and a dialog box opens. Once done, select mean, median and mode. Result below. Note: all values represent the mode since no value is repeated. Further note: You could have also found this using analyze – descriptive statistics - descriptives. Statistics seconds N Valid 30 Missing 0 Mean 12.0300 Median 11.5000 Mode 5.90a a. Multiple modes exist. The smallest value is shown LO: 2 Page: 77 b) Find the 25th, 50th, 75th percentile of the distribution. In this way you have just found the first quartile, the median, and the third quartile respectively. Ans: Using analyze – descriptive statistics - frequencies and moving the variable to the right side as selected, the dialog box opens. Once open click on statistics. Once done, check off the box for percentiles and add 25, 59 and 75. The result is given below. Statistics seconds N Valid 30 Missing 0 Percentiles 25 8.8250 50 11.5000 75 14.8250 Note: You could have also checked off the box for quartiles and median giving you the 25, 75, and 50 respectively. LO: 5 Page: 85 c) Determine the variance and standard deviation. Ans: Using analyze – descriptive statistics - descriptives and moving the variable to the right side as selected, the dialog box opens. Once open check off the boxes for variance and standard deviation. The result is below. Descriptive Statistics N Std. Deviation Variance seconds 30 3.96242 15.701 Valid N (listwise) 30 LO: 5 Page: 88-90 d) Prepare a histogram of the values. Once done, indicate the skewness and the type of kurtosis. Ans: Using Graphs – legacy dialogs – histogram and plcing the appropriate variable in the box under variable, we see the following histogram. From this, it appears to be positively skewed and leptokurtic. LO: 6, 7 Page: 94-97 Equations from Chapter 3 Scott R. Colwell and Edward M. Carter c 2012 Equation 3.1: The Mean x¯ = Pnx Where: x¯ = mean xP== summation or sum the values of x n = the sample size Equation 3.4: The Variance s2 = P(x −x¯) 2 (n −1) Where: s2 = the variance x¯ = mean xP== summation or sum the values of x n = the sample size Equation 3.7: The Standard Deviation s = sP(x −x¯) 2 (n −1) Where: s = the standard deviation x¯ = mean xP== summation or sum the values of x n = the sample size Solution Manual for Introduction to Statistics for Social Sciences Scott R. Colwell, Edward M. Carter 9780071319126
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