COURSE: JUST 3900 INTRODUCTORY STATISTICS FOR CRIMINAL JUSTICE Chapter 2: Frequency Distributions Peer Tutor Slides Instructor: Mr. Ethan W. Cooper, Lead Tutor © PLEASE DO NOT CITE, QUOTE, OR REPRODUCE WITHOUT THE WRITTEN PERMISSION OF THE AUTHOR. FOR PERMISSION OR QUESTIONS, PLEASE MR. COOPER AT THE FOLLWING:

Key Terms: Don’t Forget Notecards Frequency Distribution (p. 39) Frequency Distribution (p. 39) ΣX (p. 40) ΣX (p. 40) Proportions (p. 41) Proportions (p. 41) Percentages (p. 41) Percentages (p. 41) Grouped Frequency Distribution (p. 42) Grouped Frequency Distribution (p. 42) Class Intervals (p. 42) Class Intervals (p. 42) Histogram (p. 46) Histogram (p. 46) Polygon (p. 47) Polygon (p. 47) Bar Graphs (p. 48) Bar Graphs (p. 48)

More Key Terms: Think Notecards Relative Frequency (p. 49) Relative Frequency (p. 49) Smooth Curves (p. 49) Smooth Curves (p. 49) Symmetrical Distribution (p. 50) Symmetrical Distribution (p. 50) Skewed Distribution (p. 50) Skewed Distribution (p. 50) Tail (p. 51) Tail (p. 51) Positively Skewed (p. 51) Positively Skewed (p. 51) Negatively Skewed (p. 51) Negatively Skewed (p. 51) Percentile Rank (p. 53) Percentile Rank (p. 53) Percentile (p. 53) Percentile (p. 53)

More Key Terms: Think Notecards Cumulative Frequency (p. 54) Cumulative Frequency (p. 54) Cumulative Percentage (p. 54) Cumulative Percentage (p. 54) Interpolation (p. 55) Interpolation (p. 55) Stem and Leaf Displays (p. 60) Stem and Leaf Displays (p. 60)

Question 1: Construct a frequency distribution for the following set of scores. (see next slide for answer) Question 1: Construct a frequency distribution for the following set of scores. (see next slide for answer) Scores: 3, 2, 3, 2, 4, 1, 3, 3, 5 Scores: 3, 2, 3, 2, 4, 1, 3, 3, 5 Question 2: Find each of the following values for the sample in the following frequency distribution table. (see p. 42, #2 for answers) Question 2: Find each of the following values for the sample in the following frequency distribution table. (see p. 42, #2 for answers) 1) n 2) ΣX 3) ΣX 3) ΣX 2 Frequency Distributions X f

Question 1 Answers: Question 1 Answers: XfProportionPercentage % % % % % REMEMBER: The values in the X column should always be listed from the highest to the lowest!

Frequency Distributions Question 2: Find each of the following values for the sample in the following frequency distribution table. Question 2: Find each of the following values for the sample in the following frequency distribution table. 1) n 2) ΣX 3) ΣX 3) ΣX 2 X f

Frequency Distributions Question 2 Answers: Question 2 Answers: 1) n=10 2) ΣX=28 3) ΣX 2 =92 (square then add all 10 scores)

Grouped Frequency Distributions Question 3: An instructor has obtained the set of N=25 exam scores. To help organize these scores, place them in a frequency distribution table. The scores are: 82, 75, 88, 93, 53, 84, 87, 58, 72, 94, 69, 84, 61, 91, 64, 87, 84, 70, 76, 89, 75, 80, 73, 78, 60 See next slide for answer.

Grouped Frequency Distributions XfProportionPercentage % % % % % % % % % Question 3 Answer:

Question 4: A researcher records the gender and academic major for each student at a college basketball game. If the distribution of majors is shown in a frequency distribution graph, what type of graph should be used? (see next slide for answer) Question 4: A researcher records the gender and academic major for each student at a college basketball game. If the distribution of majors is shown in a frequency distribution graph, what type of graph should be used? (see next slide for answer) Frequency Distribution Graphs

Question 4 Answer: Question 4 Answer: In this instance, a bar graph would be used. Remember, bar graphs are used to represent nominal data. Gender and academic major are nominal categories. In this instance, a bar graph would be used. Remember, bar graphs are used to represent nominal data. Gender and academic major are nominal categories.

Frequency Distribution Graphs Question 5: If the results from a research study are presented in a frequency distribution histogram, would it also be appropriate to show the same results in a polygon? Explain your answer. (see next slide for answer) Question 5: If the results from a research study are presented in a frequency distribution histogram, would it also be appropriate to show the same results in a polygon? Explain your answer. (see next slide for answer)

Frequency Distribution Graphs Question 5 Answer: Question 5 Answer: Yes. Histograms and polygons are both used for data from interval or ratio scales. Yes. Histograms and polygons are both used for data from interval or ratio scales.

Grouped Frequency Distributions Question 6: Create a histogram using the following grouped frequency distribution table: (see next slide for answers) Question 6: Create a histogram using the following grouped frequency distribution table: (see next slide for answers) Question 7: What is the shape of the distribution? Question 7: What is the shape of the distribution? Xf

Grouped Frequency Distributions Question 6 Answer: Question 7 Answer: The distribution is positively skewed. Please note that when a distribution is positively skewed, the tail gets thinner at the higher scores.

Grouped Frequency Distributions Question 7: What is the shape of the distribution? Question 7: What is the shape of the distribution? Xf

Grouped Frequency Distributions Question 7 Answer: Question 7 Answer: The distribution is positively skewed. Please note that when a distribution is positively skewed, the tail gets thinner at the higher scores. The distribution is positively skewed. Please note that when a distribution is positively skewed, the tail gets thinner at the higher scores.

WARNING!!!! WATCH OUT!! WATCH OUT!! APLIA WILL TAKE POINTS OFF WHEN YOU CREATE HISTOGRAMS AND BAR CHARTS THAT DO NOT CONNECT THE BOTTOM OF THE BAR TO THE X-AXIS. APLIA WILL TAKE POINTS OFF WHEN YOU CREATE HISTOGRAMS AND BAR CHARTS THAT DO NOT CONNECT THE BOTTOM OF THE BAR TO THE X-AXIS. IF YOU LEAVE ANY GAP BETWEEN THE BOTTOM OF A BAR AND THE X-AXIS, YOU WILL NOT GET THE PROBLEM CORRECT. IF YOU LEAVE ANY GAP BETWEEN THE BOTTOM OF A BAR AND THE X-AXIS, YOU WILL NOT GET THE PROBLEM CORRECT. WHEN DRAWING A BAR, ALWAYS MAKE SURE THAT YOU RELEASE THE MOUSE BUTTON WHEN THE CURSOR ACTUALLY CONTACTS THE X-AXIS WITHOUT GAPS. WHEN DRAWING A BAR, ALWAYS MAKE SURE THAT YOU RELEASE THE MOUSE BUTTON WHEN THE CURSOR ACTUALLY CONTACTS THE X-AXIS WITHOUT GAPS. SEE ILLUSTRATIONS ON NEXT SLIDE SEE ILLUSTRATIONS ON NEXT SLIDE

WARNING!!!! HISTOGRAM EXAMPLES HISTOGRAM EXAMPLES HOW SHOULD THEY LOOK? HOW SHOULD THEY LOOK? NOTE: SAME RULES APPLY TO POLYGONS NOTE: SAME RULES APPLY TO POLYGONS WRONG WAY NOTICE THAT THERE IS A GAP BETWEEN THE BAR AND THE X-AXIS CORRECT WAY NOTICE THAT THERE IS NO GAP BETWEEN THE BAR AND THE X-AXIS

WARNING!!!! WATCH OUT!! WATCH OUT!! REMEMBER WHEN USING INTERPOLATION THAT CHAPTER 2 ONLY USES REAL LIMITS WITH THE X-VALUE SCALE AND NOT ON THE PERCENT SCALE. REMEMBER WHEN USING INTERPOLATION THAT CHAPTER 2 ONLY USES REAL LIMITS WITH THE X-VALUE SCALE AND NOT ON THE PERCENT SCALE. THE SAME APPLIES TO APLIA FOR THIS WEEK’S ASSIGNMENT. THE SAME APPLIES TO APLIA FOR THIS WEEK’S ASSIGNMENT.

Cumulative Frequencies & Interpolation Question 8: Fill in the cumulative frequencies and cumulative percentages for the following table: a) Find the 70 th percentile b) Find the percentile rank for X=9.5 c) Find the 15 th percentile d) Find the percentile rank for X=13 Xfcfc%c%

Cumulative Frequencies & Interpolation Xfcfc% % % % % % Question 8 Answer:

Cumulative Frequencies & Interpolation Question 8 Answers: Question 8 Answers: a) X=14.5 is the 70 th percentile b) X=9.5 has a rank of 20% c) Because 15% is between the values of 10% and 20% in the table, you must use interpolation. The score corresponding to a rank of 15% is X=7. d) Because X=13 is between the real limits of 9.5 and 14.5, you must use interpolation. The percentile rank for X=13 is 55%.

Cumulative Frequencies & Interpolation Question 8C Step-by step: Find the 15 th percentile. Question 8C Step-by step: Find the 15 th percentile. Step 1: Find the width of the interval on both scales Step 1: Find the width of the interval on both scales 5 and 10 points, respectively 5 and 10 points, respectively Step 2: Locate position of intermediate value Step 2: Locate position of intermediate value 15% is located 5 points from top (5/10 = ½ of interval) 15% is located 5 points from top (5/10 = ½ of interval) Step 3: Use same fraction to determine corresponding position on other scale. First, determine the distance from the top of the interval Step 3: Use same fraction to determine corresponding position on other scale. First, determine the distance from the top of the interval Distance = Fraction x Width = (1/2) * (5 points) = 2.5 Points Distance = Fraction x Width = (1/2) * (5 points) = 2.5 Points Step 4: Use distance from top to determine the position on the other scale Step 4: Use distance from top to determine the position on the other scale 9.5 – 2.5 = – 2.5 = 7 Thus, 15 th percentile for X is 7. Thus, 15 th percentile for X is 7.

Cumulative Frequencies & Interpolation Question 8D Step-by step: Find the percentile rank for X=13. Question 8D Step-by step: Find the percentile rank for X=13. Step 1: Find the width of the interval on both scales Step 1: Find the width of the interval on both scales 5 and 50 points, respectively 5 and 50 points, respectively Step 2: Locate position of intermediate value Step 2: Locate position of intermediate value 13 is located 1.5 points from top (1.5/5 =.3) 13 is located 1.5 points from top (1.5/5 =.3) Step 3: Use same value to determine corresponding position on other scale. First, determine the distance from the top of the interval Step 3: Use same value to determine corresponding position on other scale. First, determine the distance from the top of the interval Distance = Intermediate Vaule x Width = (.3) * (50 points) = 15 Points Distance = Intermediate Vaule x Width = (.3) * (50 points) = 15 Points Step 4: Use distance from top to determine the position on the other scale Step 4: Use distance from top to determine the position on the other scale 70 – 15 = – 15 = 55 Thus, X=7 has a percentile rank of 55%. Thus, X=7 has a percentile rank of 55%.

Stem and Leaf Displays Question 10: Use a stem and leaf display to organize the following set of scores: (Please see next slide for answers) Question 10: Use a stem and leaf display to organize the following set of scores: (Please see next slide for answers) 74, 103, 95, 98, 81, 117, 105, 99, 63, 86, 94, 107, 96, 100, 98, 118, 107, 82, 84, 71, 91, 107, 84, 77

Stem and Leaf Displays Question 10 Answer: Question 10 Answer: StemLeaf

Stem and Leaf Displays Question 11: Explain how a stem and leaf display contains more information than a grouped frequency distribution. (Please see next slide for answers) Question 11: Explain how a stem and leaf display contains more information than a grouped frequency distribution. (Please see next slide for answers)

Stem and Leaf Displays Question 11 Answer: Question 11 Answer: A grouped frequency distribution table tells only the number of scores in each interval; it does not identify the exact value for each score. The stem and leaf display identifies the individual scores as well as the number of scores in each interval. A grouped frequency distribution table tells only the number of scores in each interval; it does not identify the exact value for each score. The stem and leaf display identifies the individual scores as well as the number of scores in each interval.

Frequently Asked Questions: Suggestion from Dr. Kerbs! Question: Is there a way of visually understanding interpolation? Question: Is there a way of visually understanding interpolation? Answer: Yes. Sometimes it helps to draw a visual depiction of the x-value intervals and the % intervals. For Question 8C, consider the following images. Answer: Yes. Sometimes it helps to draw a visual depiction of the x-value intervals and the % intervals. For Question 8C, consider the following images. X-Scale%-Scale X-Scale%-Scale % % > ???. - ->15% about here. - - > ???. - ->15% about here STEP #2: 15% is 5 points down from the top of a 10-point span that begins at 20% and ends at 10%. We turn this into a fraction of the total interval with 5/10 = ½ STEP #3: We use the fraction Identified for the % scale (½) to identify the corresponding position on the X-scale. Apply fraction to the X-scale interval, which spans 5 points ( =5.0). ½ * (5.0) = 2.5 Thus, the 15 th percentile is 2.5 points down from the top of the X-Scale 15 th Percentile = = 7.0 STEP #1: DRAW PICTURES FOR BOTH SCALES. THEN START CALCULATIONS WITH % SCALE FOR STEP#2 BECAUSE THE QUESTION GIVES PERCENTILE RANK IN SEARCH OF X-SCALE VALUE REMEMBER: ALWAYS WORK FROM THE TOP DOWN IN THE SCALES