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COURSE: JUST 3900 TIPS FOR APLIA Developed By: Ethan Cooper (Lead Tutor) John Lohman

COURSE: JUST 3900 TIPS FOR APLIA Developed By: Ethan Cooper (Lead Tutor) John Lohman Michael Mattocks Aubrey Urwick. Chapter 2: Frequency Distributions. Key Terms: Don’t Forget Notecards. Frequency Distribution (p. 39) ΣX (p. 40) Proportions (p. 41) Percentages (p. 41)

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COURSE: JUST 3900 TIPS FOR APLIA Developed By: Ethan Cooper (Lead Tutor) John Lohman

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  1. COURSE: JUST 3900 TIPS FOR APLIA Developed By: Ethan Cooper (Lead Tutor) John Lohman Michael Mattocks Aubrey Urwick Chapter 2: Frequency Distributions

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

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

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

  5. Frequency Distributions • 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 • Question 2: Find each of the following values for the sample in the following frequency distribution table. (see p. 42, #2 for answers) • n • ΣX • ΣX2

  6. Frequency Distributions • Question 1 Answers: • REMEMBER: The values in the X column should always be listed from the • highest to the lowest! • Question 2 Answers: • 1) n=10 • 2) ΣX=28 • 3) ΣX2 =92 (square then add all 10 scores)

  7. Frequency Distribution Graphs Question: Sketch a frequency distribution histogram and a frequency distribution polygon for the data in the following table, then describe the shape of the distribution: (see p. 53, #1 & 2 for answers)

  8. 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.

  9. Grouped Frequency Distributions Question 3 Answer:

  10. Frequency Distribution Graphs 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 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)

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

  12. Grouped Frequency Distributions • 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?

  13. 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.

  14. WARNING!!!! • 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. • 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. • SEE ILLUSTRATIONS ON NEXT SLIDE

  15. WARNING!!!! • HISTOGRAM EXAMPLES • HOW SHOULD THEY LOOK? • 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

  16. WARNING!!!! • WATCH OUT!! • 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.

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

  18. Cumulative Frequencies & Interpolation Question 8 Answer:

  19. Cumulative Frequencies & Interpolation • Question 8 Answers: a) X=14.5 is the 70th 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%.

  20. Cumulative Frequencies & Interpolation • Question 9C Step-by step: Find the 15th percentile. • Step 1: Find the width of the interval on both scales • 5 and 10 points, respectively • Step 2: Locate position of intermediate value • 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 • 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 • 9.5 – 2.5 = 7 • Thus, 15th percentile for X is 7.

  21. Cumulative Frequencies & Interpolation • Question 9D Step-by step: Find the percentile rank for X=13. • Step 1: Find the width of the interval on both scales • 5 and 50 points, respectively • Step 2: Locate position of intermediate value • 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 • Distance = Intermediate Vaule x Width = (.3) * (50 points) = 15 Points • Step 4: Use distance from top to determine the position on the other scale • 70 – 15 = 55 • Thus, X=7 has a percentile rank of 55%.

  22. 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) 74, 103, 95, 98, 81, 117, 105, 99, 63, 86, 94, 107, 96, 100, 98, 118, 107, 82, 84, 71, 91, 107, 84, 77 • Question 11: Explain how a stem and leaf display contains more information than a grouped frequency distribution. (Please see next slide for answers)

  23. Stem and Leaf Displays • Question 10 Answer:

  24. Stem and Leaf Displays • 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.

  25. Frequently Asked Questions:Suggestion from Dr. Kerbs! • 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. • X-Scale %-Scale 9.5 20% . . . . .- - > ???.- ->15% about here . . . . 4.5 10 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 (9.5-4.5=5.0). ½ * (5.0) = 2.5 Thus, the 15th percentile is 2.5 points down from the topof the X-Scale 15th Percentile = 9.5-2.5 = 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

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