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

COURSE: JUST 3900TIPS FOR APLIA

Developed By: Ethan Cooper (Lead Tutor)

John LohmanMichael Mattocks

Aubrey Urwick

Chapter 2: Chapter 2: Frequency DistributionsFrequency Distributions

Key Terms: Don’t Forget Key Terms: Don’t Forget NotecardsNotecards

Frequency Distribution (p. 39)Frequency Distribution (p. 39) ΣXΣX (p. 40) (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 NotecardsMore 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 NotecardsMore 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 Question 1: Construct a frequency distribution for the following set of scores. (see next slide for answer)following set of scores. (see next slide for answer) Scores: 3, 2, 3, 2, 4, 1, 3, 3, 5Scores: 3, 2, 3, 2, 4, 1, 3, 3, 5

Question 2: Find each of the following values for the sample Question 2: Find each of the following values for the sample in the following frequency distribution table. (see p. 42, #2 for in the following frequency distribution table. (see p. 42, #2 for answers)answers)1)1) nn2)2) ΣXΣX3)3) ΣXΣX2

Frequency DistributionsFrequency Distributions

X f 5 1

4 2 3 2 2 4

1 1

Frequency DistributionsFrequency Distributions

X f Proportion Percentage5 1 .11 11%4 1 .11 11%3 4 .44 44%

2 2 .22 22%1 1 .11 11%

• 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=102) ΣX=282) ΣX=283) ΣX3) ΣX22 =92 (square then add all 10 scores) =92 (square then add all 10 scores)

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

Frequency Distribution GraphsFrequency Distribution Graphs

X f5 44 6

3 32 11 1

Grouped Frequency Grouped Frequency DistributionsDistributions

Question 3: An instructor has obtained the set of N=25 Question 3: An instructor has obtained the set of N=25 exam scores. To help organize these scores, place them in exam scores. To help organize these scores, place them in a frequency distribution table. The scores are:a frequency distribution table. The scores are:

82, 75, 88, 93, 53, 84, 87, 58, 72, 94, 69, 84, 61, 91, 64, 87, 84, 82, 75, 88, 93, 53, 84, 87, 58, 72, 94, 69, 84, 61, 91, 64, 87, 84, 70, 76, 89, 75, 80, 73, 78, 6070, 76, 89, 75, 80, 73, 78, 60

See next slide for answer.See next slide for answer.


X f Proportion Percentage90-94 3 .12 12%

85-89 4 .16 16%80-84 5 .20 20%75-79 4 .16 16%70-74 3 .12 12%

65-69 1 .04 4%60-64 3 .12 12%

55-59 1 .04 4%

50-54 1 .04 4%

Question 3 Answer:

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

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

Frequency Distribution GraphsFrequency Distribution Graphs

Frequency Distribution GraphsFrequency Distribution Graphs Question 4 Answer:Question 4 Answer:

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

Question 5 Answer:Question 5 Answer: Yes. Histograms and polygons are both used for data from Yes. Histograms and polygons are both used for data from

interval or ratio scales.interval or ratio scales.


Question 6: Create a histogram using the following Question 6: Create a histogram using the following grouped frequency distribution table: (see next slide for grouped frequency distribution table: (see next slide for answers)answers)

Question 7: What is the shape of the distribution?Question 7: What is the shape of the distribution?

X f25-29 120-24 115-19 5

10-14 95-9 4


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 higherscores.

WARNING!!!!WARNING!!!! WATCH OUT!!WATCH OUT!!

APLIA WILL TAKE POINTS OFF WHEN YOU CREATE APLIA WILL TAKE POINTS OFF WHEN YOU CREATE HISTOGRAMS AND BAR CHARTS THAT DO NOT CONNECT HISTOGRAMS AND BAR CHARTS THAT DO NOT CONNECT THE BOTTOM OF THE BAR TO THE X-AXIS. THE BOTTOM OF THE BAR TO THE X-AXIS.

IF YOU LEAVE ANY GAP BETWEEN THE BOTTOM OF A IF YOU LEAVE ANY GAP BETWEEN THE BOTTOM OF A BAR AND THE X-AXIS, YOU WILL NOT GET THE PROBLEM BAR AND THE X-AXIS, YOU WILL NOT GET THE PROBLEM CORRECT.CORRECT.

WHEN DRAWING A BAR, ALWAYS MAKE SURE THAT YOU WHEN DRAWING A BAR, ALWAYS MAKE SURE THAT YOU RELEASE THE MOUSE BUTTON WHEN THE CURSOR RELEASE THE MOUSE BUTTON WHEN THE CURSOR ACTUALLY CONTACTS THE X-AXIS WITHOUT GAPS.ACTUALLY CONTACTS THE X-AXIS WITHOUT GAPS.

SEE ILLUSTRATIONS ON NEXT SLIDESEE ILLUSTRATIONS ON NEXT SLIDE

WARNING!!!!WARNING!!!! HISTOGRAM EXAMPLES HISTOGRAM EXAMPLES

HOW SHOULD THEY LOOK?HOW SHOULD THEY LOOK? NOTE: SAME RULES APPLY TO POLYGONSNOTE: SAME RULES APPLY TO POLYGONS

WRONG WAY

NOTICE THAT THERE ISA GAP BETWEEN THE BAR AND THE X-AXIS

CORRECT WAY

NOTICE THAT THERE IS NO GAP BETWEEN THEBAR AND THE X-AXIS

WARNING!!!!WARNING!!!! WATCH OUT!!WATCH OUT!!

REMEMBER WHEN USING INTERPOLATION REMEMBER WHEN USING INTERPOLATION THAT CHAPTER 2 ONLY USES REAL LIMITS THAT CHAPTER 2 ONLY USES REAL LIMITS WITH THE X-VALUE SCALE AND WITH THE X-VALUE SCALE AND NOTNOT ON THE ON THE PERCENT SCALE.PERCENT SCALE.

THE SAME APPLIES TO APLIA FOR THIS WEEK’S THE SAME APPLIES TO APLIA FOR THIS WEEK’S ASSIGNMENT.ASSIGNMENT.

Cumulative Frequencies & Cumulative Frequencies & InterpolationInterpolation

Question 8: Fill in the cumulative frequencies and Question 8: Fill in the cumulative frequencies and cumulative percentages for the following table:cumulative percentages for the following table:

a) Find the 70a) Find the 70thth percentile percentileb) Find the percentile rank for X=9.5b) Find the percentile rank for X=9.5c) Find the 15c) Find the 15thth percentile percentiled) Find the percentile rank for X=13d) Find the percentile rank for X=13

X f cf c%20-24 115-19 510-14 8

5-9 4

0-4 2


X f cf c%20-24 1 20 100%15-19 5 19 95%10-14 8 14 70%

5-9 4 6 20%

0-4 2 2 10%

Question 8 Answer:


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


Question 9C Step-by step: Find the 15Question 9C Step-by step: Find the 15thth percentile. percentile. Step 1: Find the width of the interval on both scalesStep 1: Find the width of the interval on both scales

5 and 10 points, respectively5 and 10 points, respectively Step 2: Locate position of intermediate valueStep 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 Step 3: Use same fraction to determine corresponding position

on other scale. First, determine the distance from the top of the on other scale. First, determine the distance from the top of the intervalinterval

Distance = Fraction x Width = (1/2) * (5 points) = 2.5 PointsDistance = Fraction x Width = (1/2) * (5 points) = 2.5 Points Step 4: Use distance from top to determine the position on the Step 4: Use distance from top to determine the position on the

other scaleother scale 9.5 – 2.5 = 79.5 – 2.5 = 7 Thus, 15Thus, 15thth percentile for X is 7. percentile for X is 7.


Question 9D Step-by step: Find the percentile rank for Question 9D Step-by step: Find the percentile rank for X=13.X=13. Step 1: Find the width of the interval on both scalesStep 1: Find the width of the interval on both scales

5 and 50 points, respectively5 and 50 points, respectively Step 2: Locate position of intermediate valueStep 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 Step 3: Use same value to determine corresponding position on

other scale. First, determine the distance from the top of the intervalother scale. First, determine the distance from the top of the interval Distance = Intermediate Vaule x Width = (.3) * (50 points) = 15 PointsDistance = Intermediate Vaule x Width = (.3) * (50 points) = 15 Points

Step 4: Use distance from top to determine the position on the other Step 4: Use distance from top to determine the position on the other scalescale

70 – 15 = 5570 – 15 = 55 Thus, X=7 has a percentile rank of 55%.Thus, X=7 has a percentile rank of 55%.

Stem and Leaf DisplaysStem and Leaf Displays Question 10: Use a stem and leaf display to organize the Question 10: Use a stem and leaf display to organize the

following set of scores: (Please see next slide for following set of scores: (Please see next slide for answers)answers)

74, 103, 95, 98, 81, 117, 105, 99, 63, 86, 94, 107, 96, 100, 98, 74, 103, 95, 98, 81, 117, 105, 99, 63, 86, 94, 107, 96, 100, 98, 118, 107, 82, 84, 71, 91, 107, 84, 77118, 107, 82, 84, 71, 91, 107, 84, 77

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

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

Stem Leaf6 3

7 417

8 16244

9 5894681

10 357077

11 78

Stem and Leaf DisplaysStem and Leaf Displays Question 11 Answer:Question 11 Answer:

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

Frequently Asked Questions:Frequently Asked Questions:Suggestion from Dr. Kerbs!Suggestion from Dr. Kerbs!

Question: Is there a way of visually understanding Question: Is there a way of visually understanding interpolation?interpolation?

Answer: Yes. Sometimes it helps to draw a visual depiction Answer: Yes. Sometimes it helps to draw a visual depiction of the x-value intervals and the % intervals. For Question of the x-value intervals and the % intervals. For Question 8C, consider the following images. 8C, consider the following images.

X-ScaleX-Scale %-Scale%-Scale 9.59.5 20% 20% .. . . .. . . .. - - > ???- - > ??? .. - ->15% about here- ->15% about here .. . . .. . . 4.54.5 10 10

STEP #2: 15% is 5 points down from the top of a 10-point span that beginsat 20% and endsat 10%. We turn this into a fractionof the total intervalwith 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 2.5points down from the top of 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

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

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