course: just 3900 introductory statistics for criminal justice chapter 2: frequency distributions...

COURSE: JUST 3900INTRODUCTORY STATISTICS

FOR CRIMINAL JUSTICEChapter 2: Frequency Distributions

Peer Tutor SlidesInstructor:

Mr. Ethan W. Cooper, Lead Tutor© 2013 - - PLEASE DO NOT CITE, QUOTE, OR REPRODUCE WITHOUT THE

WRITTEN PERMISSION OF THE AUTHOR. FOR PERMISSION OR QUESTIONS, PLEASE EMAIL MR. COOPER AT THE FOLLWING: [email protected]

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)

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)

More Key Terms: Think Notecards

Cumulative Frequency (p. 54) Cumulative Percentage (p. 54) Interpolation (p. 55) Stem and Leaf Displays (p. 60)

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)1) n

2) ΣX3) ΣX2

Frequency Distributions

X f

5 1

4 2

3 2

2 4

1 1


Question 1 Answers: X f Proportion Percentage

5 1 .11 11%

4 1 .11 11%

3 4 .44 44%

2 2 .22 22%

1 1 .11 11%

REMEMBER: The values in the X column should always be listed from the highest to the lowest!


Question 2: Find each of the following values for the sample in the following frequency distribution table.1) n

2) ΣX3) ΣX2

X f

5 1

4 2

3 2

2 4

1 1


Question 2 Answers:1) n=10

2) ΣX=28

3) ΣX2 =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.


X f Proportion Percentage

90-94 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 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: 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: 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 Answer: Yes. Histograms and polygons are both used for data from

interval or ratio scales.


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?

X f

25-29 1

20-24 1

15-19 5

10-14 9

5-9 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.


Question 7: What is the shape of the distribution?

X f

25-29 1

20-24 1

15-19 5

10-14 9

5-9 4


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.

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

WARNING!!!! HISTOGRAM EXAMPLES

HOW SHOULD THEY LOOK? NOTE: 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!!!!

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.

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

X f cf c%

20-24 1

15-19 5

10-14 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:

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%.


Question 8C 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.


Question 8D 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%.

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 10 Answer:

Stem Leaf

6 3

7 417

8 16244

9 5894681

10 357077

11 78


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 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.

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 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 introductory statistics for criminal justice chapter 2: frequency distributions...

Documents