COURSE: JUST 3900 INTRODUCTORY STATISTICS FOR CRIMINAL JUSTICE Chapter 2: Frequency Distributions...
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Transcript of 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 Ethan W. Cooper, Lead Tutor© 2013 - - PLEASE DO NOT CITE, QUOTE, OR REPRODUCE WITHOUT THE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
Frequency Distributions
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!
Frequency Distributions
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
Frequency Distributions
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.
Grouped Frequency Distributions
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
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.
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)
Frequency Distribution Graphs
Question 5 Answer: 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 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
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 higherscores.
Grouped Frequency Distributions
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
Grouped Frequency Distributions
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
Cumulative Frequencies & Interpolation
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:
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%.
Cumulative Frequencies & Interpolation
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.
Cumulative Frequencies & Interpolation
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
Stem and Leaf Displays
Question 10 Answer:
Stem Leaf
6 3
7 417
8 16244
9 5894681
10 357077
11 78
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)
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.
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