1. DEFINITIONS
Population
Sample
Unit of analysis
Case
Sampling frame

2. Some essential definitions
Population
Largest group to which we intend to project (apply) the findings of a study
All the prisoners in Jay’s prison
Sample
Any subgroup of the population
Samples intended to represent a population must be selected in special ways (will come up later)
Unit of analysis
The “container” for the variables
Here, the variables under study are sentence length and type of crime (property or violent)
What “contains” them? Prisoners!
Case
A single occurrence of a unit of analysis
Here, it’s any one prisoner
Cases are “members” or “elements” of the population from which one or more samples are drawn
Sampling frame
A list of all “elements” or members of the population
Jay’s prison
Population
Sample

3. A more complicated situation
Hypothesis: Youth demeanor  officer disposition
Research Question: Do officers treat uncooperative youths more harshly?
Hypothesis: Uncooperative youths are more likely to be arrested
Graduate students rode around in police cars and coded interactions between officers and youths where arrest was not mandatory
So…
What is the independent (causal) variable?
What is the dependent (effect) variable?
What is the population?
What is the sample?
What is the unit of analysis? (container for the variables)
What is a case? (a single occurrence of a unit of analysis)
Officer’s disposition
From “Police Encounters With Juveniles,” Irving Piliavin and Scott Briar, The American Journal of Sociology, Vol. 70, No. 2 (Sep., 1964)

4. depicting THE DISTRIBUTION OF CATEGORICAL VARIABLES
Frequency
“X” axis and “Y” axis
Bar graph
Table

5. Depicting distribution of a categorical variable: the bar graph
Distributions depict the frequency (number of cases) at each value of a variable. Here there is one variable, gender, with two values, M/F. Thirty-two students are the “population.” Each student is a case. Frequency means the number of cases – students – at a single value of a variable. Frequencies are always on the Y axis Values of the variable are always on the X axis
Y - axis
N = 32
n=17
n=15
How many at each value/score
X - axis
Value or score of variable
Bars are “made” of cases. Here they’re made of students, who are arranged by the variable gender
Distributions depict how cases “distribute” along the values or scores of the variable. Here the proportions of male and female seem nearly equal.

6. Using a table to display the distribution of two categorical variables
Value or score of variable
“cells”
Value or score of variable
Officer’s disposition
Number of cases (frequency) at each value/score

7. depicting THE DISTRIBUTION OF CONTINUOUS VARIABLES
Histogram
Trend line

8. Depicting the distribution of continuous variables: the histogram
Distributions depict the frequency (number of cases) at each value of a variable. Here there is one variable: age, measured on a scale of A case is a single unit that “contains” all the variables of interest. Here each student is a case Frequency means the number of cases – students – at a single value of a variable. Frequencies are always on the Y axis Values of the variable are always on the X axis
“Trend” line
Y - axis
How many at each value/score
X - axis
Value or score of variable
What is the area under the trend line “made of”? Cases, meaning students (arranged by age)

9. Sometimes, bar graphs are used for continuous variables
Y - axis
How many at each value/score
Value or score of variable
X - axis
What are the bars “made of”? Cases, meaning homicides (arranged by the variable homicides per year)

10. Continuous variables: What “makes up” the areas under the trend lines?
Cases, that’s what!
Trend line
This graph displays the distribution of one continuous variable: youths murdered with guns each month
How many at each value/score
Each murdered youth is one “case”
Value or score of variable
Value or score of variable
How many at each value/score
Trend line
This graph displays distributions of two continuous variables: violent crime rate and imprisonment rate
Each violent crime is one “case”
Each commitment to prison is one “case”

11. Categorical variables
Summarizing the distribution of
Categorical variables

12. Summarizing the distribution of categorical variables using percentage
Instead of using graphs or a lot of words, is there a single statistic that can convey what a distribution “looks like”?
Percentage is a “statistic.” It’s a proportion with a denominator of 100.
Percentages are used to summarize categorical data
70 percent of students are employed; 60 percent of parolees recidivate
Since per cent means per 100, any decimal can be converted to a percentage by multiplying it by 100 (moving the decimal point two places to the right)
.20 = .20 X 100 = 20 percent (twenty per hundred)
.368 = .368 X 100 = 36.8 percent (thirty-six point eight per hundred)
When converting, remember that there can be fractions of one percent
.0020 = X 100 = .20 percent (two tenths of one percent)
To obtain a percentage for a category, divide the number of cases in the category by the total number of cases in the sample
50,000 persons were asked whether crime is a serious problem: 32,700 said “yes.” What percentage said “yes”?

13. Using percentages to compare datasets
Percentages are “normalized” numbers (e.g., per 100), so they can be used to compare datasets of different size
Last year, 10,000 people were polled. Eight-thousand said crime is a serious problem
This year 12,000 people were polled. Nine-thousand said crime is a serious problem.
Calculate the second percentage and compare it to the first

14. Draw a bar graph for each class depicting proportions for gender, then compare the proportions
TTH
Class
Friday
Class

15. TTH class
Friday class

16. Calculating increases in percentage
Increases in percentage are computed off the base amount
Example: Jail with 120 inmates. How many will there be …with a 100 percent increase?
100 percent of the base amount, 120, is 120 (120 X 100/100)
120 base increase = 240 (2 times the base amount)
…with a 150 percent increase?
150 percent of 120 is 180 (120 X 150/100)
120 base plus 180 increase = 300 (2½ times the base amount)
How many will there be with a percent increase?
200% % Original larger larger
2 times times
larger (2X) larger (3X)

17. Percentage changes can mislead
Answer to preceding slide – jail with 120 inmates 200 percent increase
200 percent of 120 is 240 (120 X 200/100)
120 base plus 240 = 360 (3 times the base amount)
Percentages can make changes seem large when bases are small Example: Increase from 1 to 3 convictions is two-hundred percent
3-1 = 2
2/base = 2/1 = 2
2 X 100 = 200%
Percentages can make changes seem small when bases are large Example: Increase from 5,000 to 6,000 convictions is 20 (twenty) percent 6, ,000 = 1,000 1,000/base = 1000/5,000 = .20 = 20%

18. Summarizing the distribution of
Continuous variables

19. Four summary statistics for continuous variables
Continuous variables – review
Can take on an infinite number of values (e.g., age, height, weight, sentence length)
Precise differences between cases
Equivalent differences: Distances between years same as years
Summary statistics for continuous variables
Mean: arithmetic average of scores
Median: midpoint of scores (half higher, half lower)
Mode: most frequent score (or scores, if tied)
Range: Difference between low and high scores
3.5
1.3

20. Summarizing the distribution of continuous variables - the mean
Arithmetic average of scores
Add up all the scores
Divide the result by the number of scores
Example: Compare arrest productivity for officers in two precincts, each with 20 officers, during dayshift
Method: Use mean to summarize arrests at each precinct, then compare the means
Variable: no. of arrests per officer
Unit of analysis: police precincts
Case: one precinct
arrests per officer
number of officers
arrests
arrests
1 X X 1 +
4 X X 3…=
60 /20 = 3
1 X X 1 +
4 X X 3…=
70 /20 = 3.5
arrests
arrests
Mean 3.0
Mean 3.5
0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6
0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 16
Means are pulled in the direction, of extreme scores, possibly distorting comparisons

21. Severity of disposition mean = 2.24
Transforming categorical/ordinal variables into continuous variables, then using the mean
Ordinal variables are categorical variables with an inherent order
Small, medium, large
Cooperative, uncooperative
Can summarize in the ordinary way: proportions / percentages
Can also transform them into continuous variables by assigning categories points on a scale, then calculating a mean
Not always recommended because “distances” between points on scale may not be equal, causing misleading results
Is the distance between “Admonished” and “Informal” same as between “Informal and Citation”? “Citation” and “Arrest”?
Value
Severity of Disposition
Youths
Freq. % 4
Arrested 16 24 3
Citation or official reprimand 9 14 2
Informal reprimand 1
Admonished & released 25 38
Total (N) 66
100
Severity of disposition mean = 2.24
(25 X 1) + (16 X 2) + (9 X 3) + (16 X 4) / 66

22. Summarizing the distribution of continuous variables - the median
Median is the physical center score (when there are two, use their average)
Median can be used with continuous or ordinal variables
Median is an especially useful summary statistic when extreme scores are present, as they tend to make the mean misleading
Although one distribution has an outlier (16), which pulls the mean higher (3.0 to 3.5), the medians for both distributions are the same (3.0)
arrests
Mean 3.0
Mean 3.5
0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6
0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 16
3 + 3 / 2 = 3
3 + 3 / 2 = 3

23. Summarizing the distribution of continuous variables - the median
Median is the physical center score (when there are two, use their average)
Exercise 1: 2, 3, 5, 5, 8, 12, 17, 19, 21
Exercise 2: 2, 3, 5, 5, 8, 12, 17, 19, 21, 21
Compute the median for each sample...

24. Summarizing the distribution of continuous variables - the median
Answers to preceding slide
Exercise 1: 2, 3, 5, 5, 8, 12, 17, 19, 21 Answer: 8
Exercise 2: 2, 3, 5, 5, 8, 12, 17, 19, 21, 21 Answer: 10 ( / 2)

25. Summarizing the distribution of continuous variables - the mode
Score that occurs most often (with the greatest frequency)
Here the mode is 3
Modes are a useful summary statistic when cases cluster at particular scores – an interesting condition that might otherwise be overlooked
Symmetrical distributions, like this one, are called “normal” distributions. In such distributions the mean, mode and median are the same. Near-normal distributions are common.
There can be more than one mode (bi-modal, tri-modal, etc.). Identify the modes:
Exercise 1: 2, 3, 5, 5, 8, 12, 17, 19, 21
Exercise 2: 2, 3, 5, 5, 8, 12, 17, 19, 21, 21
arrests

26. Summarizing the distribution of continuous variables - the mode
Answers to preceding slide
Exercise 1: 2, 3, 5, 5, 8, 12, 17, 19, 21
Exercise 2: 2, 3, 5, 5, 8, 12, 17, 19, 21, 21

27. A final way to depict the distribution of continuous variables - the range
Depicts the lowest and highest scores in a distribution 2, 3, 5, 5, 8, 12, 17, 19, 21 – range is “2 to 21”
Range can also be defined as the difference between the scores (21-2 = 19). If so, minimum and maximum scores should also be given.
Useful to cite range if there are outliers (extreme scores) that misleadingly distort the shape of the distribution

28. Practical exercise
Calculate summary statistics for age and height – mean, median, mode and range
Pictorially depict the distributions for age and height, placing the variables and frequencies on the correct axes
TTH
Class
Friday
Class

29. TTH
Class

30.
Friday
Class

31. A preview about dispersal
TTH Class
Friday Class
For age, cases (students) “cluster” at the lower values in both classes.
Even so, the means are “pulled” to the right by a sizeable number of relatively old students.
Height is far more “dispersed.” So which is a more accurate descriptor: mean age or mean height?
“Dispersal” is measured with two statistics: variance and standard deviation. We’ll deal with them later…
Mean 23.41
Without 25+
mean 21.65
Mean 22.83
Without 25+
mean 21.88
Mean 65.53
Mean 65.85

32. Next week – Every week:
Without fail – bring an approved calculator – the same one you will use for the exam. It must be a basic calculator with a square root key. NOT a scientific or graphing calculator. NOT a cell phone, etc.

33. (link on weekly schedule)
HOMEWORK
(link on weekly schedule)
1. Calculate all appropriate summary statistics for each distribution
2. Pictorially depict the distribution of arrests
3. Pictorially depict the distribution of gender
Case
No.
Income
No. of arrests
Gender 1
15600 4 M 2
21380 3 F
17220 5
18765
23220 6
44500 7
34255 8
21620 9
14890 10
16650 11 12
16730 13
23980 14
14005 15
21550 16
26780 17
18050 18
34500 19
33785 20
21450