DEFINITIONS Population Sample Unit of analysis Case Sampling frame

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 Population Sample Jay’s prison

A more complicated situation 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 Hypothesis: Youth demeanor  officer disposition From “Police Encounters With Juveniles,” Irving Piliavin and Scott Briar, The American Journal of Sociology, Vol. 70, No. 2 (Sep., 1964)

DEPICTING THE DISTRIBUTION OF CATEGORICAL VARIABLES Frequency “X” axis and “Y” axis Bar graph Table

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 Distributions depict how cases “distribute” along the values or scores of the variable. Here the proportions of male and female seem nearly equal. n=15 n=17 How many at each value/score Bars are “made” of cases. Here they’re made of students, who are arranged by the variable gender N = 32 Y - axis Value or score of variable X - axis

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

DEPICTING THE DISTRIBUTION OF CONTINUOUS VARIABLES Histogram Trend line

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 What is the area under the trend line “made of”? Cases, meaning students (arranged by age) “Trend” line How many at each value/score Y - axis Value or score of variable X - axis

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

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

CATEGORICAL VARIABLES Summarizing the distribution of

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 =.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”?

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

Class 1Class 2 Draw a bar graph for each class depicting proportions for gender, then compare the proportions Practical exercise

Increases in percentage are computed off the base amount Example: Jail with 120 prisoners. How many prisoners 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 200 percent increase? Calculating increases in percentage 2 times 3 times larger (2X) larger (3X) 200% 100% Original larger larger

Percentage changes can mislead Percentages can make changes seem large when bases are small Example: Increase from 1 to 3 convictions is 200 (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%

CONTINUOUS VARIABLES Summarizing the distribution of

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

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 Mean 3.0 Mean 3.5 arrests Variable: number of arrests per officer Unit of analysis: police precincts Case: one precinct Issue: Means are pulled in the direction of extreme scores, creating a possibly misleading comparison 1 X X X X 3…= 60 /20 = 3 number of officers 1 X X X X 3…= 70 /20 = 3.5 arrests per officer arrests

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 Citation or official reprimand Informal reprimand Admonished & released 2538 Total (N) Severity of disposition mean = 2.24 (25 X 1) + (16 X 2) + (9 X 3) + (16 X 4) / 66

0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6 Exercise 1: 2, 3, 5, 5, 8, 12, 17, 19, 21 Exercise 2: 2, 3, 5, 5, 8, 12, 17, 19, 21, 21 Compute / 2 = 3 arrests Summarizing the distribution of continuous variables - the median Median can be used with continuous or ordinal variables Median is a useful summary statistic when there are extreme scores, making the mean misleading In this example, which is identical to the preceding page except for one outlier (16), the mean is 3.5 –.5 higher But the medians (3.0) are the same

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 Summarizing the distribution of continuous variables - the mode

Range: a simple way to convey the distribution of a continuous variable – 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 A final way to depict the distribution of continuous variables - the range

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

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.

Case No. Income No. of arrests Gender M F F M F M F F M F F M F F M M F M F F 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