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DEPICTING DISTRIBUTIONS

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How many at each value/score Value or score of variable

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What is a “distribution”? One distribution for a single variable. Each youth homicide is a case. There is one variable: the number each month. Two distributions, each for a single variable: violent crime or imprisonment. Each violent crime is a case. The variable is their number each year (divided by 100,000) Each prisoner is a case. The variable is their number each year (divided by 100,000). One distribution for TWO variables: Youth’s demeanor (two categories) Officer disposition (four categories) Each police encounter with a youth is a case. An arrangement of cases in a sample or population according to their values or scores on one or more variables (A case is a single unit that “contains” all the variables of interest) Distributions can be visually depicted. How that is done depends on the kind of variable, categorical or continuous.

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Depicting the distribution of categorical variables: the bar graph Distributions depict the frequency (number of cases) at each value of a variable. Here there is one: gender. 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 X - axis Y - axis Distributions illustrate how cases cluster or spread out according to the value or score of the variable. Here the proportions of men and women seem about equal. n=15 n=17

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Depicting the distribution of continuous variables: the histogram Y - axis X - axis Frequency means the number of cases at a single value of a variable Distributions depict the frequency (number of cases) at each value of a variable Frequencies (“counts”) are always on the Y axis Values of the variable are always on the X axis

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CATEGORICAL VARIABLES Summarizing the distribution of

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Summarizing the distribution of categorical variables using percentage 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”?

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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

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Class 1Class 2 Draw two bar graphs, one for each class, depicting proportions for gender Practical exercise

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Wed. classThurs. class 15 Females 15/31 =.483 X 100 = 48% 16 Males 16/31 =.516 X 100 = 52% _____ 100% 20 Females 20/31 =.645 X 100 = 65% 11 Males 11/31 =.354 X 100 = 35% _____ 100% ____ 100%

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Calculating increases in percentage 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) 150 percent increase? – 150 percent of 120 is 180 (120 X 150 / 100) – 120 base plus 180 increase = 300 (2 ½ times the base amount) How many will there be with a 200 percent increase?

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Percentage changes can mislead Answer to preceding slide – prison with 120 prisoners 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 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%

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CONTINUOUS VARIABLES Summarizing the distribution of

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Four summary statistics 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

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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 numbers of arrests for twenty police precincts during a certain shift Method: Use mean to summarize arrests at each precinct, then compare the means Mean 3.0 Mean 3.5 arrests Variable: number of arrests Unit of analysis: police precincts Case: one precinct Issue: Means are pulled in the direction of extreme scores, possibly misleading the comparison

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Using the mean for ordinal variables Ordinal variables are categorical variables with an inherent order – Small, medium, large – Cooperative, uncooperative Can summarize in the ordinary way: proportions / percentages Or, treat categories as points on a continuous scale and calculate 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”? Rank Severity of Disposition Youths Freq.% 4 Arrested Citation or official reprimand Informal reprimand Admonished & release 2538 Total Severity of disposition mean = 2.24 (25 X 1) + (16 X 2) + (9 X 3) + (16 X 4) / 66

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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

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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) Median can be used with continuous or ordinal variables Median is a useful summary statistic when there are extreme scores, making the mean misleading 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, / 2 = 3 arrests

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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

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Answers to preceding side Exercise 1: 2, 3, 5, 5, 8, 12, 17, 19, 21Mode = 5 (unimodal) Exercise 2: 2, 3, 5, 5, 8, 12, 17, 19, 21, 21Modes = 5, 21 (bimodal) 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

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In-class exercise Calculate your class 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 Case no.

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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.

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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 EXERCISE (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

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