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Chapter 2

Chapter 2- Basic Descriptive Statistics.docx

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Weiguo Zhang

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Chapter 2: Basic Descriptive Statistics 2.1 Percentages and Proportions  Percentages and proportions supply frame of reference for reporting research results in the sense that they standardize the raw data: percentages to the base 100 and proportions’ to the base 1.00  Percentages and proportions are easier to read and comprehend than frequencies o Particularly obvious when attempting to compare groups of different sizes  Computing percentages eliminates the difference in size of the two groups by standardizing both distributions to the base of 100 Guidelines on use of percentages and proportions: 1. When working with a small number of cases (i.e. less than 20), it is preferable to report the actual frequencies rather than percentages or proportions  Percentages can change drastically with relatively minor changes in data 2. Always report number of observations along with proportions and percentages  Permits reader to judge adequacy of the sample size 3. Percentages and proportions can be calculated for variables at ordinal and nominal levels of measurement, even though they require division  Percentages and proportions don’t require division of scores of variable (as would be the case in computing average score on a test for example) but rather the number of cases in particular category (f) of the variable by the total number of cases in the sample (n) 2.2 Ratios and Rates  Provide some additional ways of summarizing results simply and clearly  Ratios specially useful for comparing categories of a variable in terms of relative frequency o Determine ratios by dividing frequency of one category by frequency in another o Express relative size of categories: they tell us exactly how much one category outnumbers the other o Often multiplied by some power of 10 to eliminate decimal points  Rates are another way of summarizing distribution of a single variable o Defined as number of actual occurrences of some phenomenon divided by number of possible occurrences per some unit of time o Usually multiplied by some power of 10 to eliminate decimal points o Often multiplied by 100,00 when number of actual occurrences of some phenomenon is extremely small relative to size of population (i.e. homicides in Canada) 2.3 Frequency Distributions Introduction  Frequency distribution is a table that summarizes distribution of a variable by reporting number of cases contained in each category of the variable  very helpful and commonly used way of organizing and working with data  construction of a frequency table is almost always the first step in any statistical analysis  one general rule that applies to all frequency distributions is that the categories of frequency distribution must be exhaustive and mutually exclusive o categories must be stated in a way that principle applies to construction of frequency distributions for variables measured at all three levels of measurement 2.4 Frequency Distributions for variables measured at nominal and ordinal levels  Nominal Level Variables- for each category of the variable being displayed, the occurrences are counted and the subtotals, along with total number of cases (n)are reported o table has descriptive title, clearly labeled categories and a report of total number of cases at bottom of frequency column  must be included in all tables regardless of variable or level of measurement o when categories are collapsed (i.e. non-medical doctors could include counselor or a psychologist), information and detail will be lost  Ordinal-level Variables- frequency distributions constructed following same routines used for nominal-level variables o Column of percentages by category has been added to table (increase clarify of table and are common adjuncts to basic frequency distribution for variables measured at all levels) 2.5 Frequency distributions for variables measured at the internal-ratio level  In general, construction of frequency distributions for variables measured at interval-ration level is more complex than for nominal and ordinal variables  Interval-ratio variables usually have a large number of possible scores o Large number of scores requires some collapsing or grouping of categories to produce reasonably compact frequency distributions o Must decide how many categories to use and how wide these categories should be  Always involve a trade-off between more detail (greater number of narrow categories) or more compactness (small number f wide categories) Constructing the Frequency Distribution  Categories often called intervals when working with interval-ratio data  Frequency distribution constructed by listing categories in order (i.e. smallest to largest, youngest to oldest), counting the number of times each score occurs and then totaling the number of scores for each category  Midpoints- need to be used when constructing or interpreting certain graphs like the frequency polygon o Midpoints defined as the points exactly halfway between upper and lower limits and can be found for any interval by dividing the sum of the upper and lower limits by two  Real Limits- for certain purposes, you must eliminate the gap between intervals and treat a distribution as a continuous series of categories that border each other o Necessary in constructing some graphs such as the histogram  Stated Limits (intervals of a frequency distribution when stated as discrete categories)- organize scores of the variable into a series of discrete, non-overlapping intervals  To treat the variables as continuous, we must use real limits o To find the real limit of any interval, divide the distance between the stated limits (the gap) in half and add the result to all upper stated limits and subtract it from all lower
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