basic descriptive statistics chapter 2. percentages and proportions most used statistics could say...
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Basic Descriptive Basic Descriptive StatisticsStatisticsChapter 2
Percentages and Percentages and ProportionsProportionsMost used statisticsCould say that 927 out of 1,516 people
surveyed said that hard work determines who gets ahead in America
Or say that 61% of people feel that hard work is most important
Percentages and proportions give you a frame of reference◦ It standardizes raw data◦ Percentages use a base of 100, and
Proportions use a base of 1.00◦ Need to distinguish among frequencies,
proportions, and percentagesPercent change will be on the exam, so
review this in your chapter summary
Frequency TablesFrequency Tables
Frequency◦ Number of cases (most often people) in
each categoryProportion
◦ f/N◦ F stands for frequency (number of cases
in any category)◦ N stands for number (of cases in all
categories◦ So, the sum of the proportions for any
distribution of cases will be equal to 1.00Percentage
◦ Is the proportion multiplied by 100◦ Your author leaves four decimal places for
proportions and two for percentages
Usefulness and Usefulness and LimitationsLimitationsPercentages and proportions are particularly
useful when you want to compare groups of different sizes on the same variable◦ Like comparing one college to another
Rules on the use of percentages and proportions◦ Don’t use on very small samples or with
small denominators (N of less than 20) Particularly with percentage change from
year to year Example: a small city will have a larger
percentage increase in crime than a large city, but it is still more dangerous to live in a large city
◦ Always report your sample size along with proportions or percentages
Ratios Ratios Useful for comparing categories in terms of
relative frequency—extent that one category outnumbers the other
Will divide the number (frequency) in one category by the number (frequency) in another
Ratio = f1/f2 The number of cases (people) in the first
category divided by the number of cases (people) in the second
Report it: for every case in the denominator, the ratio is the number of cases in the numerator
Example: 1370 Protestants and 930 Catholics◦ The ratio is 1370 divided by 930 = 1.47
(rounded off)◦ So you say, for every Catholic family, there are
1.47 Protestant families
RatesRatesMuch more often usedIt is the number of actual
occurrences divided by the number of possible occurrences per some unit of time
Usually multiplied by 1,000 to eliminate the decimal points
Example of a RateExample of a RateThe crude death rate for a population
◦ The number of deaths in that population (actual occurrences)
◦ Divided by the number of people in the population (possible occurrences) per year
◦ This is then multiplied by 1,000◦ If the crude death rate is 14.29, then for
every 1,000 people, there were 14.29 deaths during this particular year
Rates are most often used to compare crime rates for different cities, or rates for different countries◦ Also used for infant mortality rates and
death rates for different countries
Crime Rate ExampleCrime Rate ExampleFor crime rates, the percentage is so small,
it is often multiplied by 100,000◦ Example of 50.63 auto thefts per 100,000 people
So .05 percent, or .05 per 100 people It seems to be greater as a rate than as a percent
◦ It doesn’t mean much, until you compare it with another country or with the rate for last year
Frequency DistributionsFrequency DistributionsThese are tables that show the
number of people in each category◦All computer programs will construct
these for you◦The categories in the frequency
distribution must be exhaustive and mutually exclusive
Nominal Level VariablesNominal Level VariablesNeed to construct a frequency
distribution table for each variable
The total number in each category is referred to as the Frequency (f)
Rules for All TablesRules for All Tables
Need a descriptive titleNeed clearly labeled categoriesWill report the total number of cases at
the bottom of the frequency columnMay want to “collapse” some categories
◦ Will lose information, so you will always collapse categories only after the survey is done
◦ To collapse is to combine two or more categories together Example, “single” and “divorced” into a
new category of “not married” if your hypothesis only needs to determine if a person is married or not
Ordinal Level VariablesOrdinal Level VariablesDone the same way as nominal
variablesWill want to include a column of
percentages by category to give the reader a better understanding of the results◦Do this for nominal level variables as
well
Interval Level VariablesInterval Level Variables
Will have too many categories to make sense of, so need to collapse or group the data◦ Need to decide how many categories to
use and how wide these categories should be
◦ Will need a balance between detail and conciseness Many categories give more detail Fewer are more concise, and more
easily understood◦ The purpose of the research will
determine how many categories needed
Cumulative Frequency and Cumulative Frequency and Cumulative PercentageCumulative PercentageTheir primary purpose is to allow
the researcher to tell at a glance how many cases fall below a score in the distribution
Cumulative percentage is more important than the cum. Freq.
Charts and GraphsCharts and Graphs
Pie ChartsPie Charts
All charts and graphs are used to present data in a more visually dramatic way
Pie charts◦ Used for nominal and ordinal-level data◦ Used for discrete variables◦ Can only show the frequency distribution
of one variable at a time (usually reported in percentages in each category)
Bar ChartsBar ChartsUsed for nominal and ordinal-level dataUsed for discrete variablesThe categories of the variable are put along
the horizontal axis (or abscissa)The frequencies, or percentages, are put
along the vertical axis (or ordinate)The width of each bar will be equal, and the
height will correspond to the number of people in the category
If a variable has more than 4 or 5 categories, the bar chart is preferred over a pie chart
HistogramsHistogramsUsed for continuous variablesUsed for inter-ratio dataThe bars are contiguous to each
other (meaning they have to touch)
Otherwise, it is the same as the bar chart
Frequency PolygonsFrequency PolygonsUsed for continuous variablesUsed for interval-ratio dataAnother way to look at is is to put
a dot in the middle of the line at the top of a bar in a histogram and connect the dots