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 TablesFrequency

◦ Number of cases (most often people) in each category

Proportion◦ 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 TablesNeed 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 VariablesWill 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 ChartsAll charts and graphs are used to present

data in a more visually dramatic wayPie 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