Basic Descriptive Statistics Chapter 2

Percentages and Proportions Most used statistics Could 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 percentages Percent change will be on the exam, so review this in your chapter summary

Frequency Tables Frequency ◦ 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 1.00 Percentage ◦ Is the proportion multiplied by 100 ◦ Your author leaves four decimal places for proportions and two for percentages

Usefulness and Limitations Percentages 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 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 = f 1 /f 2 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

Rates Much more often used It 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 Rate The 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 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 Example For crime rates, the percentage is so small, it is often multiplied by 100,000 ◦ Example of 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 Distributions These 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 Variables Need 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 Tables Need a descriptive title Need clearly labeled categories Will report the total number of cases at the bottom of the frequency column May 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 Variables Done the same way as nominal variables Will 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 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 Percentage Their 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 Graphs

Pie 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 Charts Used for nominal and ordinal-level data Used for discrete variables The 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

Histograms Used for continuous variables Used for inter-ratio data The bars are contiguous to each other (meaning they have to touch) Otherwise, it is the same as the bar chart

Frequency Polygons Used for continuous variables Used for interval-ratio data Another 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