Basic Descriptive Statistics

1. Basic Descriptive Statistics Chapter 2

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

3. 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

4. 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

5. 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

6. 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

7. 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 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

8. Crime Rate Example • For 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

9. 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

10. 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)

11. 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

12. 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

13. 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

14. 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.

15. Charts and Graphs

16. 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)

17. 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

18. 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

19. 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