Statistics and Probability periodicals and newspapers or reports from industries government agencies and research bureaus 2 by setting up a survey and col lecting data thirsthand from individuals or organizations and 3 by conducting scientific experiments and measuring or counting under controlled conditions Basic information must be collected in such a way that it is accurate representative and as comprehensive as possible Statistical treatment cannot in any way improve the basic validity or accuracy of the raw data Methods of collecting data are therefore basic to the whole field of statistics POPULATIONS The term population ordinarily means the whole number of people in a specific place The stat istician speaks of a population of automobiles salaries accidents ballots blood pressures or any other characteristic For the statistician a population also called a universe means the entire group of items in the class being considered Usually it is not possible to gather observa tions from all the possible cases in a population Some populations are infinite For example 12
STATISTICAL DATA THE LAW OF LARGE NUMBERS In statistics the law of large numbers is the theorem that as the number of identically distributed randomly generated variables increases their sample mean aver age approaches their theoretical mean The theorem was first proved by the Swiss mathema tician Jakob Bernoulli in 1713 He and his contemporaries were developing a formal probability theory with a view toward analyzing games of chance Bernoulli envis aged an endless sequence of repetitions of a game of pure chance with only two outcomes a win or a loss Labeling the probability of a win p Bernoulli considered the fraction of times that such a game would be won in a large number of repetitions It was commonly believed that this fraction should eventually be close to p This is what Bernoulli proved in a precise manner by showing that as the number of repetitions increases indefinitely the probability of this fraction being within any pre specified distance from p approaches 1 if you should want to count how many times heads will turn up when a penny is tossed you would have to set a limit to the number of throws because even a billion would not exhaust the infinite universe of possible trials 13
d REDERER DRER Here an employee of the U S Census Bureau visits a household to collect information The answers to census questions are used to measure statistically the many changes in American life U S Census Bureau 14
STATISTICAL DATA Other populations though finite are so large that it would take too long or cost too much to collect data on each unit in them Every ten years the United States government conducts a census of the entire population of the United States but this is a gigantic and costly undertaking SAMPLES The statistician usually gets information from a relatively small number of cases called a sam ple From the measurements or observations of the individuals sampled generalizations can be made about the population from which the sample was selected The individuals in a sample must be repre sentative of the larger population otherwise the conclusions drawn from the sample would not be valid for the larger population For example one cannot draw valid con clusions about the probable outcome of a national election from interviewing a sample of 10 000 registered voters unless they are representative of the people who will actu ally vote in the election The size of the sample is also a factor Other things being equal a larger sample is better than a smaller one However excellent results 15
