Beginning Statistics with Data Analysis

Frederick Mosteller, Stephen E. Fienberg, Robert E.K. Rourke

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486782256
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

This introduction to the world of statistics draws examples from a broad range of applications, including business, government, medicine, social sciences, biology, and everyday life. The first four chapters present a brief course in exploratory data analysis. The authors place special emphasis on methods for collecting data and offer an elementary treatment of multiple regression. A concluding chapter on the design of comparative experiments links the basic ideas of design to previously discussed methods of analysis. Each chapter and section concludes with review problems. Suitable for undergraduate students, the text requires only one year of high school algebra.
This authoritative treatment was co-written by Frederick Mosteller, one of the twentieth century's preeminent statisticians and the founding chairman of Harvard's Department of Statistics, and Stephen Fienberg, a leader of statistics departments at the University of Minnesota and Carnegie Mellon University and who is internationally known for innovative statistical research and its application to social science and public policy.

Frequency Distributions	1

Learning Objectives

1. Reaping the benefits from organizing data and displaying them in histogram form

2. Gaining skill in interpreting frequency histograms

3. Constructing a cumulative frequency diagram

4. Finding quartiles from a cumulative graph

1-1 DATA: AN AID TO ACTION

Much human progress grows from the practice of keeping and analyzing records. Important examples are the records leading to the calendar, an appreciation of the seasons, the credit, banking, and insurance systems, much of modern production processes, and our health and medical systems. We shall, therefore, begin by explaining how to organize, display, and interpret data.

What does this book do? This text primarily equips the reader with the skills

1. to analyze and display a set of data,

2. to interpret data provided by others,

3. to gather data,

4. to relate variables and make estimates and predictions.

ORGANIZING DATA: THE USE OF PICTURES

Data often come to us as a set of measurements or observations along with the number of times each measurement or observation occurs. Such an array is called a frequency distribution.

To display a frequency distribution and disclose its information effectively, we often use a type of diagram called a frequency histogram. Let us look at some histograms. Examples 1 and 2 suggest some benefits we get from organizing data and displaying them in histogram form.

EXAMPLE 1 Allocation of police. The chief of police of New York City has enlisted as many men as his budget permits. He has divided his forces about equally to cover three daily shifts; the first shift runs from midnight until 8 A.M., the second from 8 A.M. until 4 P.M., and the third from 4 P.M. until midnight. This system is bringing many complaints, and it is clear that during certain hours of the day the police calls require more police than are available. The chief selects a certain Sunday in August as a guide to action and makes a histogram (Fig. 1-1) showing the frequency distribution of emergency police calls, hour by hour, in New York City during this Sunday. Using this Sunday in August as a guide for late summer Sundays, what action, short of hiring additional police, does the histogram suggest?

Discussion. The histogram vividly exhibits the changes in the numbers of police calls from hour to hour during the 24-hour period. The day’s calls appear to divide into three types of periods:

Figure 1-1 Frequency distribution of New York City police calls during a Sunday in August.

Source: Adapted from R. C. Larson (1972). Improving the effectiveness of New York City’s 911. In Analysis of Public Systems, edited by A. W. Drake, R. L. Keeney, and P. M. Morse, p. 161. Cambridge, Mass.: M.I.T. Press.

1. A connected interval of peak demand, about 700 or more calls per hour from 0 to 4 hours (midnight to 4 A.M.), and another interval from 15 to 24 hours (3 P.M. to midnight); these two intervals join to become one period as we run from one day to the next starting at 15 hours and running through to 4 hours the next day (3 P.M. to 4 A.M.).

2. A period of medium demand, over 400 calls to less than 700 calls per hour from 11 A.M. to 3 P.M.

3. A period of low demand, about 400 or fewer calls per hour from 4 A.M. to 11 A.M.

A study of such histograms over a period of time gives important information about the number of police required for emergency duty during a given shift. Figure 1-1 shows that there are over twice as many calls during the 7-hour period between 6 P.M. and 1 A.M. as during the 7-hour period between 4 A.M. and 11 A.M.

If we assume that the chief does not want to change the shift times, then he might reallocate his force so that the numbers on the shifts are more nearly proportional to the numbers of calls received during the shifts. A practical nonstatistical problem then arises. How can arrangements be made with the union to reschedule the shifts? In the actual event, the rescheduling was worked out. Note that the guide to action came from looking at the whole distribution of calls, not just the average number of calls per hour.

EXAMPLE 2 A gap in the histogram. The quality control expert W. E. Deming reported that part of a manufacturing process involves making steel rods. These rods have a lower specification limit (LSL) of 1.000 cm on their diameters; rods smaller than 1.000 cm are too loose in their bearings. Such rods are rejected, or thrown out, which implies losses of labor and material and of the overhead expenses incurred up to this point in the manufacturing process. As an aid to action, the quality-control engineer constructs Fig. 1-2, a histogram displaying the inspectors’ m...