Mathematics in the Modern World

1. Data
Management
Statistics

2. Statistics
• Statistics involves the collection, organization, summarization, presentation, and
interpretation of data.

3. PRESENTATION OF DATA
• There are many ways to depict a large quantity of data. One way is to list out all
the information. For example, a theatre that recorded the age of all attendees
might come up with the following: 27, 27, 22, 47, 54, 28, 32, 38, 44, 50, 14, 68,
63, 66, 34, 36, 38, 27, 58, 35, 30, 24, 19, 33, 35, 44, and 45.

4. One way to simplify this data would be to group the data into ranges and record
the number in each group with a frequency chart. For example, the information
above could be put into age ranges that spanned ten years each, as shown in Table
10-1.

5. Another way to depict this data would be with a histogram: a sequence of bars
proportional in height to the frequency of each range, as shown in Fig. 10-1.

6. Another way would be to calculate the percentage in each range. For example,
there were 27 total people who attended the theatre, so the eight in the 31–40
range make up
8
27
≈ 30 % of the whole. The percentage chart for the ongoing
example is in Table 10-2.

7. A pie chart depicts these percentages with proportional sectors (wedges) of a
circle. For example, the 26% of the people in the 21–30 age range will be
represented by a sector with an angle that measures 360° × 0.26 = 93.6°. The pie
chart for the ongoing example is in Fig. 10-2.

8. The Measures of Central Tendency
The Arithmetic Mean
• One of the most basic statistical concepts involves finding measures of central tendency of a set
of numerical data. Here is a scenario in which it would be helpful to find numerical values that
locate, in some sense, the center of a set of data. Elle is a senior at a university. In a few months
she plans to graduate and start a career as a landscape architect. A survey of five landscape
architects from last year’s senior class shows that they received job offers with the following
yearly salaries.
$43,750 $39,500 $38,000 $41,250 $44,000
Before Elle interviews for a job, she wishes to determine an average of these 5 salaries. This average should be a
“central” number around which the salaries cluster. We will consider three types of averages, known as the
arithmetic mean, the median, and the mode. Each of these averages is a measure of central tendency for
numerical data.

9. The arithmetic mean is the most commonly used measure of central
tendency. The arithmetic mean of a set of numbers is often referred to
as simply the mean. To find the mean for a set of data, find the sum of
the data values and divide by the number of data values. For instance,
to find the mean of the 5 salaries listed above, Elle would divide the
sum of the salaries by 5.
The mean suggests that Elle can reasonably expect a job offer at a salary of about $41,300.

10. In statistics it is often necessary to find the sum of a set of numbers. The traditional
symbol used to indicate a summation is the Greek letter sigma,∑ . Thus the
notation ∑x , called summation notation, denotes the sum of all the numbers in a
given set. The use of summation notation enables us to define the mean as follows.
Statisticians often collect data from small portions of a large group in order to determine information about the
group. In such situations the entire group under consideration is known as the population, and any subset of
the population is called a sample. It is traditional to denote the mean of a sample by ( which is read as “x
bar”) and to denote the mean of a population by the Greek letter µ (lowercase mu)

11. Example 1:

12. The Median
Another type of average is the median. Essentially, the median is the middle
number or the mean of the two middle numbers in a list of numbers that have been
arranged in numerical order from smallest to largest or from largest to smallest.
Any list of numbers that is arranged in numerical order from smallest to largest or
from largest to smallest is a ranked list.

13. Example 2

14. The mode
A third type of average is the mode.
Some lists of numbers do not have a mode. For instance, in the list 1, 6, 8, 10, 32, 15, 49, each number occurs
exactly once. Because no number occurs more often than the other numbers, there is no mode.

15. A list of numerical data can have more than one mode. For instance, in the list 4, 2,
6, 2, 7, 9, 2, 4, 9, 8, 9, 7, the number 2 occurs three times and the number 9 occurs
three times. Each of the other numbers occurs less than three times. Thus 2 and 9
are both modes for the data.
Example 3:

16. The Weighted Mean
A value called the weighted mean is often used when some data values are more important than
others. For instance, many professors determine a student’s course grade from the student’s tests
and the final examination. Consider the situation in which a professor counts the final examination
score as 2 test scores. To find the weighted mean of the student’s scores, the professor first assigns
a weight to each score. In this case the professor could assign each of the test scores a weight of 1
and the final exam score a weight of 2. A student with test scores of 65, 70, and 75 and a final
examination score of 90 has a weighted mean of

17. Note that the numerator of the weighted mean on the previous page is the sum of
the products of each test score and its corresponding weight. The number 5 in the
denominator is the sum of all the weights (1 + 1 + 1 + 2 = 5). The procedure on the
previous page can be generalized as follows.

18. Many colleges use the 4-point grading system:
A student’s grade point average (GPA) is calculated as a weighted mean, where the student’s grade in each
course is given a weight equal to the number of units (or credits) that course is worth. Use this 4-point grading
system for Example 4