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Statistics
Statistics is the branch of
mathematics that deals with
collecting, organizing,
analyzing, and reporting
quantitative information.
Data
Recorded observations that
can be measured or counted
Population
A population is an entire set
of objects sharing similar
characteristics, such as
human beings, automobiles,
or measurements, from which
data can be collected and
analyzed.
Sample
A sample is a portion of a
population from which data is
collected to estimate the
characteristics of the entire
population.
Statistic
A statistic is a measure
calculated from a sample of
data.
Parameter
A parameter is a measure
calculated from data for an
entire population.
Example 1
Identify as a sample or a
population: the snacks
dispensed from a vending
machine during its existence.
population
All snacks are being
considered.
Example 1
Identify as a sample or a
population: 100 freshman
from among those enrolled
at a local college for the fall
of 2010.
sample
A small number of students
is being considered.
Example 1
Identify as a sample or a
population: the automobiles
built by General Motors in
the 1990s.
population
All of the automobiles
produced by the company
are being considered.
Example
Identify as a sample or a
population: a school official
gathers data about all
students at Union University
when studying the majors
offered at the university.
population
Example
Identify as a sample or a
population: a magazine
wishes to determine the
political viewpoint of college
students, so it sends a
questionnaire to all students
at Union University.
sample
Four Types of
Population
Sampling
1. Random—A random
number generator is used
to determine page number,
column, and row of each
person in the sample.
2. Systematic—Every
thirtieth person listed in
the phone book is
included in the sample.
3. Convenience—A
questionnaire with a return
envelope is mailed out
with the phone bill.
4. Cluster—Everyone whose
address indicates that he
lives in a particular section
of town is included in the
sample.
Example 2
Identify the sample as
random, systematic,
convenience, or cluster:
math teachers who attend a
workshop at the regional
conference.
convenience
Example 2
Identify the sample as
random, systematic,
convenience, or cluster:
every fifth person on the
class roster.
systematic
Example 2
Identify the sample as random,
systematic, convenience, or
cluster: 500 voters, based on
phone numbers chosen by
computer-generated selection
of page number and column in
the phone book.
random
Example
Identify the sample as
random, systematic,
convenience, or cluster: A
worker selects the first
phone number from every
page of the phone book for a
phone survey.
systematic
Example
Identify the sample as random,
systematic, convenience, or
cluster: To evaluate rates for
the countywide school system,
the school board randomly
selects two high schools and
evaluates the data at those two
schools. cluster
Range
The range is the difference
between the largest and
smallest numbers in a set of
data.
Mean
The mean is the arithmetic
average of a set of numbers
(sum of data divided by the
number of data).
Median
The median is the middle
number in a set of data
arranged in numerical order.
If there is an even number of
data, the median is the
average of the two middle
numbers.
Mode
The mode is the number or
numbers that occur most
frequently in a set of data. If no
number occurs more than once,
there is no mode. Data sets in
which two values occur most
frequently are called bimodal,
while other data sets may have
no mode.
Mean
Average
Median
Middle
Mode
Most
Example 3
Find the range, mean,
median, and mode of the
following set of data:
{6, 9, 7, 3, 6, 7, 10, 5}.
10, 9, 7, 7, 6, 6, 5, 3
range: 10 – 3 = 7
mean: sum of data = 53
average = 53 ÷ 8 ≈ 6.6
Find the range, mean,
median, and mode of the
following set of data:
{6, 9, 7, 3, 6, 7, 10, 5}.
median: 10, 9, 7, 7
7, 6
6, 6, 5, 3
7 + 6 = 13 = 6.5
2
2
mode: 6 and 7
Example 4
Find the range, mean,
median, and mode of the
following set of data: {58, 38,
60, 44, 45, 42, 49, 50, 41}.
38, 41, 42, 44, 45, 49,
50, 58, 60
range: 60 – 38 = 22
Find the range, mean,
median, and mode of the
following set of data: {58, 38,
60, 44, 45, 42, 49, 50, 41}.
mean: sum of data = 427
average = 427 ÷ 9 ≈ 47.4
median: 38, 41, 42, 44, 45
45, 49,
50, 58, 60
mode: none
Example 5
The following are daily low
temperatures from the
preceding week. Find the
range, mean, median, and
mode of the temperatures.
Day
Temperature (°F)
Monday
25°
Tuesday
21°
Wednesday
20°
Thursday
29°
Friday
25°
Saturday
22°
Sunday
19°
29, 25, 25, 22, 21, 20, 19
range: 29 – 19 = 10°
mean: sum of data = 161
average = 161 ÷ 7 = 23°
median: 29, 25, 25, 22
22, 21,
20, 19
mode: 25°
Example
Use this data to solve the
following problems:
A = {13, 25, 22, 18, 17, 17, 14,
16, 22}
B = {19, 22, 35, 39, 35, 37, 40,
100}
Example
Arrange the data for each set
in ascending order.
A = {13, 14, 16, 17, 17, 18,
22, 22, 25}
B = {19, 22, 35, 35, 37, 39,
40, 100}
Example
Find the range of each set.
A = {13, 14, 16, 17, 17, 18,
22, 22, 25}
B = {19, 22, 35, 35, 37, 39,
40, 100}
A: 12; B: 81
Example
Find the median of each set.
A = {13, 14, 16, 17, 17, 18,
22, 22, 25}
B = {19, 22, 35, 35, 37, 39,
40, 100}
A: 17; B: 36
Example
Find the mode of each set.
A = {13, 14, 16, 17, 17, 18,
22, 22, 25}
B = {19, 22, 35, 35, 37, 39,
40, 100}
A: 17 and 22; B: 35
Example
Find the mean of each set.
A = {13, 14, 16, 17, 17, 18,
22, 22, 25}
B = {19, 22, 35, 35, 37, 39,
40, 100}
A: 18.22; B: 40.875
Example
Would changing a single
value in A always change the
mean? Is this true for any set
of data?
A = {13, 14, 16, 17, 17, 18,
22, 22, 25}
yes; yes
Example
Would changing a single
value in A always change the
median?
A = {13, 14, 16, 17, 17, 18,
22, 22, 25}
no
Example
If C = {12, 13, 14}, does C
have three modes or no
modes?
no modes