Download Standard Deviation - Avon Community School Corporation

Avon High School
Section:
12.2
ACE COLLEGE ALGEBRA II - NOTES
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Samples and Surveys
Analyzing Data Shapes
Standard Deviation
Mr. Record: Room ALC-129
Semester 2 - Day 53
Samples and Surveys
When doing a survey, it is usually not practical to get the opinion of every member of a population. You
can get a fairly accurate picture of the opinion of a population by surveying a sample of the population.
A sample is a smaller group than the whole population. There are several ways to choose a sample.
Convenience
Self-selection
Systematic
Random
choosing any people easily available
having people volunteer to participate in the survey
ordering the population and choosing participants at regular intervals (such as chosing
every fifth person from the telephone book)
all memebers of the population have an equal chance of being asked to particpate
The way you choose the sample can introduce bias, or systematic error, into the survey. When a survey is
biased, the results are inaccurate.
Example 1
Analyzing Sampling Methods
A newspaper wants to find out what percent of the city population favors a property tax increase to
raise money for local parks. What is the sampling method used for each situation? Does the sample
have bias? Explain.
a. A newspaper article on the tax increase invites readers to call the paper and express their opinions.
b. A reporter interviews people leaving the city’s largest park.
c. A survery service calls every 50th listing from the local phone book.
You can introduce bias into a survey by simply using poorly written questions. Survey question s should
NOT be
Confusing
by asking about more than one issue or by using double negatives
Ambiguous
by offering answer choices that overlap
Loaded
by using words that might provoke strong reaction in all or some people
Leading
by suggesting that one particular answer is correct
Analyzing Survey Questions
Example 2
Determine if there is any bias in the survery questions below.
a. Do you think farmers should use poison to control insects on crops?
b. Don’t you agree that most childcare workers are overpaid?
c. Do you think teachers should communicate frequently with students and their parents about class grades?
Variation
In the previous section, you studied range and inner quartile range. Each of these is a measure of variation.
A measure of variation describes how the data in a data set are spread out.
Variance and standard deviation are measures showing how much data values deviate from the mean.
The Greek letter  (lower case sigma) represents standard deviation.  2 (sigma squared) is the variance if
the data being analyzed is considered to be a total population. For the purposes of this section, we will
anayze data that is considered only to be a sample of the total population and therefore we will use the letter
s to represent standard deviation and s2 to represent the variance.
Finding Variance and Standard Deviation
Step
1
2
Procedure
Formula
Find the mean of the n values in the dats
set.
We will call the mean, x .
Find the difference between each value
and the mean.
This will give you n values that are each
referred to as x  x .
3
This will give you n values that are each

Square each difference from Step 2.

2
referred to as x  x .
4
This is the variance.
Find the average (mean) of the squares
from Step 3.
5
Take the square root of the variance.
s
2
  x  x

2
n 1
This is the standard deviation.
s
 x  x
 x   

2
n 1
2
and  
n
 represents the average of the data across the entire population.
Note: When anayzing data from a total population, 
2
 x   
n
2
.
Example 4
Finding Variance and Standard Deviation
Consider the following set of data.
6.9
8.7
7.6
4.8
a. Find the mean.
9.0
b. Find the variance.
c. Find the standard deviation.
Finding Variance and Standard Deviation on the TI-Nspire
--------------------------------------------------------------------------------------------------------------------------------Step 1
------------------------------------------------------------Step 3
-------------------------------------------------------------Step 5
Step 2
------------------------------------------------------Step 4
-----------------------------------------------------------
--------------------------------------------------------------------------------------------------------------------------------Use stat.sx for finding standard deviation of a sample size. You could also read this value from the
List and Spreadsheet Page.
Example 5
Using a Calculator to Find Standard Deviation
Number of U.S. Hurricane Strikes by Decade
Decade 1 signifies the 1850’s, Decade 15 signifies the 1990’s.
Decade
1
19
Strikes
2
15
3
20
4
22
5
21
6
18
7
21
8
13
9
19
10
24
11
17
12
14
13
12
14
15
15
14
Source: National Hurricane Center
a. Find the mean of this data set.
b. Find the standard deviation of this data set.
Example 6
Using Standard Deviation to Describe Data
Use the U.S. hurricane-strike data from Example 2. Draw a number line representing this data according to
the plan below.
KNOW
The data
values, their
mean, and
their standard
deviation.
NEED
The number of
standard
deviations
from the mean
that include all
the data
PLAN
- Draw a number line.
- Plot the dats values and the
mean.
- Mark off intervals of s on each
side of the mean.
a. Within how many standard deviations of the mean do all the values fall?
b. How many values fall within one standard deviation of the mean?