Margin of Error
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Often used in surveys or sampling of a population, a margin of error asserts the
likelihood that the sample will represent the entire population.
It is the “give or take” numbers seen in a set of data.
The larger the amount of people used in the survey, the smaller the margin of error.
1 divided by the square root of the number of people in the sample
Equation:
M = 1÷√n
Example:
 Out of the student body, a survey is taken by 600 students.
The margin of error is calculated with the equation: M = 1÷√n
 The margin of error would be +/- 4%
Another version of Margin of Error
Definition: In a confidence interval, the range of values above and below the sample statistic is called
the margin of error. In order to compute the margin of error, it must be represented as: Margin of error
= Critical value x Standard deviation of the statistic or Margin of error = Critical value x Standard error
of the statistic. If you know the standard deviation of the statistic, use the first equation to compute the
margin of error. Otherwise, use the second equation.
Example: Nine hundred (900) high school freshmen were randomly selected for a national survey.
Among survey participants, the mean grade-point average (GPA) was 2.7, and the standard deviation
was 0.4. What is the margin of error, assuming a 95% confidence level?
Solution: The correct answer is (B). To compute the margin of error, we need to find the critical value
and the standard error of the mean. To find the critical value, we take the following steps.
 Compute alpha (α): α = 1 - (confidence level / 100) = 1 - 0.95 = 0.05
 Find the critical probability (p*): p* = 1 - α/2 = 1 - 0.05/2 = 0.975
 Find the critical z score. Since the sample size is large, the sampling distribution will be roughly
normal in shape. Therefore, we can express the critical value as a z score. For this problem, it
will be the z score having a cumulative probability equal to 0.975. Using the Normal Distribution
Calculator, we find that the critical value is 1.96.
Next, we find the standard error of the mean, using the following equation:
SEx = s / sqrt( n ) = 0.4 / sqrt( 900 ) = 0.4 / 30 = 0.013
And finally, we compute the margin of error (ME).
ME = Critical value x Standard error = 1.96 * 0.013 = 0.025
This means we can be 95% confident that the mean grade point average in the population is 2.7 plus or
minus 0.025, since the margin of error is 0.025.
Use the confidence interval to find the margin of error and the sample mean:
(0.256,0.380)
The margin of error is equal to the radius of the confidence interval or half the width of the confidence
interval.
Margin of Error:
ME = (b-a)/2, where the confidence interval is (a,b)
ME = (0.380-0.256)/2
ME = 0.062
So the margin of error is 0.062
Stratified Sampling
Stratum: A subgroup, class, or level formed on the basis of common attributes or
characteristics between variables.
Stratified Sampling: A probability sampling technique that has data separated into different
strata, then the final subjects are randomly selected from each of the strata.
Proportionate:
● The sample size (amount of final subjects) is determined by a sampling fraction that is
consistent across all stratums, which makes the stratum size proportionate to the total
population size.
Disproportionate:
● The sample size (amount of final subjects) is determined by a sampling fraction that
may vary from one stratum to the next.
Example: At PHS there are 20 students in grade 9, 30 students in grade 10, 26 students in
grade 11, and 24 students in grade 12, for a total of 100 students. The data would be split up
by grades. From there the researcher would select a set number of students from each class
to represent the final population. If you wanted to take a sample of 50 kids, 20% would be from
grade 9, 30% from grade 10, 26% from grade 11 and 24% from grade 12. That would be 10
from grade 9, 15 from grade 10, 13 from grade 11, 12 from in grade 12.
SYSTEMATIC SAMPLING
Definition: A method or technique of sampling that involves collecting data and research from
a larger group by selecting a starting point, then an interval/integer is chosen to pick
information equally from the starting point onward within the sampling population until the ideal
sample size is achieved. Creates a pattern that is predetermined creating a system to interpret
the data.
Examples:
-Every 10th person at a supermarket to observe buying habits.
-Starting point 3, equal interval of 7, 10 individual members ( sample size ), population 100
3, 10, 17, 24, 31, 38, 45, 52, 59, 66
Open Question:
a question that cannot be answered with a yes or no but requires a
developed answer (What is your best quality?)
Closed Question: a question that can be answered in only one word or very short phrase
(Are you feeling better today?
Leading Question: a question phrased in a way that suggests the answer (How fast was the
red car going when it hit the blue car?)
Loaded Question: a question containing a hidden trap or implication (Have you stopped
cheating on tests?)
SRS & VRS
Simple Random Samples:
A simple random sample requires that: all sections and combinations of
selections have an equal chance.
A random sample may not end up being representative of the population, but
any deviations are due only to chance.
Example: In a medical study, the population might be all adults over the age of
50 who have high blood pressure.
Example: In another study, the population might be all hospitals in the U.S. that
perform heart bypass surgery.
Voluntary Response Samples:
 A voluntary response sample is made up of people who self-select into the
survey.
 Example: A news show asks viewers to participate in an online poll. The sample is
chosen by the viewers.
 Example: Determining an outcome based on the votes of the people; who wore
it best at the red carpet at the Oscars?
o Dakota Johnson
o Jennifer Lopez
o Lady Gaga
o Anna Kendrick
o Felicity Jones
Longitudinal Study & Cross-Sectional Study
Longitudinal Study:
Definition:
-Researchers conduct many observations on the same subject over a period of time while
not interfering with their subjects. (Observational Study)
-Likely to suggest a cause-and-effect relationship
Example:
-Cholesterol levels among women over 40 years old who walk daily for 20 years.
-The longitudinal aspects are viewing the cholesterol level over an extended amount of
time due to the women walking
Cross-Sectional Study:
Definition:
-Researchers record information on their subjects without manipulating the environment.
(Observational Study) This is found in a single moment in time.
Example:
-Measure the cholesterol levels of subjects who are daily walkers’ and subjects who are
non-walkers without influencing or interfering with the subjects in a single moment in
time
Ex. Researchers are trying to determine if the
quality of the weld is good enough to withstand
extreme pressure or to verify the properties of a
material. The samples are put under different
loads and stress. That way the researcher can
analyze in which point the material cracks under
pressure.
The three the
street
names
represent
the individual
clusters.
Population
Cluster sampling refers to the “natural” method in which the
population is divided into homogenous groupings or in other
words known as clusters. Later on, random sampling would be
selected from within the cluster that has been previously
chosen.
Destructive sampling determines the service life
and weakness from testing the product or
component by operating the most severe
conditions until it fails.
Simple or Systematic
Random sampling is
taken place to choose
one of the clusters to sample