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### Margin of Error: Who Figures into It?

```Date: 11/09/2011 at 17:10:38
From: Aoi
Subject: How to get Margin of Error in a survey

How would I calculate a margin of error for a survey I made?

The total population size is 45,000.

The sample size is 100.

The question asked has 3 possible options: Yes, No, Abstain.

I'm not too familiar with this type of math, and I'm just asking because
I'm curious. I do not know where to start.

Any example showing how to arrive at the answer would be fine.

Thank you.

```

```
Date: 11/10/2011 at 04:53:39
From: Aoi
Subject: How to get Margin of Error in a survey

I did a bit of research on how to find margin of errors on my own. Using a
different set of values, I think I might have a start on it, maybe?

Here's the example I took as a template:

Sample Size (s): 42
Population (P): 25,318
Proportion (p'): 77%
Confidence Level (CL): 95%

The math would be:

Standard Deviation of Proportion (SD) = sqrt((p' * (1 - p')) / s)
Standard Error (SE) = s/sqrt(SD)
CL = 95%

So the confidence interval (CI) should be ~1.96.

Margin of Error (MoE) = SE * CI

And this should be around 1.95%, I think.

Is this correct?

Am I supposed to use the Population (P) somewhere in this?

I'm seeking help if I'm even on the right path to understanding this.

Thank you.

```

```
Date: 11/10/2011 at 22:35:04
From: Doctor Wilko
Subject: Re: How to get Margin of Error in a survey

Hi Aoi,

Thanks for writing to Ask Dr. Math!

Your question is ultimately concerned with statistical error. There's
always going to be some error in an estimate, but the question is how much
is acceptable?

Basically, the more people you survey, the less statistical error in your
results. If you asked five people your question, you can't be very certain
what 45,000 people really think. If by contrast you ask 10,000 people, you
can probably be pretty certain of what the 45,000 think. The trick is to
find the "sweet spot," i.e., how many people to survey to get a reasonable
and accurate estimate of the true population parameter you're interested
in.

Now, I can't answer your question directly yet, but once you get a result
from the survey, then you can talk about the margin of error (and
confidence interval) of the result you're interested in. Keep reading; I
should be able to clear this up!

The point of taking a survey is to estimate some value of the whole
population, right? For instance, you're wondering how many out of 45,000
people would, for example, choose "yes."

Let's say you already conducted your survey, and you found that 50 people
in your sample of 100 respondents answered "yes." The logical next
question is about the accuracy of your survey, i.e.,

In a survey of 100 people, which included 50 "yes" responses,
what's the 95% Confidence Interval (CI) of the true proportion
of people who would say "yes"?

A CI is your sample proportion, i.e., your estimate of 50% "yes"
responses, plus and minus the margin of error of your estimate.

To construct the CI, you first need the margin of error, denoted by E:

E = z * sqrt((p'*(1 - p')/n)

In this example, z = 1.96, since the Central Limit theorem tells us that
for a large sample, about 95% of the sample means will fall within 1.96
standard errors of the population mean. Since we want a 95% CI, we'll use
z = 1.96 in this example. (For a 99% CI, z = 2.58; there are tables that
give you different values depending on what CI you want to calculate.)

The only other values you need are p', the proportion of "yes" responses
from the survey; and n, the sample size.

Plugging in p' and n, the error of the estimate is:

E = 1.96 * sqrt((.50*.50)/100)
= 1.96 * 0.05
= 0.098 (or 9.8% accurate in either direction)

We can use the margin of error to get a CI:

CI around p' = 0.50 (+/-) 0.098, or

0.402 < p' < 0.598

Even though you got a point estimate of 50% "yes" responses, you can be
95% confident that the true population "yes" response is between about 40%
and 60%, or about 10% in either direction. With only 100 surveys, that's
the best guess you can make at this point!

That's actually a pretty wide CI. The way you can tighten up the accuracy
of a survey is to administer more surveys. For example, if you gave the
same survey to 1000 people and got 500 "yes" responses (still 50% "yes"),
now your CI would be

0.469 < p' < 0.530

In this case, the proportion of the population that would choose "yes" is
likely between 47% to 53%. That makes for a margin of error closer to 3%
-- a much tighter estimate!

There is actually a sample size formula that tells you how many surveys to
administer to be within some Margin of Error. For instance, if you want to
be within 2% of the true proportion of the population, then you'd need to
administer ... 2,401 surveys! I'm not going into the details here, but
it's basically using the equation above and solving for n, the sample
size.

As you noticed, the population size doesn't matter. In your case, it
wouldn't matter if the population had 4,500 people or 45,000 people!
That's why for a Presidential election, you'll often see polls where they
survey only a few thousand voters and keep a margin of error to within
about 2-3%. You don't have to survey millions of people to get a good
estimate of the population's voting preference (assuming you took a good
random sample).

Does this help?

Please write back if you need anything else.  :-)

- Doctor Wilko, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Statistics

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