Margin of Error
A survey of a sample population gathers information from a few people and then the results are used to
reflect the opinions of a larger population. The reason that researchers and pollsters use sample
population is that it is cheaper and easier to poll a few people rather than everybody. One key to
successful surveys of sample populations is finding the appropriate size for the sample that will give
accurate results without spending too much time or money.
Suppose that 900 American teens were surveyed about their favorite ski category of the 2002 Winter
Olympics in Park City, Utah. Ski jumping was the favorite for 20% of those surveyed.
This result can be used to predict how many of all 31 million American teens favor ski jumping.
How?
To determine how accurately the results of surveying 900 American teens truly reflect the results of
surveying all 31 million American teens, a margin of error should be given. When pollsters report the
margin or error for their surveys, they are stating their confidence mathematically in the data they have
1
collected. The margin of error can be calculated by using the formula , where n is the number in a
√𝑛
sample size.
1
1
For the above sample, the margin of error would be
=
= 0.03, 𝑜𝑟 3%,. Since the actual statistic
30
√900
could be larger or smaller than the true amount, the margin of error can be expressed as ± 3%.
1. Find the margin of error for a survey of 100 American teens.
2. Compare that margin of error to the margin of error of 900.
3. Find the margin of error for a survey of 9,000 teens.
4. Find the margin of error for a survey of 90,000 teens.
5. Draw a conclusion about the margin of error based on the size of the sample. Why do you think this
is so?
6. If you want to cut your margin of error in half, what would you have to do to the sample size?
Why?
7. What if you wanted your margin of error to be 5%? How large would your sample need to be?
8. Find the sample size needed to create a margin of error of 2%.
Answers:
1. 1⁄
= ± 0.10 or ± 10%
√100
2. 1⁄
= ± 0.033 or ± 3.3%
√900
3. 1⁄
= ± 0.0105 or ± 1.05%
√9000
4. 1⁄
= ± 0.0033 or ± 0.33%
√90000
5. As the sample size increases, the margin of error decreases. You need a smaller margin of error because a
larger sample makes you more sure of your results.
6. To cut the margin of error in half, you need to multiply the sample size by 4. The sample size is under the
square root, so it will need to be the square of whatever you want to do to the margin of error.
7. 0.05 = 1⁄ ; n = 400
√𝑛
8. 0.02 = 1⁄ ; n = 2500
√𝑛