Tool for measuring the upper and lower bounds of the confidence interval attributable to a survey. The 95% or 99% confidence interval makes it possible to better qualify the quality of a survey.

Confidence Interval of a Survey - dCode

Tag(s) : Statistics

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Definition: A **confidence interval** is a lower and upper bound that defines a margin of error for the raw results of a survey. The **confidence interval** evaluates the quality and accuracy of the estimate obtained with the sample surveyed.

This interval applies to all types of survey / sampling (street interviews, online surveys) in order to evaluate the trust score that can be attributed to it.

The frequency observed in a sample is noted $ f $ and the probability in the total population is noted $ p $, these values are often confused.

For a survey of $ N $ people resulting in the frequency $ f $ and the probability $ p $, then the 95% **confidence interval** is $$ \left[p-1.96\frac{\sqrt{f(1-p)}}{\sqrt n},p+1.96\frac{\sqrt{p(1-p)}}{\sqrt n}\right] $$

With 1.96 the value of the 2.5 percentile of the normal distribution (for 99%, the value will be 2.58).

__Example:__ For a poll with a sample of 80 people of whom 60 declare voting YES, the frequency is $ f = 60/80 $ and the probability $ p = 0.75 $, the **confidence interval** is $ \left[0.75-1.96\frac{\sqrt{0.75(1-0.75)}}{\sqrt 80},0.75+1.96\frac{\sqrt{0.75(1-0.75)}}{\sqrt 80}\right] = \left[ 0.655, 0.845 \right] $. This means that there is 95% chance in the final vote the YES result is between 65.5% and 84.5%.

In order to reduce/improve a **confidence interval**, it is necessary to increase $ N $ (the number of items or people participating in the survey).

When the probability is close to 0, the calculation of the **confidence interval** can lead to probabilities outside the interval $ [0,1] $ which is impossible. One rule is to use the limit as $ 3 / N $.

__Example:__ A poll of $ N = 100 $ people gives a probability of 0, then the **confidence interval** is $ [0, 0.03] $, a percentage between 0 and 3%.

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