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Confidence Interval of a Survey

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.

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Confidence Interval of a Survey -

Tag(s) : Statistics

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# Confidence Interval of a Survey

## Survey Confidence Interval Calculator

 Confidence Level 95% (5% error) 99% (1% error)

### What is a confidence interval? (Definition)

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.

### How to calculate a confidence interval?

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%.

### How to reduce a confidence interval?

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

### What is the rule of three?

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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