Margin of Error & Confidence Intervals by Formula
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Introduction

    Margin of error is the quantification of inaccuracy that is built in to the data gathering process. It is an important consideration when reporting or deciphering information.

    This lesson will inform you how to calculate margin of error (and confidence intervals) by using formulas. Here are the sections within this lesson page:




    This sections that follow require a basic understanding of statistics. If you are not familiar with margin of error and confidence intervals, please review this lesson before proceeding.

    esson: Margin of Error and Confidence Intervals


    Here are the formulas for a margin of error when calculating a 95% confidence interval.

margin of error 95% formula proportion

    The p-hat is the proportion and again the n-value is the number of data points we have.

margin of error 95% formula sample standard deviation

    The s-value is the sample standard deviation. The n-value is the number of data points we have.

    These formulas can be adapted for different confidence levels (other than 95% confidence intervals). However, doing so is an advanced topic for another time that involves understanding z-values to modify the 1.96 coefficient.


    The best way to understand how to use the formula is by looking at an example.

Example: A certain poll says 35% of registered voters favor Mr. Dimos in the upcoming senate race. If 600 people were polled, determine a 95% confidence interval for this poll result.

    First, we need to identify the values we were given. The 35% is the p-hat (the proportion). The 600 is the n-value.

    Now, we can plug this information into the formula.

margin of error 95% formula proportion example

    We can take this margin of error and we can build our confidence interval, like so.

High End: 35% + 3.82% = 38.82%
Low End: 35% - 3.82% = 31.18%

    So, the confidence interval is…

confidence interval 95%

    ideo: Margin of Error and Confidence Intervals Using a Formula
    uiz: Confidence Intervals Using a Formula


    If given a list of data from a survey, experiment, or some other activity, we will use a different formula for determining the margin of error. We will look at a specific example to see how this formula is used.

Example: Sentences were chosen at random from Michael Medved's book The American Miracle: Divine Providence in the Rise of the Republic. The number of words in the sentences were counted. This is the result.

data words sentences michael medved the American Miracle

    Use the data above to: a) determine the mean and sample standard deviation, b) calculate a 95% margin of error, c) determine the respective confidence interval.

    Whether by hand or by calculator, the sample standard deviation must be determined. The result is 6.496, rounded to the nearest thousandth.

    Now, we can use the formula to calculate the margin of error. We will plug in the sample standard deviation (s), which is 6.496. The n-value is the number of data points and in this case it is 24 (see the 24 data points in the table).

    Here is the information entered into the formula.

margin of error 95% formula sample standard deviation

    The mean of the data, which was collected by calculator, is 12.75. Now, we can determine the confidence interval.

High End: 12.75 + 2.599 = 15.349
Low End: 12.75 – 2.599 = 10.151

    So, the confidence interval for this data is…

confidence interval

    uiz: Mean, Standard Deviation, and Margin of Error


    Try this video, which can help you learn the content above.

    ideo: Margin of Error and Confidence Intervals Using a Formula


    Try this quiz, which can help you learn the content above.

    uiz: Confidence Intervals Using a Formula
    uiz: Mean, Standard Deviation, and Margin of Error


    Try these lessons and the quizmaster, which are related to the sections above.

    esson: Margin of Error & Confidence Intervals
    esson: Frequency Tables
    esson: Standard Deviation
    esson: Bell Curve: 68-95-99 Rule

    uiz: Dice Roller: Mean, Standard Deviation