Skip to main content
Mathematics LibreTexts

6.3: The Central Limit Theorem for Sample Proportions

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    If \(X\) is a binomial random variable, then

    \[X \sim B(n, p)\nonumber \]

    where \(n\) is the number of trials and \(p\) is the probability of a success.

    To form a proportion, take \(X\), the random variable for the number of successes and divide it by \(n\), the number of trials (or the sample size). The random variable \(\hat{P}\) (read "P hat") is that proportion,

    \[\hat{P} = \dfrac{X}{n}\nonumber \]

    When \(n\) is large and \(p\) is not close to zero or one, we can use the normal distribution to approximate the binomial.

    \[X \sim N(np, \sqrt{npq})\nonumber \]

    If we divide the random variable, the mean, and the standard deviation by \(n\), we get a normal distribution of proportions with \(\hat{P}\), called the estimated proportion, as the random variable. (Recall that a proportion as the number of successes divided by \(n\).)

    \[\dfrac{X}{n} = \hat{P} \sim N\left(\dfrac{np}{n}, \dfrac{\sqrt{npq}}{n}\right)\nonumber \]

    Using algebra to simplify:

    \[\dfrac{\sqrt{npq}}{n} = \sqrt{\dfrac{pq}{n}}\nonumber \]

    \(\hat{P}\) follows a normal distribution for proportions:

    \[\dfrac{X}{n} = \hat{P}\ \sim N\left(p, \sqrt{\dfrac{pq}{n}}\right)\nonumber \]


    The Central Limit Theorem for Proportions:


    1. Sample is random with independent observations.

    2. Sample is large:  The sample size, n, is considered large enough when the sample expects at least 10 successes (yes) and 10 failures (no); i.e. 

     \(n\cdot\ p\)≥10 and \(n\cdot(1-p)\)≥10

    3. Big population:  If sampling is done without replacement, the population must be at least 10 times larger than the sample size.

    then \(\hat{P}\) follows a normal distribution with

    1. The mean of sampling distribution of sample proportions \(\mu_{\hat{P}}\) is the population proportion, p, \(\mu_{\hat{P}}=p\)
    2. The standard deviation of the sampling distribution of sample proportions, \(\sigma_{\hat{P}}\)=\(\sqrt{\dfrac{pq}{n}}=\sqrt{\dfrac{p(1-p)}{n}}\)

    Formula Review

    This page titled 6.3: The Central Limit Theorem for Sample Proportions is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.