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Formula

Central limit theorem for sample means

X ~ N ( μ X , σ X n ) The Mean ( X ) : μ X

Formula

Central limit theorem for sample means z-score and standard error of the mean

z = x - μ X ( σ X n ) Standard Error of the Mean (Standard Deviation ( X ) ): σ X n

Formula

Central limit theorem for sums

ΣX ~ N [ ( n ) μ X , n σ X ] Mean for Sums ( ΣX ) : n μ X

Formula

Central limit theorem for sums z-score and standard deviation for sums

z = Σx n μ X n σ X Standard Deviation for Sums ( ΣX ) : n σ X

Definitions

    Average

  • A number that describes the central tendency of the data. There are a number of specialized averages, including the arithmetic mean, weighted mean, median, mode, and geometric mean.

    Central limit theorem

  • Given a random variable (RV) with known mean μ and known standard deviation σ. We are sampling with size n and we are interested in two new RVs - the sample mean, x , and the sample sum, ΣX.If the size n of the sample is sufficiently large, then X ~ N ( μ X , σ X n ) and ΣX ~ N ( n μ X , n σ X ) . If the size n of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population. The mean of the sample means will equal the population mean and the mean of the sample sums will equal n times the population mean. The standard deviation of the distribution of the sample means,, is called the standard error of the mean

    Mean

  • A number that measures the central tendency. A common name for mean is 'average.' The term 'mean' is a shortened form of 'arithmetic mean.' By definition, the mean for a sample (denoted by x ) is x (the sum of all values in the sample divided by the number of values in the sample), and the mean for a population (denoted byμ) is μ (the sum of all the values in the population divided by the number of values in the population).

    Standard error of the mean

  • The standard deviation of the distribution of the sample means, σ n

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Source:  OpenStax, Collaborative statistics using spreadsheets. OpenStax CNX. Jan 05, 2016 Download for free at http://legacy.cnx.org/content/col11521/1.23
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