7.8 Central limit theorem: summary of formulas

Central limit theorem for sample means

$\overline{X}$ ~ $\mathrm{N}({\mu }_X,\frac{{\sigma }_X}{\sqrt{n}})$ The Mean $\left(\overline{X}\right)$ : ${\mu }_X$

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

$z=\frac{\overline{x}-{\mu }_X}{\left(\frac{{\sigma }_X}{\sqrt{n}}\right)}$ Standard Error of the Mean (Standard Deviation $\left(\overline{X}\right)$ ): $\frac{{\sigma }_X}{\sqrt{n}}$

Central limit theorem for sums

$\mathrm{\Sigma X}$ ~ $N\left[\right(n)\cdot {\mu }_X,\sqrt{n}\cdot {\sigma }_X]$ Mean for Sums $\left(\mathrm{\Sigma X}\right)$ : $n\cdot {\mu }_X$

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

$z=\frac{\mathrm{\Sigma x}-n\cdot {\mu }_X}{\sqrt{n}\cdot {\sigma }_X}$ Standard Deviation for Sums $\left(\mathrm{\Sigma X}\right)$ : $\sqrt{n}\cdot {\sigma }_X$

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, $\overline{x}$ , and the sample sum, ΣX.If the size n of the sample is sufficiently large, then $\overline{X}$ ~ $\mathrm{N}({\mu }_X,\frac{{\sigma }_X}{\sqrt{n}})$ and $\mathrm{\Sigma X}$ ~ $N(n\cdot {\mu }_X,\sqrt{n}\cdot {\sigma }_X)$ . 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 $\overline{x}$ ) is $\overline{x}$ (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, $\frac{{\sigma }_{}}{\sqrt{n}}$