Formula
Central limit theorem for sample means
$\overline{X}$ ~
$\mathrm{N}({\mu}_{X},\frac{{\sigma}_{X}}{\sqrt{n}})\phantom{\rule{35pt}{0ex}}$
The Mean
$\left(\overline{X}\right)$ :
$\phantom{\rule{10pt}{0ex}}{\mu}_{X}$
Formula
Central limit theorem for sample means z-score and standard error of the mean
$z=\frac{\overline{x}-{\mu}_{X}}{\left(\frac{{\sigma}_{X}}{\sqrt{n}}\right)}\phantom{\rule{25pt}{0ex}}$
Standard Error of the Mean (Standard Deviation
$\left(\overline{X}\right)$ ):
$\phantom{\rule{10pt}{0ex}}\frac{{\sigma}_{X}}{\sqrt{n}}$
Formula
Central limit theorem for sums
$\mathrm{\Sigma X}$ ~
$N\left[\right(n)\cdot {\mu}_{X},\sqrt{n}\cdot {\sigma}_{X}]\phantom{\rule{10pt}{0ex}}$
Mean for Sums
$\left(\mathrm{\Sigma X}\right)$ :
$\phantom{\rule{10pt}{0ex}}n\cdot {\mu}_{X}$
Formula
Central limit theorem for sums z-score and standard deviation for sums
$z=\frac{\mathrm{\Sigma x}-n\cdot {\mu}_{X}}{\sqrt{n}\cdot {\sigma}_{X}}\phantom{\rule{25pt}{0ex}}$
Standard Deviation for Sums
$\left(\mathrm{\Sigma X}\right)$ :
$\phantom{\rule{25pt}{0ex}}\sqrt{n}\cdot {\sigma}_{X}$
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,
$\frac{{\sigma}_{}}{\sqrt{n}}$