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Mean, Variance and Standard Deviation

Mean: The mean of a data is the sum of the data entries divided by the number of entrie.

\mathbf{Population\,Mean:\quad \mu =\frac{{\sum(x)}}{N}}

\mathbf{Sample\,Mean:\quad \bar x =\frac{{\sum(x)}}{n}}

Standard Deviation: In mathematics and statistics, deviation is a measure of difference between the observed value of a variable and some other value, often that variable's mean. The standard deviation is a non-negative number that measures how far data values are from their mean.

\mathbf{Standard\,Deviation(Population):\quad \sigma =\sqrt{\frac{{\sum(x-\mu)}^{2}}{N}}}

\mathbf{Standard\,Deviation(Sample):\quad s=\sqrt{\frac{{\sum(x-\bar x)}^{2}}{n}}}

Variance: The average of the "squared" differences from the Mean.

\mathbf{Variance(Sample):\quad s^2=\frac{{\sum(x-\bar x)}^{2}}{n}}

\mathbf{Variance(Population):\quad \sigma^2=\frac{{\sum(x-\mu)}^{2}}{N}}

The Pronunciation of Variables and Notations & Introduction of Formulas (Dr.Licwinko, Susan)

Finding the standard deviation using the definition formula (Dr. Lee, Jae Ki)

More example: Finding the standard deviation using the definition formula (Luis Cruz)

Frequency Distribution: The frequency distributions are helpful for computing the mean and standard deviation.

\mathbf{Standard\,Deviation(Population):\quad \sigma =\sqrt{\frac{N(\sum(x^2\cdot f))-(\sum(x\cdot f))^2}{N(N-1))}}}

\mathbf{Standard\,Deviation(Sample):\quad s =\sqrt{\frac{n(\sum(x^2\cdot f))-(\sum(x\cdot f))^2}{n(n-1))}}}

Standard deviation calculated using a frequency table(Dr. Lee, Jae Ki ; Example #1)

Standard deviation calculated using a frequency table with midpoint(Dr. Lee, Jae Ki ; Example #2)

More example: Standard deviation calculated using a frequency table with midpoint(Luis Cruz)

The Computational Formula: The Computational(Computing) Formula helps if data are not whole numbers and the mean is not one decimal.

\mathbf{Standard\,Deviation(Population):\quad \sigma =\sqrt{\frac{\sum(x^2)-\frac{(\sum(x))^2}{N}}{N}}}

\mathbf{Standard\,Deviation(Sample):\quad s =\sqrt{\frac{\sum(x^2)-\frac{(\sum(x))^2}{n}}{n-1}}}

Finding the standard deviation using the computational formula