Statistical Equations

Central Tendency

Mean

The mean can be measured as follows

\overset{―}{x} = \frac{\sum x}{n}

where:
$\bar{x} =$ the mean
$\sum x =$ the sum of all data values
$n =$ the number of data values

Mean of Frequencies

For data in a cumulative frequency table, the mean is calculated as follows

\overset{―}{x} = \frac{\sum x f}{\sum f}

where:
$\sum x f =$ sum of the products of the data values and their frequencies
$\sum f =$ the sum of frequencies

Quartiles

Discrete Data

To find upper and lower quartiles for discrete data

Divide $n$ by 4 (lower quartile) or find $\frac{3}{4}$ of $n$ (upper quartile)
If this is a whole number, the lower or higher quartile is the midpoint between this and the datapoint above. If it is not, round up and pick this number.

Frequency Table Quartiles

When data is in a grouped frequency table you can use interpolation to estimate the means, quartiles, and percentiles. This method assumes that data values are equally distributed across each class.

Q_{1} = \frac{n}{4} t h d a t a v a l u e

Q_{2} = \frac{n}{2} t h d a t a v a l u e

Q_{3} = \frac{3 n}{4} t h d a t a v a l u e

Variance

There are three versions of the formulae to find variance $σ^{2}$
1)

σ^{2} = \frac{\sum (x - \overset{―}{x})^{2}}{n - 1}

This is easier to use when given raw data. You can remember this as "mean of squares minus square of means"

σ^{2} = \frac{\sum x^{2}}{n} - (\frac{\sum x}{n})^{2}

$σ^{2} = \frac{S_{x x}}{n}$

where $S_{x x} = \sum (x - \bar{x})^{2} = \sum x^{2} - \frac{(\sum x)^{2}}{n}$
$S_{x x}$ is a summary statistic used to simplify formulae.

Variance of Frequencies

For data displayed in a frequency table:

σ^{2} = \frac{\sum f (x - \overset{―}{x})^{2}}{\sum f}

σ^{2} = \frac{\sum f x^{2}}{\sum f} - (\frac{\sum f x}{\sum f})^{2}

Standard Deviation

The standard deviation $(σ)$ is the square root of variance; larger value = more spread

$\sqrt{σ^{2} = \frac{\sum (x - \overset{―}{x})^{2}}{n}} =$ $\sqrt{σ^{2} = \frac{\sum x^{2}}{n} - (\frac{\sum x}{n})^{2}} =$ $\sqrt{σ^{2} = \frac{S_{x x}}{n}}$ $= σ$

Standard Deviation of Frequencies

For data displayed in a frequency table:

σ = \sqrt{σ^{2} = \frac{\sum f (x - \overset{―}{x})^{2}}{\sum f}}

σ = \sqrt{σ^{2} = \frac{\sum f x^{2}}{\sum f} - (\frac{\sum f x}{\sum f})^{2}}

Source

See Individual Pages

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