2.11 Univariate descriptive statistics: summary of formulas

Commonly used symbols

• The symbol $\Sigma$ means to add or to find the sum.
• $n$ = the number of data values in a sample
• $N$ = the number of people, things, etc. in the population
• $\overline{x}$ = the sample mean
• $s$ = the sample standard deviation
• $\mu$ = the population mean
• $\sigma$ = the population standard deviation
• $f$ = frequency
• $x$ = numerical value

Commonly used expressions

• $x*f$ = A value multiplied by its respective frequency
• $\sum x$ = The sum of the values
• $\sum \mathrm{(x}*\mathrm{f)}$ = The sum of values multiplied by their respective frequencies
• $(x-\overline{x})$ or $(x-\mu )$ = Deviations from the mean (how far a value is from the mean)
• ${(x-\overline{x})}^2$ or ${(x-\mu )}^2$ = Deviations squared
• ${f(x-\overline{x})}^2$ or ${f(x-\mu )}^2$ = The deviations squared and multiplied by their frequencies

Box plot with outliers formulas

• IQR=Q3 - Q1
• LOWER FENCE=Q1 – 1.5(IQR)
• UPPER FENCE=Q3 + 1.5(IQR)
• OUTLIERS=greater than 1.5(IQR) but less than 3.0(IQR), this is indicated by: o
• Far OUTLIERS=3.0(IQR) or greater, this is indicated by: *

Mean Formulas:

• $\overline{x}=$ $\frac{\sum x}{n}$ or $\overline{x}=$ $\frac{\sum \mathrm{(f}·\mathrm{x)}}{n}$
• $\mu =$ $\frac{\sum x}{N}$ or $\mu$ = $\frac{\sum \mathrm{(f}·\mathrm{x)}}{N}$

Standard Deviation Formulas:

• $s=$ $\sqrt{\frac{\Sigma {(x-\overline{x})}^2}{n-1}}$ or $s=$ $\sqrt{\frac{\Sigma {f·(x-\overline{x})}^2}{n-1}}$
• $\sigma =$ $\sqrt{\frac{\Sigma {(x-\overline{\mu })}^2}{N}}$ or $\sigma =$ $\sqrt{\frac{\Sigma {f·(x-\overline{\mu })}^2}{N}}$

Formulas Relating a Value, the Mean, and the Standard Deviation:

• value = mean + (#ofSTDEVs)(standard deviation), where #ofSTDEVs = the number of standard deviations
• $x$ = $\overline{x}$ + (#ofSTDEVs)( $s$ )
• $x$ = $\mu$ + (#ofSTDEVs)( $\sigma$ )

Glossary

Frequency

Interquartile range (irq)

Mean

Median

Mode

Outlier

Percentile

Quartiles

Relative frequency

Standard deviation

Variance