Chapter 3 Formulas and Vocabulary

• Mean (p. 108)
• Measure of center (p. 108)
• Median (p. 112)
• Mode (p. 114)
• Population mean (p. 109). $\mu =\sum x/N$.
• Population size (p. 109). Denoted by $N$.
• Sample mean (p. 109). $\overline{x}=\sum x/n$.
• Sample size (p. 109). Denoted by $n$.

• Chebyshev's rule (p. 138). The proportion of values from a data set that will fall within $k$ standard deviations of the mean will be at least $\left(1-\frac{1}{k^2}\right)100\%$, where $k>1$.
• Deviation (p. 128). $x-\overline{x}$.
• Empirical rule (p. 136). If the data distribution is bell-shaped:
  • About 68% of the data values will fall within 1 standard deviation of the mean.
  • About 95% of the data values will fall within 2 standard deviations of the mean.
  • About 99.7% of the data values will fall within 3 standard deviations of the mean.
• Measure of variability (measure of spread, measure of dispersion) (p. 127)
• Population standard deviation (p. 132).
  

  $\sigma =\sqrt{\frac{{\sum \left(x - \mu \right)}^2}{\mathrm{N}}}$

• Population variance (p. 130).
  

  ${\sigma }^2=\frac{{\sum \left(x - \mu \right)}^2}{\mathrm{N}}$

• Range (p. 127)
• Sample standard deviation (p. 133).
  

  $s^2=\sqrt{\frac{{\sum \left(x-\overline{x}\right)}^2}{n-1}}$

  Page 183
• Sample variance (p. 133).
  

  $s^2=\frac{{\sum \left(x-\overline{x}\right)}^2}{n-1}$

• Standard deviation (p. 128)

• Estimated mean for data grouped into a frequency distribution (p. 150).
  

  $x=\frac{\sum \left(\mathrm{f}·x\right)}{\sum \mathrm{f}}$

• Estimated standard deviation for data grouped into a frequency distribution (p. 151).
  

  $s=\sqrt{s^2}=\sqrt{\frac{{\displaystyle \sum {\left(x-x\right)}^2\cdot f}}{{\displaystyle \sum f}}}$

• Estimated variance for data grouped into a frequency distribution (p. 151).
  

  $s^2=\frac{{\displaystyle \sum {\left(x-\overline{x}\right)}^2\cdot f}}{{\displaystyle \sum f}}$

• Weighted mean (p. 151).
  

  $x=\frac{\sum \left(w·x\right)}{\sum \mathrm{w}}$

• Finding a data value $x$ given its $z$-score (p. 157)
  

  $\begin{array}{ll}\text{sample}:\hfill & =\text{z-score}·\mathrm{s}+\overline{x}\hfill \\ \text{population}:\hfill & =\mathrm{z-}\text{score}·\sigma +\mu \hfill \end{array}$

• interquartile range (IQR) (p. 166).
  

  $\text{IQR}=\text{Q3}-\text{Q1}$

• outlier (p. 158)
• percentile (p. 159)
• Percentile rank (p. 161)
• Quartiles (p. 162)
• $z$-Score (p. 155)
  1. Sample:
     

     $\mathrm{z}-\text{score}=\frac{\text{data value }-\text{ mean}}{\text{standard deviation}}=\frac{x-\overline{x}}{s}$

  2. Population:
     

     $\mathrm{z}-\text{score}=\frac{\text{data value }-\text{ mean}}{\text{standard deviation}}=\frac{x-\mu }{\sigma }$

• boxplot (p. 173)
• five-number summary (p. 172)
• IQR method of detecting outliers (p. 177)