Categorical Variables

• categorical variable
  • a variable that falls into one of two or more categories
    • categories like blood type, each person only falls in one category
  • frequency = number of occurences
  • relative frequency = proportion/percentage
  • example
    • roadrage accidents in germany
    • 
      • pie chart = good for seeing how big the bigger is, bad when there are multiple tiny slices
• ways of organizing graphs with categorical variables
  • ordinal
    • ordered from most frequent category to less frequent
  • nominal
    • random order?, cause its not alphabetical
  • side-by-side bar charts
    • 
  • stacked bar charts
    • 

Quantitative Variables

• quantitative variables
  • a numeric variable that represents a measurable quantity
  • we plot the values of the variables and how often these values occur
    • days after receiving virus that guinea pigs died
    • 
• can be illustrated in a variety of ways
  • histograms
    • 
  • box plots
  • dot plots
  • scatterplots
    • when we have 2+ quantitive variables
    • analyzed using simple ___ regression and correlation
    • 
  • time series plots
    • used to visualize observations that occur in time order
    • 
• important considerations when analyzing the data
  • where the center of the distribution is located
    • mean, median
  • how much variability there is
    • variance, standard deviation
  • are there any outliers?
    • extreme values
  • the shape of the distribution
    • is it skewed?
• distribution shapes
  • symmetric distributions
    • 
  • bimodal distributions
  • skewed distributions
    • to the left
      • like grades, lot of people ~90
      • 
    • to the right/positively skewed
      • common with time from something
      • can always stretch to the right, but bounded at 0
      • 

Numerical Measures

• assume that the data is from a sample, and does not represent the entire population

Measures of Central Tendency

• Measures of Central Tendency
  • mean
    • more impacted by outliers/extreme values
    • represented by $\overline{x}$ = sample mean
      • estimates population mean (mu) $\mu$
    • $$\overline{x}=\frac{\sum _{i=1}^n{x}_i}{n}$$
  • median
    • splits the observation in the middle
      • 
    • the value that falls in the middle when the data are ordered from smallest to largest
      • if n is odd, the median is the middle value
      • if n is even, the median is the average of the two middle values
  • mode
    • the most frequently occurring observation
    • not used very often
  • trimmed mean
  • weighted mean
  • midrange
  • harmonic mean
  • geometric mean

Measures of variability

• Measures of variability
  • range
    • maximum - minimum
    • or both max and min
      • R gives the values themselves
  • the best measures of variability are based on deviations from the mean
    • for any data set, the sum of the deviations (values - mean) = 0
    • Mean Absolute Deviation (MAD)
      • the average distance from the mean
        • not very useful
        • $$MAD=\frac{\sum |x_i-x|}{n}$$
      • usually just uses the squared distance from the mean (sample variance)
    • $Var(X)$ = $\text{sigma { <a class="tag" onclick="toggleTagSearch(this)" data-content="\#2">\#2</a>}}$
      • $Var(X)=E[(X-\mathrm{\mu }{)}^2]$
        • $E[(X-\mathrm{\mu }{)}^2]=E(X^2)-[E(X){]}^2$
        • average squared distance from the true mean
          • something distant = exaggerated effect
      • $Var(X)=\frac{\sum (x_i-\mu {)}^2}{n}$
    • $s^2$ (sample variance) estimates $\text{sigma { <a class="tag" onclick="toggleTagSearch(this)" data-content="\#2">\#2</a>}}$ (population variance)
      • $s^2=\frac{\sum (x_i-\overline{x}{)}^2}{n-1}$
      • n-1 instead of n
        • sample mean makes the number the smallest possible value, but true mean could be anywhere, so just have the -1 to have a better estimation
    • sample standard deviation
      • square root of the sample variance
      • s always $\ge$ MAD
      • $\overline{x}$ = sample mean, estimation of population mean
      • $$s=\sqrt{s^2}=\sqrt{\frac{\sum (x_i-\overline{x}{)}^2}{n-1}}$$
        • $s^2$ estimates $\text{sigma { <a class="tag" onclick="toggleTagSearch(this)" data-content="\#2">\#2</a>}}$ (population variance)
      • empirical rule
        • for mount shaped distributions:
          • Approximately 68% of the observations lie within 1 standard deviation of the mean
          • Approximately 95% of the observations lie within 2 standard deviations of the mean
          • All or almost all of the observations lie within 3 standard deviations of the mean
          • 
    • Z-score
      • the sample z-score for the ith observation
        • zi = (xi - xbar)/s
        • measures how many standard deviations the observation is above or below the mean
        • 
      • average z score = 0
        • theres negatives and positives
      • sd of z-scores = 1
      • rephrasing empirical rule
        • ~68% of z-scores lie between -1 and 1
        • ~95% of z-scores lie between -2 and 2
        • ~almost all z-scores lie between -3 and 3
          Percentiles

Linear Transformations

• x* = a + bx
• some measures change with the transformation (F to C)
  • new mean = a + b * original mean
    • times 4 or -4 = increases distance between the mean by 4 times
    • times 1 or -1 doesnt change sd
  • new median = a + b * original median
• additive constant (a) does not affect the measures of variability
  • new sd = |b| x original sd
  • new variance = b^2 x original variance
  • new IQR = |b| x original IQR
  • 
• not all transformations
  • log, sqrt are calculated differently, these are just simple bc its + and *