Basic Statistics

Basic Statistics
- Terms of Endearment
  - pop'n vs. sample, parameter (true value) vs. statistic (estimate)
  - mean, median, mode
  - standard deviation vs. coefficient of variation
- Middle Aged Spread (or, does this graph make me look fat?)
  - shape (distribution)
    - samples means
      - normal
        
        what normal "looks like"
        
        what exactly is normal?
        
        if nature is not normal ...
    - use properties of normal
  - centre
    - typically the mean
    - THE most important statistic
  - spread
    - variation
    - ... linked to sampling error & confidence
    - SD & CV - for pop'n
    - SE & SE% - for the sample
    - pop'n data & sample data
      - likely distr'bns
        
        skewed
        
        bimodal
        
        strange
      - helps to understand the pop'n
      - sample data distr'bn will be similar
- It's Magic
  - SD & CV
  - SE & SE%
  - the "magic table"
  - column 1: estimates of pop'n; too big? ... "live with it"
  - column 2: indicates "uncertainty" of the sample; too big? ... do more plots
- Final vs. True
  - What's the diff?
    - e.g. consider a small sample size ... likely the real answer?
    - what happens if a add a couple more samples?
    - what if I have a very large sample size?
    - what if I add a couple more samples?
    - technique? eqpt? sample selection?
  - What happens as more samples taken
    - mean
    - SD (CV)
    - SE
  - Bias
    - UNbiased def'n: long run average from repeated samples = real (pop'n) value
    - biased: sampling process results in final <> true
    - reasons for bias
      - measures (technique, eqpt)
      - sample selection
      - calculation/ improper weighting
  - How to ensure Final ~ True
  - Accuracy & Precision
    - Accuracy
      - how close sample mean is to True
      - stats give NO clue
      - "check cruise"
    - Precision
      - how close sample mean is to Final
      - this IS stats (SE & E)
    - accurate but NOT precise
    - precise but NOT accurate
- I'll Have a Guiness
  - ± 1SE ~68%, ± 2 SE ~ 95%
  - magic table - add 3rd column
  - beer??
    - small sample size? ... ± 2 SE ~ 95% NO longer true
    - Student's t-table ... Guiness
  - What does ±15% @ 95% mean??
    - 2 parts to statement of certainty
      - sampling error (range)
      - confidence level
  - Sampling Error vs. Confidence Interval
    - sampling error = std error at some other confidence level
    - confidence interval
      - a) sampling error in units for silv surveys (can be %)
      - b) expressed as a range
  - Assignment
    - ave wt of 2nd yr forestry students
    - calc: mean, sd, cv, se and e (units & %)
    - values on board
    - hand-in lab when done
- Stop Already!
  - when to stop sampling
    - becomes inefficient, extra plots don't help (see graph)
    - becomes too expensive, not worth $$/effort to do more
    - you already answered the question to your satisfaction
    - "you met stats" - met necessary precision (not accuracy)
      - OR reached the required maximum plot density (e.g. 1.5 plots/ ha)
  - Rule of Thumb for Silv Surveys
    - 80% of strata need 10 or less plots
    - 15% of strata need 11-25 plots
    - 5% may need >25 plots
  - Meeting Stats
    - sampling error < threshold
    - eq'n for sampling error ...
    - rearrange for n
    - but
      - what about t-value?
      - process
        
        use t=2 (or existing t if sample didn't meet stats)
        
        get a value for n
        
        revise t-value, and calculate again
        
        then add 2 or 10% for good measure (remember CV is an estimate)
- Break It Up!
  - What
    - divide pop'n into homogeneous groups
    - each stratum is (typically) less variable than the pop'n as a whole
    - each strata is sampled independently, then ...
      - ... the results are combined (inventory & cruising)
      - ... the results are considered independently (silv. surveys)
    - typically based on geography (space), but ...
      - could be other parameter, e.g. species
  - Why
    - reduce variation ... leads to reduced std. error (& sampling error)
      - gymnasium of "basketball players"
      - high volume rich site vs. low volume scrub
      - patch of "happy" seedlings vs. patch of marginal survival
      - variation btwn groups "disappears" from the calculations
    - simply to recognize that separate sub-pop'ns exist
      - i.e. HB & CH types
      - even though they have similar stocking (or volumes ...)
      - ... "typing" provides a better description
    - isolate a variable (or problem) sub-pop'n
      - e.g. OG patch in a 2nd growth cutblock
      - e.g. patch of "borderline NSR scrub" adjacent to "typical stand"
    - allows for different sampling methods
      - 3P in riparian corridor
      - prism for rest of block
- Silviculture Surveys & Stats
  - Number of Plots
    - 5 plot minimum per stratum
    - maximum density of 1.5 plots/ ha - regardless of stats
    - Rule of Thumb for Silv Surveys
      - 80% of strata need 10 or less plots
      - 15% of strata need 11-25 plots
      - 5% may need >25 plots
  - Filling in the card (FS 1138A)
    - Opening - "the block"
    - Strata - usually a SU
    - Area - important for max. plots
    - No. of Plots
    - mean (stems per plot & per ha)
    - std dev (S)
    - std error (Sx)
    - t (@90%) n-1
    - CI (t* Sx)
    - LCL (x - CI)
    - MSS (sph)
    - e (precision, <0.5 or 100 sph)
    - (new) n
  - Decision Rules
    - 1) if mean & LCL > MSS ... SR/FG
    - 2) if mean < MSS ... NSR/NFG
    - 3) if MSS is btwn mean and LCL ... oh crap, gotta think
      - a) if CI < e (0.5 or 100 sph), use mean ... SR/FG
      - b) if CI > e ... need more plots
        
        calculate new n
        
        new n - plots already done = additional plots (to a max of 1.5/ha)
        
        re-calculate mean with extra plots
        
        compare mean to MSS to decide (no longer look at LCL)