Basic Statistics

Basic Statistics
- Terms of Endearment
  - pop'n vs. sample
  - parameter (true value of pop'n) vs. statistic (estimate from sample)
  - sample size = n
  - "centre": mean
  - "variability": range, standard deviation, coefficient of variation
  - frequency distributions of data (e.g. grades)
- What's Normal Anyways?
  - what normal "looks like"
    - bell shaped
    - centred on mean
    - symmetrical
    - inflection point = 1 "SD"
    - +- 1SE = 68.3%, ... 95.4%, ... 99.7%
  - what follows a normal distribution?
    - sample means, thousands of 'em
    - a.k.a. Central Limit Theorum
  - what if your pop'n is NOT normal ...
    - it's OK 'cause sample means will be
    - need a large n (~30) ...
    - what if n <30? ... (beer is the answer)
- Middle Aged Spread (or, does this graph make me look fat?)
  - shape (distribution)
    - pop'n data & sample data
      - likely pop'n distr'bns
        
        skewed
        
        bimodal
        
        "strange"
      - helps to understand the pop'n
      - but usu. not available or considered
    - samples MEANS
      - follow normal distribution ...
      - ... so we use properties of normal (+- 1se = 68%) for large "n"
      - ... what about small "n"? (... beer)
  - centre
    - typically the mean (vs. median or mode)
    - THE most important statistic
  - spread
    - variation
    - basic measure is SD
    - SD (or "fatness" of the curve)
- It's Magic
  - the "magic table" to calculate
    - SD & CV (pop'n)
    - SE & SE% (sample)
- Finally! ... but is it True?
  - What's the diff?
    - e.g. consider a small "n" (e.g. 8) ... likelihood of mean = real answer?
    - n = 16 ? (i.e. I did 8 more) ...
    - n = 2,000 ? ...
    - n = 2,008 ?(i.e. I did 8 more) ...
  - What happens as more samples taken
    - mean
    - SD (CV)
    - SE (why is this one different?)
  - Bias
    - biased: SYSTEMATIC error ... final ≠ true
    - reasons
      - measures (technique, eqpt)
        
        drunken diameters
        
        kinked tape
      - sample selection
        
        the closest tree
        
        the train
        
        choosing your plot location
      - calculation/ improper weighting
        
        silv survey with concern for roadside
    - bias is result, not intent
  - How to ensure Final ~ True?
    - quality check ("check cruise") ...
    - checks for bias (systematic error)
  - Accuracy & Precision
    - Accuracy
      - NO bias
      - ... Final = True
      - stats give NO clue
      - "check cruise"
    - Precision
      - this IS stats (SE & E)
      - a measure of certainty
- I Need a Drink
  - ± 1SE ~68%, ± 2 SE ~ 95%
    - actually +-2 SE = 95.4%
    - +- 1.96 = 95.0%
    - +- 1.645 = 90.0%
  - beer??
    - small sample size? ... ± 1.96 SE = 95% NO longer true
    - Student's t-table ... Guiness (beer)
    - t distribution
      - looks like normal, but
      - it's "flatter" and more spread out
      - there is a separate curve for each sample size
      - for large n we use +/- 2se for 95%
      - for small n we use t ... Student's t-table
      - ... smaller the n, the bigger the t (or flatter the curve)
      - refer to t tables
  - magic table - add 3rd column
    - sampling error (E)
    - it's standard error expressed at a different confidence level
  - What does ±15% @ 95% mean??
    - 2 parts to statement of certainty
      - sampling error (i.e. the 15%)
      - confidence level (i.e. the 95%)
    - in words ...
      - we are 95% sure that ...
      - ... the real answer is within 15% of the sample mean
      - alternately
      - there is a 5% chance that ...
      - ... the real answer is actually >15% away from the sample mean
    - example
      - n = 6, mean = 550sph, se = 30sph
      - se% = ?
      - e% @90% = ?
      - e% @95% = ?
  - Sampling Error vs. Confidence Interval
    - sampling error as before
    - confidence INTERVAL
      - really is the "full range" (= mean +- E)
      - sometimes expressed as a range (i.e. LCL - UCL)
      - survey card is incorrect (CI = E on card)
    - example
      - n = 6, mean = 550sph, se = 30sph
      - E @ 90% in units = ??
      - confidence interval = ??
  - Assignment
    - ave wt of forestry students
    - calc: mean, sd, cv, se, se%, t and e (units & %)
    - n = 6
    - values on board
    - hand-in lab when done
- Stop Doing That
  - when to stop sampling
    - 1) becomes inefficient, extra plots don't help (see graph)
    - 2) too expensive, not worth $$/effort to do more
    - 3) you already answered the question to your satisfaction
      - "good enough for me"
      - "you met stats" (or max. plots)
  - Meeting Sampling Objective
    - max plots done (usu. max plots/ha)
    - sampling error (E%) < threshold
      - cruising <15% @95% confid.
      - silv surveys
        
        LCL > MSS (... SR)
        
        X < MSS (... NSR)
        
        CI < e ... really E < 100sph
    - how to ensure enough plots to meet stats ...
      - eq'n for sampling error ...
      - rearrange for n
- Break It Up
  - What (a.k.a. Stratification)
    - divide pop'n into homogeneous groups
    - each strata is sampled, then ...
      - ... the results are combined (inventory & cruising)
      - ... the results are kept independent (silv. surveys)
  - Why
    - reduce variation ... leads to reduced std. error (& less plots)
    - better description: separate sub-pop'ns do exist
    - isolate a variable (or problem) stratum
  - "The Mechanics" (for cruising)
    - How to combine strata & get sampling error
      - 1) convert SE to total units (i.e. m3, NOT m3/ha nor a %)
      - 2) use "Pythagoras theorum" ... A^2 +B^2 = C^2 ... want C
      - 3) convert C (i.e. SE combined) to %
      - 4) SE% * t = E% ... but t ...
    - Data (m3/ha, 1 plot/ha)
      - 850 (F)
      - 710 (F)
      - 476 (FPl)
      - 398 (FPl)
      - 567 (FPl)
      - 974 (F)
      - 766 (F)
      - 616 (F)
      - 630 (FPl)
      - stats
        
        x = 655.2
        
        sd = 181.0
        
        se, se%, t & e% ??
        
        se = 60.3
        
        se% = 9.2%
        
        t = 2.306
        
        e% = 21.2%
    - stratified
      - F type
        
        x = 738.2
        
        sd = 136.5
        
        se (m3/ha) = ?
        
        se (total) = ??
      - FPl type
        
        x = 517.8
        
        sd = 101.8
        
        se (m3/ha) = ?
        
        se (total m3) = ??
      - combining ...
        
        SE (total)
        
        (F) = 61.0 m3/ha * 5ha = 305m3
        
        (FPl) = 50.8m3/ha * 4ha = 203m3
        
        Pythagoras
        
        A^2 + B^2 = C^2
        
        (305)^2 + (203)^2 = C^2
        
        C^2 = 134,234
        
        then sqrt C^2
        
        C = 366 m3
        
        this is SE(combined) in units
        
        want SE(combined) in %
        
        we have SE in m3
        
        need total volume in m3
        
        (738.2 m3/ha * 5ha) + (517.8 m3/ha * 4ha) = total vol. = 5,762.2 m3
        
        SE% = 366 m3 / 5762.2 m3 = 6.4%
        
        finally getting E% @ 95%
        
        t(n-s) = ?
        
        SE% * t = ?
- Who Cares?
  - 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
  - Plot Layout
    - Grid
      - variable strata
      - need lotsa plots
      - want to "cover the ground"
      - stats
    - Vector
      - odd shaped blocks
      - less variable strata
      - fewer plots
      - stats
    - Representative
      - obvious stocking status
      - few plots done
      - you pick the location ... caution!
      - skilled & experienced surveyor
      - stats
    - Visual
      - obvious status
      - estimate
      - skilled & experienced surveyor
      - no plots ... no stats
  - 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) ... = sampling error
    - 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)
        
        do the extra plots
        
        re-calculate mean with extra plots
        
        compare mean to MSS to decide (no longer look at LCL)
    - example (3, 2, 4, 1, 4, 5)
    - area = 7 ha, MSS = 500
  - End of Survey
    - map with table
      - SU & stratum
      - Area: Gross, NP, NAR
      - Status: SR/NSR or FG/NFG
      - Inventory Label
      - Silviculture Label
      - Recomendations