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BASIC STATISTICAL CONCEPTS

BASIC STATISTICAL CONCEPTS. Ocean is not “stationary”. “Stationary” - statistical properties remain constant in time. Data collected have signal and noise. Both signal and noise are assumed to have random behavior. Most basic descriptive parameter :. Sample Mean.

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BASIC STATISTICAL CONCEPTS

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  1. BASIC STATISTICAL CONCEPTS Ocean is not “stationary” “Stationary” - statistical properties remain constant in time Data collected have signal and noise Both signal and noise are assumed to have random behavior

  2. Most basic descriptive parameter : Sample Mean over the duration of a time series – “time average” or over an ensemble of measurements – “ensemble mean” It is an unbiased estimate of the population mean ‘’ The population mean, μ, can be regarded as the expected outcome E(y) of an event y. If the measurement is executed many times, μ would be the most common outcome, i.e., it’d be E(y) (e.g. the weight printed on a bag of chips)

  3. Sample Mean - locates center of mass of data distribution such that: Weighted Sample Mean relative frequency of occurrence of ith value

  4. Variance - describes spread about the mean or sample variability Sample variance Sample standard deviation Computationally more efficient (only one pass through the data) Population variance (unbiased) typical difference from the mean N needs to be > 1 to define variance and std dev Only for N < 30 s’ and are significantly different

  5. Sample variance has one degree of freedom (dof) < Population variance because we estimate population variance with sample variance (one less dependent measure) d.o.f. :  = # of independent pieces of data being used to make a calculation.  = measure of how certain we are that our sample population is representative of the entire population The larger  the more certain we are that we have sampled the entire population

  6. Other values of Importance 0.66 N = 1601 range (1.27) -0.61 Median – equal number of values above and below = -0.007 Mode – value occurring most often

  7. Mode = -0.3 Two Modes Bimodal

  8. Probability Provides procedures to infer population distribution from sample distribution and to determine how good the inference is The probability of a particular event to occur is the ratio of the number of occurrences of that event and the total number of occurrences for all possible events P (a dice showing ‘6’) = 1/6 0  P (x) 1 The probability of a continuous variable is defined by a PROBABILITY DENSITY FUNCTION -- PDF

  9. Probability is measured by the area underneath PDF

  10. Probability Density Function Gaussor Normalor Bell  3 2 68.3% 99.7% 95.4% 1

  11. Probability Density Function Gaussor Normalor Bell  3 2 standardized normal variable 68.3% 1 99.7% 95.4%

  12. Probability Density Function Gamma  = 1  = 1  = 2  = 3  = 4

  13. Probability Density Function Gamma  = 1  = 1  = 2  = 3  = 4

  14. Probability Density Function Chi Square  = 2  = /2  = 4  = 6 82  = 2 122 42 162  = 8

  15. CONFIDENCE INTERVALS Confidence Interval for  with  known For N > 30 (large enough sample) the 100 (1 - )% confidence intervalis: standardized normal variable /2 /2 1 - 

  16. (1 - /2) = 0.975 http://statistics.laerd.com/statistical-guides/normal-distribution-calculations.php

  17. 100 (1 - )% C.I.is: If  = 0.05, z/2 = 1.96 Suppose we have a CT sensor at the outlet of a spring into the ocean. We obtain a burst sample of 50 measurements, once per second, with a sample mean of 26.5 ºC and a stdev of 1.2 ºC for the burst. What is the range of possible values, at the 95% confidence, for the population mean?

  18. CONFIDENCE INTERVALS Confidence Interval for  with  unknown For N < 30 (small samples) the 100 (1 - )% confidence intervalis: Student’s t-distribution with  = (N-1) degrees of freedom /2 /2 1 - 

  19. /2 = 0.025 d.o.f.= 19

  20. 100 (1 - )% C.I.is: Suppose we do 20 CTD profiles at one station in St Augustine Inlet. We obtain a mean at the surface of 16.5 ºC and a stdev of 0.7 ºC . What is the range of possible values, at the 95% confidence, for the population mean? If  = 0.05, t0.025,19 = 2.093 /2 /2 1 - 

  21. CONFIDENCE INTERVALS Confidence Interval for 2 To determine reliability of spectral peaks Need to know C.I. for 2 on the basis of s2  = (N-1) degrees of freedom /2 1 -  /2

  22. 100 (1 - )% C.I.is: Suppose that we have  = 10 spectral estimates of a tidal record. The background variance near a distinct spectral peak is 0.3 m2 95% C.I. for variance? How large would the peak have to be to stand out, statistically, from background level? /2 /2 1 -  /2 = 0.025; 1 - /2 = 0.975 Look at Chi square table:

  23. The background variance lies in this range The spectral peak has to be greater than 0.92 m2 to distinguish it from background levels Chi Square Table

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