1. BASIC STATISTICAL CONCEPTS Statistical Moments & Probability Density Functions 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 Population Sample

2. Most basic descriptive parameter for any set of measurements: Sample Mean over the duration of a time series – “time average” or over an ensemble of measurements – “ensemble mean” Sample mean 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 i th value

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

5. Population variance has one degree of freedom (dof) < Sample 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 is representative of the entire population The larger the more certain we are that we have sampled the entire population Example: we have 2 observations, when estimating the mean we have 2 independent observations: = 2 But when estimating the variance, we have one independent observation because the two observations are at the same distance from the mean: =1

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

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 The probability of a continuous variable is defined by a PROBABILITY DENSITY FUNCTION -- PDF 0  P (x)  1

9. Probability is measured by the area underneath PDF

11. Probability Density Function Gauss Normal Bell Gauss or Normal or Bell  11 22 33 erf(1/(2) ½ ) = 68.3% erf(2/(2) ½ ) = 95.4% erf(3/(2) ½ ) = 99.7%

12.  11 22 33 68.3% 95.4% 99.7% standardized normal variable Probability Density Function Gauss Normal Bell Gauss or Normal or Bell

13. Probability Density FunctionGamma  = 1  = 1  = 2  = 3  = 4

14. Probability Density FunctionGamma  = 2  = 1  = 2  = 3  = 4

15. Probability Density Function Chi Square  =  /2 Special case for  = 2  = 2  = 4  = 6  = 8 4 24 2 8 28 2  12 2  16 2

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

17. (1 -  /2) = 0.975 http://statistics.laerd.com/statistical-guides/normal-distribution-calculations.php  z  /2 = 1.96

18. C.I. 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?

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

20.  /2 = 0.025 d.o.f.= 19

21. 1 -   /2 C.I. 100 (1 -  )% C.I. is: If  = 0.05, t 0.025,19 = 2.093 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?

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

23. 1 -   /2 Suppose that we have = 10 spectral estimates of a tidal record. C.I. 100 (1 -  )% C.I. is: The background variance near a distinct spectral peak is 0.3 m 2 95% C.I. for variance? How large would the peak have to be to stand out, statistically, from background level?  /2 = 0.025; 1 -  /2 = 0.975 Look at Chi square table:

24. Chi Square Table The background variance lies in this range The spectral peak has to be greater than 0.92 m 2 to distinguish it from background levels