Counting Statistics

1. Counting Statistics Cember, Ch 9 pps 396 � 417 Other sources

2. Outline Terminology Probability Exponential Decay Binomial Distribution Poisson Distribution Normal Distribution Characterization of Data Test of Data Sets

3. Outline [cont�d] Error and Error Propagation Counting Radioactive Samples Gross Count Rates Net Count Rates Optimizing Counting Time Additional Applications

4. Types of Errors Systematic Random

5. Systemic Errors Systemic errors are from Improper calibration Changing conditins Temperature, pressure, volume (PV=nRT) Parallax errors Can be minimized Affect accuracy (more later) Sometimes can be found, solved and accounted for (without statistics) � see example

6. Random Errors Random changes in temperature over time Random nature of radioactive decay Random errors affect precision Random errors cannot be found without statistics Random errors tend to cancel each other out This is how to distinguish between systemic and random!

7. Random Errors Random errors Random errors are squared then added {Total random error}2 = {Random error in counting}2 + {Random error in measuring sample}2 + � When adding systemic and random errors, add without squaring to be conservative

8. Systemic vs. Random Error Systemic errors affect accuracy Random errors affect precision Possible to have Good precision, poor accuracy Poor precision, poor accuracy Poor precision, good accuracy Good precision, good accuracy

9. Illustration of bias, precision and accuracy

10. On the average the duck is dead

11. Statistics Purpose is to make estimates on a population based on limited sample or measurements Ideally select a sample which represents the whole, determine characteristics on sample, make estimates on the whole

12. Statistics Symbols �True� values of populations are typically given as m,s Estimators from samples are given as , s Remember that these are estimators which should be designed to represent the population or �whole�

13. Statistics Terms Mode: Most frequently occurring value; Most likely value Although most likely event, usually small probability of occurring Median: Middle value Half of the number of observations above or below this value

14. Statistics Terms Mean: Sum of data divided by the number of data points. Frequently called the average; Half of observations above, Half below Measures of a central tendency ni value of ith measurement m total number of measurements

15. Statistics Terms Example Ten one minute backgrounds were taken on a GM counting assembly. What is the mode, median and mean of the backgrounds? Assume that the counter is operating properly. 11, 9, 14, 12, 10, 11, 8, 11, 12, 10 Mode: 11: Most frequently occurring number Can have more than one mode

16. Median Median: 11.5 (half above, half below) To solve, numbers were ranked 8, 9, 10, 10, 11, 11, 11, 12, 12, 14 Since even number of points, median equals mean of central two points Mean:10.8 11 + 9 + 14 + 12 + 10 + 11 + 8 + 11 + 12 + 10 = 108 108/10 = 10.8

17. Statistics Terms Note that mode, median and mean ONLY coincide for symmetrical distributions Coincide with normal distributions Note from our experience with beta energies that the mean (average) would NOT be the same as the median (half energies above and below point) These terms do not describe data dispersion

18. Dispersion Data dispersion may be quantified by using the deviation from the mean; Need more than mean to describe data.

19. Dispersion Dispersion can also be described by range Range is strongly affected by extremes Dispersion is generally represented by using the standard deviation of the data Also can be referred to as RMS (Root Mean Square) value

20. Important Types of Distributions for HP Binomial Poisson Normal Log Normal Log Probability

21. Probability of Decay/Survival Basic equation for radioactive decay Decay is a random event Mean rate characteristic of nuclide Probability of decay, p, during time ?t is proportional to length of interval

22. Probability of Decay/Survival, cont�d If ? is the constant of proportionality, then The probability of surviving the time interval ?t is: The probability of surviving n successive periods is:

23. Probability of Decay/Survival, cont�d If n intervals of ?t is the total time, then Substituting this relationship into the previous As ?t ?0, n ?8, and

24. Exponential Decay Therefore, e-?t is the probability that a single atom survives for time t Restated: the probability (q) for a nucleus to survive the time t: q=e-lt The probability (p) for a nucleus to decay during t: p=1-q=1-e-lt There are only two alternatives for a given atom in the time t, since p+q=1.

25. Binomial Distribution Radioactive transformations and other nuclear reactions are randomly occurring events Must be described in statistical terms The sampling distribution of a series of random events is called the Binomial distribution Binomial may also be called Bernoulli trials or a Bernoulli process Only two possible outcomes for each trial Either an atom decays or not The probability of a success is the same for each trial The probability of any particular atom decaying is the same

26. Binomial Distribution There are n trials, where n is constant n is the number of atoms available for decay The n trials are independent Any atom decaying has no effect on any other atom

27. Binomial Distribution For any initial number N of identical radioactive atoms, the probability that n will disintegrate in time t is:

28. Binomial Distribution [cont�d] The distribution p+q=1 is called the binomial distribution. The expected, or mean, number of disintegrations in time t is given by the mean value m:

29. Binomial Distribution [cont�d] The result of the summation is: m=N p Repeated observations of many sets of N identical atoms for time t is expected to give the binomial probability distribution Pn for the number of disintegrations n.

30. Binomial Data: Using Tables Do not have to calculate values Tabulated values available Need to know: Number of trials, n Number of successes, x Probability of successes, p

31. Poisson Distribution When the number of trials (atoms in our case) is large, and the probability (p) of decaying is small (generally true) the Poisson distribution is used to approximate the binomial distribution Approximating binomials is only one of the uses of the Poisson distribution

32. Poisson Distribution, continued Mean = m = np (describes it all!) Standard deviation = vm Use this for counting statistics

33. Normal Distribution As probability of decay p gets small and N gets large (> 30), binomial and Poisson distributions tend to approach the shape of a normal distribution.

34. Normal (Gaussian) Distribution The mean represents the most frequently occurring result The peak of a symmetric bell-shaped curve. Where ni is the value of the ith measurement and m the total number of measurements

35. Normal Curve, continued The width of the bell-shaped curve is described by the standard deviation. Combined with the mean this completely defines the distribution. The standard deviation is calculated as:

36. Side note The standard deviation is often written with m-1 replaced by m. Although m-1 is mathematically correct, as the sample size increases m-1 � m and so the error is minimal. By definition the area under the bell curve within � 1.0 standard deviation of the mean is 68.3% of the total area under the curve

37. Normal Distribution [cont�d] Full width at half maximum (FWHM) is the width of the normal distribution at the position of half of its maximum:

38. Normal Distribution [cont�d]

39. Errors and Confidence Intervals If the half life of a nuclide is long relative to the counting time, then no appreciable decay occurs during counting. Given that situation, we want to know how close a measured value of counts is to the �true� value.

40. Errors and Confidence Intervals This can be described by: q = K(r)1/2 = Ks Where q is the deviation from the true, mean count K is the number of standard deviations from the mean r is the true, average count s is the standard deviation (= r1/2 for Poisson distributions).

41. Errors and Confidence Intervals For the case K = 1 q = s This means for a large number of measurements, 68.3% should be within 1 standard deviation of the mean. If K = 2, then 96% of all measurements should lie within 2 standard deviations of the mean.

42. Confidence Limits for Gaussian Distributions

43. Reality True count, r not known Substitute observed count, n Error is small with dealing with large numbers of counts in radioactive decay Can express deviation as a fractional (F) or percentage value, substituting n for r: F = q/n = K/n1/2 F is the fractional error, or uncertainty within which there is a certain probability (confidence limit) that the measurement will occur when n counts are measured. (whew!)

44. Addressing Statistical Error Radioactivity normally described in terms of count rate Substitute Rt for r where R is the count rate t is the count time Then equation for uncertainty in terms of counting rate and counting time can be derived

45. Addressing Statistical Error q = K(r)1/2 = Ks q = K (Rt)1/2 Let Q = q/t = K (Rt)1/2/t And Q = K (R/t)1/2 Where Q is the uncertainty or error (counts per min). When K = 1, Q is s (the std. deviation).

46. Addressing Statistical Error As with total counts, the error can also be expressed as a fractional value F = Q/R = K(R/t)1/2 /R F = K /(Rt)1/2

47. Errors from Background Background count rate is included in gross count rate Error (Q) in the count rate corrected for background is calculated from individual errors associated with the sample (s+b) and background count rate (b)

48. Errors in count rate

49. Errors in net count rate The error in the net count rate, Qn is:

50. When K is set to 1 (1 std deviation)

51. Optimum Counting Times It is typical to use a single background count to correct several sample results. For the case of a single sample and background count, the optimum division of time is:

52. Terms to Remember Critical Level Lower Limit of Detection Minimum Detectable Activity Type Errors

53. Critical Level, Lc A value at which come percentage (usually 95 %) of measurements made on samples containing no activity will show no activity above the background statistical fluctuations There is a 5% false positive indication of activity This is a level pre-established by the user!

54. Critical Level, Lc Calculated value Determined by counting system Defining Equation: Lc done in advance m is a constant associated with selected confidence level, where for CLs of 90%, 95%, and 99%, m= 1.282, 1.645, and 2.327 respectively.

55. Critical Level, Lc, for 95th percentile Lc done in advance Sample and background times must match those initially selected For 95th percentile:

56. Critical Level, Lc, for 95th percentile - Can be expressed in cpm dpm If sample and counting times are equal, then simplifies to:

57. Lower Limit of Detection (LLD) LLD is smallest sample activity that will produce a net count rate Which will be detected as being positive 95 percent of the time Only 5 percent will be counted with a false negative result For this to occur the LLD level is set one std deviation of the LLD value times the appropriate confidence level factor above the critical level

58. LLD, continued There are two 95% confidence levels associated with LLD 5% risk of obtaining a false positive on a sample with no activity 5% risk of getting a false negative on a sample with activity equal to the LLD Represents a combination of tow 95% confidence levels

59. LLD, continued Represented by replacing m in critical level equation with 2m and defining LLD as

60. LLD, continued LLD usually defined at 95 confidence interval, so can be specified as Imposing the condition that tg = tb gives

61. Minimum Detectable Activity Often used interchangeably with LLD May include terms for Collection efficiency Yield Decay Be careful!

62. Type Errors How to best state likelihood of activity in a sample Type I � probability of rejecting a true null count (no activity) when none is there (false positive) Type II � probability of rejecting activity in a sample when it is there (false negative)

63. Type Errrors Related to LLD through Where ka value for the upper percentage of the standardized normal variate corresponding to the arbitrarily chosen risk for falsely concluding that activity is present (Type I error) kb value for the arbitrarily chosen degree of confidence for detecting the presence of activity (1-Type II error) snstandard error for net sample activity

64. Type errors continued The standard deviation (error) for the net sample activity is If background and count rates are equal then

65. Type errors continued Combing the two previous equations yields the results If ka = kb = m then the LLD can be expressed as

66. Type errors continued For the 95% confidence interval (m=1.645) this leads to the result that

67. Test of Data Sets One can determine if the variations showed when successive measures of the same quantity with an apparatus can be due to other factors than nuclear radiation. The method is called �Chi-Squared Test� and it allows the evaluation of the probability that a set of data follows the normal distribution.

68. Test of Data Sets [cont�d] One defines the quantity:

69. Chi-Square Goodness of Fit test Determines, on basis of random sample from a population, the extent to which you can assume a distribution for the population Null hypothesis � experimental mean and observed values are from same distribution (e.g., normal) The test is distribution free (applies to all distributions) The degrees of freedom = n-1

70. Chi-square, continued Generally expect measurements to deviate from the mean by the standard deviation Expect ?2 to approximately equal n, the number of data points If chi-square is "a lot" bigger than expected something is wrong. For radioactive decay statistics one key assumption must be met � measured count values must be greater than 5

71. References Turner, J. E., Atoms, Radiation, and Radiation Protection, 2nd Ed., John Wiley&Sons , Inc.(1995) Tsoulfanidis, N., Measurement and Detection, Hemisphere Publishing Corp. (1983) Bevelacqua, J.J., Basic Health Physics, Wiley Interscience, (1999).