DATA & STATISTICS 101 Presented by Stu Nagourney NJDEP, OQA

Precision, Accuracy and Bias 4 Precision: Degree of agreement between a series of measured values under the same conditions 4 Accuracy: Degree of agreement between the measured and the true value 4 Bias: Error caused by some aspect of the measurement system

Precision, Accuracy and Bias

Sources of Error 4 Systematic Errors: Bias always in the same direction, and constant no matter how many measurements are made 4 Random Errors: Vary in sign and are unpredictable. Average to 0 if enough measurements are made 4 Blunders: The occasional mistake that produces erroneous results; can be minimized but never eliminated

Applying Statistics 4 One cannot sample every entity of an entire system or population. Statistics provides estimates of the behavior of an entire system or population, provided that: –Measurement system is stable –Individual measurements are all independent –Individual measurements are random representatives of the system or population

Distributions 4 Data generated by a measurement process generally have the following properties: –Results spread symmetrically around a central value –Small deviations from the central value occur more often than large deviations –The frequency distribution of a large amount of data approximates a bell-shaped curve –The mean of even small sets of data represent the overall better than individual values

Normal Distribution

Other Distributions

Issues with Distributions 4 For large amounts of data, distributions are easy to define. For smaller data sets, it is harder to define a distribution. 4 Deviations from normal distributions: –Outliers that are not representative of the population –Shifts in operational characteristics that skew the distribution –Large point-to-point variations that cause broadening

Estimation of Standard Deviation 4 The basic parameters that characterize a population are –Mean ( ) –Standard Deviation ( ) 4 Unless the entire population is examined, and cannot be known. They can only be estimated from a representative sample by –Sample Mean (X) –Estimate of Standard Deviation (s)

Measures of Central Tendency & Variability 4 Central Tendency: the value about which the individual results tend to cluster 4 Mean: X = [X 1 + X 2 + X 3 + … X n ] / n 4 Median: Middle value of an odd number of results when listed in order 4 s = [ (X i - X) 2 / n-1] 1/2

Measures of Central Tendency & Variability

Statistics 4 If you make several sets of measurements from a normal distribution, you will get different means and standard deviations 4 Even the best scientist and/or laboratory will have measurement differences when examining the same sample (system) 4 What needs to be defined is the confidence in measurement data and the significance of any differences

Estimation of Standard Deviation

Does a Measured Value Differ from an Expected Value? 4 Confidence Interval of the Mean (CI) : The probability where a sample mean lies relative to the population mean 4 CI = X ± (t) (s) / (n) 1/2 : value of t depends upon level of confidence desired & # of degrees of freedom (n-1)

Does a Measured Value Differ from an Expected Value?

Criteria for Rejecting an Observation 4 One can always reject a data point if there is an assignable cause 4 If not, evaluate using statistical techniques 4 Common Outlier Tests –Dixon (Q) Test –Grubbs Test –Youdon Test –Student t Test

Criteria for Rejecting an Observation: Dixon (Q) Test

Control Charts