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I Introductory Material A. Mathematical Concepts

I Introductory Material A. Mathematical Concepts. Scientific Notation and Significant Figures. Scientific Notation A method devised to express very small and very large numbers. Based on powers of 10 e.g 123 billion is 123,000,000,000 expressed in scientific notation = 1.23 x 10 11

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I Introductory Material A. Mathematical Concepts

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  1. I Introductory MaterialA. Mathematical Concepts Scientific Notation and Significant Figures

  2. Scientific Notation • A method devised to express very small and very large numbers. • Based on powers of 10 e.g 123 billion is 123,000,000,000 expressed in scientific notation = 1.23 x 10 11 • 1.23 is called the coefficient • It must be greater than 1 and less than 10 • 10 11 is called the base. It is a power of 10. In this example the power of 10 is the 11

  3. Significant Figures • Tells how well known a measurement is. • Using Significant Figures is a way to express the error in a measurement. • The more significant figures the more accurate the measurement. • The last number listed is the value that is uncertain • e.g. • 2.3 value known to units, estimated to tenths • 2.34 value known to tenths, estimated to hundredths

  4. Determining the number of Significant Figures • All non - zero numbers are significant • Zeros within a number are significant • Both 40.05 and 3201 have 4 significant figures • Zeros used to set the decimal point are NOT significant. • 0.00235 • 3 significant figures • 47,000 • 2 significant figures • If you want it to have more, use scientific notation: 4.7000 x 104 denotes 5 significant figures • Other possibility: 47,000. is sometimes used to denote 5 significant figures.

  5. Types of Numbers • Exact • Numbers from counting • Conversions in the same system of measurement • Defined conversions between different systems of measurement • Have an infinite number of significant figures • Inexact • Numbers from measurements • Conversions between different systems of measurement • Have a finite number of significant figures

  6. Working with Significant Figures • When doing calculations, you can not gain or lose precision. The answer can only be as good as the least precise number • Multiplication and Division • The number of significant figures in the answer depends on the number with the least number of significant digits. • Determine the number of significant figures in: (16.300)(0.00235)/1.6300 • can only have 3 digits in the answer.

  7. Determine the correct answer for:(4.10)(3.023 x 109)/(1.5) • 8.26 x 109 • 8.3 x 109 • 8.263 x 109 • 8.2628 x 109

  8. Addition and subtraction • The number of significant figures in the answer depends on the number with the least number of significant decimal places. • Determine the answer: 4.72 + 0.653 + 9.2 • 9.2 is only known to the tenths place, so the answer can only be known to the tenths place 4.72 0.653 9.2 14.573 14.6

  9. What is the sum of 9 x 103 + 7.2 x 102? A) 1 x 104 B) 9.7 x 103 C) 1.0 x 104 D) 9.72 x 103

  10. 9 x 103 known only to the thousands place 7.2 x 102 known to the tens place, so answer can only be known to the thousands place. 9000 720 9720 rounds to 10000 10 000 How many significant digits do you have? 2 10. x 103 = 1.0 x 104

  11. Rounding • When the answer contains too many digits, (more than the number of significant digits) the answer must be rounded to the correct number of significant digits. • General Rules: If the digit after the last significant digit is • 0 - 4, the significant digit stays the same • 15.635 to the tenths place is 15.6 • 6 - 9, the significant digit goes up • 15.625 to the units place is 16 • 5, the significant digit goes to an even number • 15.625 to the hundredths place is 15.62 • While 15.635 to the hundredths place is 15.64 • The reason for this is that in a series of calculations any error will be averaged out.

  12. Error Analysis • Types of Error • Determinate Error (ED) • Source of the error is known and can be determined. • e.g. a watch that is ten minutes fast • Determinate error will most often effect accuracy • It is the difference between the true value and the measured value • Indeterminate Error (EI) • Source of the error is unknown or not controllable • e.g. the vibration in a balance table • Indeterminate error will most often effect precision • Can be expressed as: • Standard deviation • Variance • Coefficient of Variation • Tolerances

  13. Total Error • Sum of all the determinate and indeterminate error • If the true value is known, this can be expressed as the percent relative error: %ER • Describing Indeterminate Error • Population • Complete set of individual data points in a study • The female population of the United states • Sample • A specific subset of the population that is assumed to be representative of the whole • The female population of Newark, De. • Members of the Data Set • Individual items (xi) within the sample of a known number (N) of items

  14. Measuring the Central Tendency of a Population or Sample • Mean • Average value of all the members in the set • A population mean is the true or expected value; it is represented by the symbol µ • A sample mean is the calculated average of the sample population; it is represented by the symbol x • When a sample is used, we assume that x is a good approximation of µ • Median • The middle value of all the members in the set • Mode • The value that appears with the greatest frequency in the set

  15. Measuring the Spread of Data • Range • Difference between the highest (largest) value and the lowest (smallest) value in the set • Range = xHi - xLo • Standard deviation(s for population; s for sample) • Measure of how closely the individual values are clustered around the mean • Variance • Square of the standard deviation • Variance = s2

  16. Accuracy • Closeness of population mean and sample mean • Precision • Closeness of individual data points to each other • Measured by range or standard deviation

  17. Determining and Describing Indeterminate Error (Uncertainty Statistics) • Standard deviation Population Sample • Where xi is the data point, x is the sample mean, n is the number of data points • Variance • s2

  18. Relative standard deviation(RSD) and Coefficient of Variation (CV) • Coefficient of Variation is a measure of the dispersion of a distribution • It allows us to compare populations or samples that have different mean values • Standard deviation of the mean,sm

  19. Collection of 10 data points • Ways of Representing Data Data Table

  20. Histogram a tabulation of the frequency of each measurement

  21. How do you decide if all you data should be used or is some can be ignored? • Sort the data from low to high Look at the Range of the data: 114.4 - 121.3 Difference = 6.5

  22. Apply Q Test Look at the gap between the smallest two numbers (Low Gap) and the largest two numbers (High Gap). Low Gap = 118.2 - 114.8 = 3.4 High Gap = 121.3 - 121.2 = 0.1 1. Calculate the Gap to Range Ratio

  23. If Qcalc < Qtable data CAN NOT be rejected 2. Compare Qcalc to Q from the table (90% confidence) Qlo = 0.523 Qhi = 0.016 Qtable = 0.41 for 10 data points So: Qlo > Qtable so can reject 114.8 Always make sure you mention the method used to reject data Calculate mean, deviation and RSD x = 119.8; s = 1.1; RSD = 0.009

  24. (X - Xi)2 X - Xi Data now consists of 9 points. x = 119.8

  25. x = 119.8 s = 1.1 RSD = s/ x = 0.009 68.3 % of data in this range 119.8 ± 1.1 119.8 ± 1  95.8 % of data in this range 119.8 ± 2  99.7 % of data in this range 119.8 ± 3 

  26. Confidence Limit • How certain you are that the true value [population mean] (µ) lies within a range around the sample mean(x) • Confidence Limits are usually calculated at 90%, 95% or 99% • Confidence Limits are calculated using the Student t table

  27. DF = degrees of freedom It is equal to n - 1 Where n is the number of data points To calculate the Confidence Limit From our data, for the 9 data points x = 119.8; s = 1.1; DF = 8, t = 2.31

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