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Week 2: Estimation to Basic Statistics

by: Justin Bartell

Week 2: Estimation to Basic Statistics ECH 3023

Marketplace > Florida State University > Engineering > ECH 3023 > Week 2 Estimation to Basic Statistics
Justin Bartell
GPA 3.977
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About this Document

Notes from Wednesday (9/7) lecture: discusses techniques for estimation and answer checking, and statistic techniques: mean, standard deviation, best fit line.
Mass & Energy Balances I
Dr. Yeboah
Class Notes
Math, Engineering




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This 3 page Class Notes was uploaded by Justin Bartell on Monday September 5, 2016. The Class Notes belongs to ECH 3023 at Florida State University taught by Dr. Yeboah in Fall 2016. Since its upload, it has received 5 views. For similar materials see Mass & Energy Balances I in Engineering at Florida State University.


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Date Created: 09/05/16
ECH 3023– Y. Yeboah Week 2 – 02 (Wednesday Lecture 9/7/2016) Chapter 2: From Estimation to Basic Statistics I. Objectives a. Identify commonly used units and convert them (conversion factors) b. Identify number of significant figures c. Validate a quantitative problem solution d. Calculate sample mean, range, variance, and standard deviation e. Explain concept of dimensional homogeneity and dimensionless quantities f. Do interpolation and extrapolation g. Curve fit data II. Review a. A measured or counted quantity has a value and a unit.  b. Base units: mass, length, time, temperature, electrical current, light intensity c. Multiple units: Terra (10^12), giga (10^9), mega (10^6), micro (10^­6), nano  (10^­9) d. Derived units: Volume, Force, Pressure, Energy/Work, Power e. Questions: i. What is the derived SI unit for velocity? The velocity unit in the CGS  system? In the American engineering system? m/s, cm/s, ft/s ii. Convert 23 (lb_m*ft) / (min)^2 to its equivalent in kg*cm/s^2 [23 lb_m*ft/min^2] * [0.453592 kg / 1lb_m] * [100cm / 3.281 ft.]  * [1^2 min^2 / 60^2 s^2] = 0.088 kg*cm/s^2 (2 sig figs. Conversion factors found in the book.) f. Number of SF provide indication of precision in the data. i. When 2 or more quantities are combined by multiplication and/or division, the number of SF in the result should equal the lowest number of SF of  any of the multiplicands or divisors. ii. When 2 or more numbers are added or subtracted, the positions of the last  SFs of each number relative to the decimal point should e compared the  farthest to the left is the position of the last permissible SF of the sum or  difference. iii. Always round­off number to be EVEN if the digit to be dropped is 5. 1.35  1.4          1.25  1.2 III. Validating Results a. When I arrive at an answer, how can I know it’s right? Three Techniques: i. Back­substitution: After you solve a set of equations, substitute your solution back into the  equations and make sure it works. ii. Order of magnitude estimation: Before solving, make a crude/easy estimation of the answer (substitute  simple integers for all numerical quantities, using powers of 10 i.e.  scientific notation). Make sure that your final answer comes near this  estimation. Ex. Y = [ 254 / (0.879*62.4) + 13 / (0.866*62.4) ] * 1 / (31.3124 * 60)        Y ~ [250 / (1 * 50) + 10 / (1*50) ]* 1 / ( (3*10) * (6*10) )        Y ~  [5 + 0 ] /  (20 * 10^2)         Y ~ 2.5 * 10^­3         (Exact: Y = 0.00230) iii. Test of reasonableness: Check if the answer makes sense. Ex. If the water flowing in a pipe is  faster than the speed of light, your answer is wrong.  IV. Estimation of Measured Values It is impossible to replicate EXACT experimental conditions and, thus, results  (random error). So, we use a variety of statistical tools in order to reflect precise data  in experiments: a. Sample Mean: X = (x1+x2+x3+…xN)/N [add all numbers and divide by amount of terms (N)] b. Range: x_max – x_min [largest number minus smallest number] c. Sample Variance: S^2 = [(x1­X)^2 + (x2­X)^2 + … (xN­X)^2] / [N – 1] [Subtract each number by mean, square it, then add them all up. Finally, divide by number of terms minus 1] d. Sample Standard Deviation (SD): S = (sqrt)(S^2) [Standard deviation is the square root of variance] For typical random variables (that create a “normal curve” : bell­shaped), ~67% of  measure values fall within 1 SD of the mean; 88% fall within 2 SD and 99% fall  within 3 SD of the mean. Data is usually reported with: average +­ SD  ex. The average is 47.6 and SD is 0.6: 47.6 +­ 0.6  i.e. (47.0 – 48.2)  V. Dimensional Homogeneity and Dimensionless Quantities a. Dimensional Homogeneity: both sides of equation having same dimensions i. All valid equations MUST be dimensionally homogeneous ex. V (m/s) = V0 (m/s) + g(m/s^2) * t (s) ii. However, an invalid equation can be dimensionally homogeneous Ex. M = 2M (except for M=0) b. Dimensional Quantities: groups of numbers having no dimensions i. Ex. Fluid Mechanics 1. Ma = V/c (flow speed / sound speed) VI. Process Data Representation and Analysis a. Calibration b. Interpolation: y = y1 + [ (y2 – y1) / (x2 – x1) ] * (x – x1)  c. Straight Line Fit (linear data) i. y = ax + b   a= (y2­y1)/(x2­x1)  b = y1 ­ a*x1 = y2 – a*x2 (Check by plugging in the point (x1,y1) and (x2,y2) in the equation.) d. Curve Fitting (nonlinear data) i. Much harder, but can sometimes plot data to get straight line. Ex. y^2 = ax^3 + b ; By plotting y^2 vs. x^3, we get a straight line. Ex. siny = a(x^2 – 4) ; y plotting siny vs x^2, we get a straight line. ii. Common nonlinear functions used in process analysis: 1. Exponential: y = ae^(b*x) = a*exp(bx) 2. Power law: y = ax^b  ln(y) = ln(ax^b)  ln(y) = ln(a) + ln(x^b)   ln(y) = ln(a) + b*ln(x) 3. Log plot vs. semilog plot a. Log plot: A plot with logarithmic scales on both axis (ex.  ln(y) vs. ln(x)) i. Therefore y = ax^b b. Semilog plot: A plot with logarithmic scales on one axis  (ex. ln(y) vs. x) i. Therefore, y = a*exp(bx) e. Linear Regression (Least Squares): statistical method of fitting LINEAR data.


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