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# ME LAB III ME 305

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This 20 page Class Notes was uploaded by Diamond Franecki on Monday September 28, 2015. The Class Notes belongs to ME 305 at Old Dominion University taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/215326/me-305-old-dominion-university in Mechanical Engineering at Old Dominion University.

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Date Created: 09/28/15

ME305 THERMO FLUIDS LAB Basic concepts in Statistics Statistics It is a mathematical science pertaining to the collection analysis interpretation and presentation of data Basic terminology Arithmetic mean E Defined as the total sum of all the measurements in the set divided by the number of measurements in the set 1 N XWxz39 where N number of readings xi each reading There is only one average for a set of measurements Median Defined as the measurement that occurs in the middle when a given set of data is arranged in an increasing order Median for a set of data having odd number of measurements is the middle measurement when arranged in an increasing order Median for a set of data having even number of measurements is the average of the two middle measurements when arranged in an increasing order Mode In a given set of data the measurement that occurs the most number of times is called the mode A given set of data can have more than one mode Deviation d It is the difference between the reading xi and the mean E of the set of measurements from which the reading is taken dxic Range R It is the difference between the lowest reading and the highest reading from a given set of measurements Variance S2 or 02 Variance of a set of N samples measurements is the ratio of the sum of the squared deviations of the measurements to N1 N 20939 x2 2 i1 N l Standard deviation 8 or 0 It is given by SJ iur f Consider readings from a pressure Trial No Pressure output Pa gauge 1 10 2 99 3 101 4 97 5 102 6 100 7 102 8 101 9 103 10 102 1 99 12 101 13 100 14 1005 15 102 Arithmetic mean or average for the above set of readings is E 11515095100633 Pa Median Number of readings 15 odd Median 1 1 Pa arrange in increasing order Mode Reading that occurs the most number of times 1amp2Pa Deviation d15 103 100633 d15 02367 Pa Range R R103 97 R06Pa Reading No Pressure output Pa 1 97 2 99 3 99 4 100 5 100 6 100 7 1005 8 101 9 101 10 101 11 102 12 102 13 102 14 102 15 103 Variance 82 N Elm 302 2 2 i1 N l S2 2 03323 15 1 52 0023731 Standard Deviation S S 0154 Pa Empirical rule for a Gaussian distribution i The interval E S to 6 S contains approximately 68 of the measurements as shown below fig a For the above example 68 of the measurements will be contained between 100633 0154 and 100633 0154 Relative l lrequency Approximately 6800 1 of the measurements I 1 I f I x lSlvSl a The interval Tr s to E s ii The interval 672 to 2S contains approximately 95 of the measurements as shown below fig b For the above example 95 of the measurements will be contained between 100633 20154 anol 100633 20154 Relative freq uency Approxmately 95 ol the measurements 139 25 lo 25 l b Yhe Interval 7 25 01 25 iii The interva73Sto x3S contains approximately 99 of the measurements as shown below fig 2 For the above example 99 of the measurements will be contained between 100633 30154 and 100633 30154 Relative Aquot or lrequency nearly aquot o 5 measurements 35 35 c The Interval 7 35 lo 7 35 Method of Least Squares The equation of a straight line is YmXb where m is the slope and b is the intercept From least square fit we have NinyiinZyi m i1 N 139le i1 Nfoin2 Zyiin2inyiin NZXZ392ZX12 b Example Obtain y as a linear function of x using the method of least squares Soution We seek an equation of the form ymxb We calculate the following quantities Yi Xi 12 10 20 16 24 34 35 40 35 52 Yi Xi XiYi x 12 10 12 10 20 16 32 256 24 34 816 1156 35 40 140 160 35 52 182 2704 Zy126 in152 inyi4476 fo5616 Thus NZ xiyi Z xix y Nle2 2x02 m 54476 152126 55816 1522 m 0540 inyiin Nix 4392 b 1265816 4476152 55816 1522 b 0879 Thus the desired relation is y 0540x 0879 A plot of this relation and the data points from which it was derived is shown in the accompanying figure 4 i o y 2 3owom l

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