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General Statistics Ch En 475 Unit Operations

This guide explores the concepts of measured and calculated variables in unit operations, focusing on error assessment and statistical methods to quantify uncertainty. It covers the determination of errors in measured variables, the significance of sample size, and the implications of sampling distributions. Additionally, we discuss how to compare means from different datasets using t-tests and how to utilize Excel for statistical analysis. Understanding these principles is essential for accurately interpreting experimental data and ensuring confidence in measurement results.

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General Statistics Ch En 475 Unit Operations

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  1. General Statistics Ch En 475 Unit Operations

  2. Quantifying variables (i.e. answering a question with a number) • Directly measure the variable. - referred to as “measured” variableex. Temperature measured with thermocouple • Calculate variable from “measured” or “tabulated” variables - referred to as “calculated” variable ex. Flow rate m = r A v (measured or tabulated) Each has some error or uncertainty

  3. Outline • Error of Measured Variables • Comparing Averages of Measured Variables

  4. Some definitions: x = sample mean s = sample standard deviation m = exact (or true) mean s = exact standard deviation As the sampling becomes larger: x  m s  st chart z chart not valid if bias exists (i.e. calibration is off) 1. Error of Measured Variable • Several measurements • are obtained for a • single variable (i.e. T). • What is the true value? • How confident are you? • Is the value different on • different days? Questions

  5. small large (n>30) How do you determine the error? • Let’s assume “normal” Gaussian distribution • For small sampling: s is known • For large sampling: s is assumed we’ll pursue this approach Use t tables for this approach Don’t often have this much data Use z tables for this approach

  6. Example

  7. Standard Deviation Summary (normal distribution) 40.9 ± (3.27) 1s: 68.3% of data are within this range 40.9 ± (3.27x2) 2s: 95.4% of data are within this range 40.9 ± (3.27x3) 3s: 99.7% of data are within this range • If normal distribution is questionable, use Chebyshev's inequality: • At least 50% of the data are within 1.4 s from the mean. • At least 75% of the data are within 2 s from the mean. • At least 89% of the data are within 3 s from the mean. Note: The above ranges don’t state how accurate the mean is - only the % of data within the given range

  8. Student t-test (gives confidence of where m (not data) is located) measured mean 5% 5% t true mean a=1- confidence r = n-1 = 6 2-tail Remember s = 3.27

  9. t-test in Excel • The one-tailed t-test function in Excel is: =T.INV(,r) • Remember to put in /2 for tests (i.e., 0.025 for 95% confidence interval) • The two-tailed t-test function in Excel is: =T.INV.2T(,r) Where •  is the probability • (i.e, .05 for 95% confidence interval for 2-tailed test) and • r is the value of the degrees of freedom

  10. T-test Summary • = exact mean 40.9 is sample mean 40.9 ± 2.4 90% confident m is somewhere in this range 40.9 ± 3.095% confident m is somewhere in this range 40.9 ± 4.6 99% confident m is somewhere in this range

  11. Attention Span Modifier

  12. Outline • Error of Measured Variables • Comparing Averages of Measured Variables

  13. Comparing averages of measured variables Experiments were completed on two separate days. When comparing means at a given confidence level (e.g. 95%), is there a difference between the means? Day 1: Day 2:

  14. Comparing averages of measuredvariables New formula: Larger |T|: More likely different Step 1 For this example, Step 2 r = nx1+nx2-2

  15. Comparing averages of measured variables At a given confidence level (e.g. 95% or a=0.05), there is a difference if: Step 3 2-tail T t 2.5 > 2.145 95% confident there is a difference! (but not 98% confident)

  16. Example (Students work in Class) • Calculate and s for both sets of data • Find range in which 95.4% of the datafall (for each set). • Determine range for m for each set at 95% probability • At the 95% confidence level are the pressures different each day?

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