6 Uncertainties in Measurement and Propagation of Errors Inserting the data ofTa

Question

6 Uncertainties in Measurement and Propagation of Errors Inserting the data ofTable 1-1 into eqs. 1-3 and 1-4 yields x-21.00 mm and .-0.23 mm. Notice that the spread of the histogram in Fig. 1-2 gives an idea of the precision of the mea- surements. We have observed that the standard deviation is also a measure of the spread of results, so the standard deviation and the histogram width must be related To make the connection between the standard deviation and the spread more quantitative, we need to digress to briefly discuss the Gaussian or normal distribution. Ifwe increase the number of observations and decrease the interval sizes in histograms such as Fig. 1-2, we can imagine that the rough histogram would approach a smooth curve. In most simple physical situations this curve is given by the so-called Gaussian curve shown in Fig. 1-3 For this curve, the width at 61% of the height is just twice the standard deviation. Returning to the example of Fig. 1-2, let us roughly take the spread to be the combined width of the 5 central intervals, or 0.5 mm. For a perfect Gaussian curve, this width would be equal to twice the standard deviation, so estimating the standard deviation from the "spread gives 0.5 mm/2 - 0.25 mm, which is very similar to the above computed 0.23 mm. For curves such as Fig. 1-3, the number of measurements in a given interval is given by the area under the curve for that interval. The area under the entire curve is the total number of measurements. The area under the curve between the limits of + and-o of the mean turns out to be 68% of the total number of measurements. Another way to express this important result is as fol- lows: There is a 68% probability that a measurement will fall within trof the mean. 20.4 20.6 20.8 21 212 21.4 21.6 Cylinder Diameter, mm Figure 1-3. The normal or Gaussian distribution that approximates the results in Fig. 1-2. The curve has width of twice the standard deviation. Exercise 1-8. Find the total number of measurements within the 5 central intervals of Fig. 1-2 by using the data of Table 1-1. Express your result as a percentage of the total number of measurements 1 This width is measured at 61% of the height of the curve.

Explanation / Answer

I need the aforementioned Table 1-1 to solve this problem. Please upload the table.

Related Questions

Navigate

• Browse (All)
• Browse 6
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.