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Chapter 6: Problem 50

Below are tabulated a number of Rockwell G hardness values that were measured on a single steel specimen. Compute average and standard deviation hardness values. $$\begin{array}{lll} 47.3 & 48.7 & 47.1 \\ 52.1 & 50.0 & 50.4 \\ 45.6 & 46.2 & 45.9 \\ 49.9 & 48.3 & 46.4 \\ 47.6 & 51.1 & 48.5 \\ 50.4 & 46.7 & 49.7 \end{array}$$

Short Answer

Expert verified
Answer: The formulas used to calculate the average and standard deviation of a given set of Rockwell G hardness values for a single steel specimen are as follows: Average: $$\frac{Sum}{N} = \frac{\sum_{i}^{} data_{i}}{N}$$ Standard Deviation: $$\sqrt{\textrm{Variance}}$$, where Variance is calculated as: $$\frac{\sum_{i}^{} (data_{i} - \textrm{Average})^2}{N}$$

Step by step solution

01

Gather all hardness values in a list

Compile a list of all the hardness values: $$data = [47.3, 48.7, 47.1, 52.1, 50.0, 50.4, 45.6, 46.2, 45.9, 49.9, 48.3, 46.4, 47.6, 51.1, 48.5, 50.4, 46.7, 49.7]$$
02

Calculate the sum and the number of values

Compute the sum of all hardness values and the number of values in the list: $$Sum = \sum_{i}^{} data_{i} = 47.3 + 48.7 + \cdots + 49.7$$ $$N = \textrm{number of values} = 18$$
03

Compute the average

Calculate the average of the hardness values: $$\textrm{Average} = \frac{Sum}{N} = \frac{\sum_{i}^{} data_{i}}{18}$$
04

Calculate the variance

Compute the variance of the hardness values: $$\textrm{Variance} = \frac{\sum_{i}^{} (data_{i} - \textrm{Average})^2}{N}$$
05

Compute the standard deviation

Calculate the standard deviation of the hardness values: $$\textrm{Standard Deviation} = \sqrt{\textrm{Variance}}$$

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Most popular questions from this chapter

A steel alloy to be used for a spring application must have a modulus of resilience of at least \(2.07 \mathrm{MPa}(300 \mathrm{psi}) .\) What must be its minimum yield strength?

A cylindrical specimen of a hypothetical metal alloy is stressed in compression. If its original and final diameters are 30.00 and \(30.04 \mathrm{mm}\) respectively, and its final length is \(105.20 \mathrm{mm}\) compute its original length if the deformation is totally elastic. The elastic and shear moduli for this alloy are 65.5 and $25.4 \mathrm{GPa}, respectively.

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