The following table gives a number of Rockwell B hardness values that were measured on a single steel specimen. Compute average and standard deviation hardness values. $$ \begin{array}{lll} 83.3 & 80.7 & 86.4 \\ 88.3 & 84.7 & 85.2 \\ 82.8 & 87.8 & 86.9 \\ 86.2 & 83.5 & 84.4 \\ 87.2 & 85.5 & 86.3 \end{array} $$

Material hardness refers to a material's resistance to local deformation, particularly indentations or scratches. One common method to assess this property is Rockwell hardness testing. In Rockwell hardness measurements, a diamond or hard metal indenter is pressed into the material under a large load, followed by a lower load. The depth of penetration under the lower load is measured, and this depth is converted into a hardness number; the shallower the indentation, the harder the material.

Rockwell tests are differentiated by scales, with the Rockwell B scale, as used in the given exercise, involving a 1/16 inch diameter steel ball indenter with a 100 kg load. The Rockwell hardness number is read directly from the testing machine. These hardness values provide a quick reference to determine a material's suitability for various applications such as machining, wear resistance, and structural support.

It's crucial for the measurements to be accurate as they can influence material selection in engineering design. Materials with higher hardness values are preferred for high wear-resistant applications, while softer materials might be selected for applications where ease of machining is a priority.

Standard deviation is a statistical measure that quantifies the variation or dispersion of a set of numerical data. In the context of material hardness measurements, standard deviation provides insight into the consistency and quality of the material. A lower standard deviation indicates that the hardness values are closely clustered around the mean, suggesting uniform material properties. Conversely, a high standard deviation could indicate a wide range of hardness values, pointing to potential inconsistencies in material uniformity or quality.

The standard deviation is particularly useful in quality control and assurance procedures. It helps to ensure that a material's properties meet the required specifications for its intended application. In the provided Rockwell B hardness testing example, the calculation of standard deviation involved comparing each individual measurement to the mean and then deriving an average squared deviation (variance). The square root of the variance gives us the standard deviation. Therefore, a standard deviation of approximately 2.70 in the example indicates that the majority of the hardness values for that steel specimen are within ±2.70 units of the mean hardness value.

Statistical analysis is a foundational tool in material science, used to evaluate and interpret the variability and performance of materials under various conditions. It involves collecting data, such as the hardness values from Rockwell hardness testing, and applying statistical methods to summarize and gain insights from the data set.

In materials science, statistical analysis can predict material behavior, assess quality control, and guide the development of new materials. For example, reliability analyses may use statistical methods to ascertain the likelihood of material failure under expected service conditions. Additionally, regression analysis can help understand how various factors, such as temperature or load, impact material properties.

In the context of the exercise provided, statistical analysis was employed to determine the mean and standard deviation of hardness measurements. Knowing that there are variations in the measurements, these analyses help ensure that the variations are within acceptable limits for the material's intended application. By using these tools, materials scientists and engineers can make informed decisions about material selection, processing methods, and quality assurance protocols.