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The sample median is a measure used in descriptive statistics to identify the middle point of a dataset when it is arranged in order. It is particularly useful when you want to understand the central tendency of your data in a way that is not skewed by extreme values.
To determine the median, arrange your dataset in ascending order. For datasets with an odd number of observations, the median is the middle value. However, when there is an even number of observations, like in our example with fracture strength measurements, the median is calculated as the average of the two middle numbers.
Here are some important points to remember:

• For 10 observations in ascending order (87, 93, 96, 98, 105, 114, 128, 131, 142, 168), locate the 5th and 6th values: 105 and 114.
• The sample median is the average of these two values: \( \frac{105 + 114}{2} = 109.5 \).

This value tells us that halfway through the data points, you would find the value 109.5. This provides a central reference point for the distribution of the dataset.

The sample mean is another measure of central tendency that gives us the average value of the data. It is calculated by adding all the data points together and then dividing by the number of observations. This method provides a quick way to understand the overall level of the data.
The steps to calculate the sample mean in our fracture strength example are as follows:

• Add all observations: \( 87 + 93 + 96 + 98 + 105 + 114 + 128 + 131 + 142 + 168 = 1162 \).
• Divide the sum by the number of observations: \( \frac{1162}{10} = 116.2 \).

The mean value of 116.2 MPa indicates that, on average, the fracture strength observations center around this point.
However, it's worth noting that the mean can be affected by outliers, or data points that are significantly different from others in the dataset. But, in many cases, it provides a useful summary of the dataset as a whole.

The sample standard deviation provides insight into the spread or dispersion of data points around the mean. It is a crucial concept in statistics that tells us how much the values in the dataset deviate from the average value. A higher standard deviation indicates more variability among the data points, whereas a lower one suggests they are closely clustered around the mean.
Here’s how to calculate the sample standard deviation:

• First, calculate the mean: from our calculations, it is 116.2.
• Compute the deviation of each observation from the mean and square it.
• Sum these squared deviations to get the total squared deviation.
• Finally, divide by the number of observations minus one (the denominator "N - 1" is used to provide an unbiased estimate), and take the square root of the result.

In this exercise:

• The sum of squared deviations equals 5987.00.
• The variance is calculated as \( \frac{5987.00}{9} \approx 664.11 \).
• The sample standard deviation thus comes out as \( \sqrt{664.11} \approx 25.77 \).

Understanding the standard deviation helps you to gauge how consistent the fracture strength measurements are. In this case, a standard deviation of 25.77 MPa suggests a moderate level of variability in the strength of the fractures studied.