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The mean, or average, of a set of data is one of the most fundamental concepts in statistics. It provides us with a single value that summarizes the entire dataset. A mean is calculated by adding up all the values in a dataset and then dividing by the number of values.In our original exercise, the mean weight of the cattle is 1152 pounds. When data is shifted uniformly, as in subtracting 1000 pounds from each cattle's weight, the mean also shifts by the same amount without affecting the distribution of the data relative to the mean. The mean is sensitive to every value in the dataset, making it a useful tool but also susceptible to outliers. However, in the example given, there are no outliers mentioned, so the mean gives us a good central tendency of the cattle weights.Let's say we have five cattle weighing 1100, 1200, 1150, 1050, and 1200 pounds. To calculate the mean, we sum the weights: 1100 + 1200 + 1150 + 1050 + 1200 = 5700 pounds. Since there are five cattle, we divide by 5, which gives us a mean weight of 1140 pounds. This process illustrates how every value contributes to the overall average.

Standard deviation is a measure of dispersion or variability in a dataset. It tells us, on average, how much each data point differs from the mean of the data set. A smaller standard deviation indicates that the data points are closer to the mean, while a larger standard deviation points to data that are more spread out.In this case, the original standard deviation of the cattle weights is 84 pounds. When we shift all weights by subtracting 1000 pounds, the standard deviation remains unchanged at 84 pounds. This is because standard deviation is concerned with the relative distance of each data point from the mean, which does not change when you add or subtract a constant to all values.In practice, calculating standard deviation involves several steps: finding the mean, calculating the difference between each data value and the mean, squaring those differences, averaging those squared differences, and then taking the square root of that average. For instance, if the weights of the cattle were identical to the mean provided, the standard deviation would be zero, indicating no variability among the cattle's weights.

Data scaling involves adjusting the range of data values using multiplication or division, which alters the scale but not the shape of the data distribution. When you sell cattle by the pound, for example, the price per pound scales the data. If the original mean weight is 1152 pounds and the price is \(0.40 per pound, we multiply the mean weight by the price to find the mean sale price.In data scaling, the new mean is calculated by multiplying the original mean by the scaling factor, resulting in a new mean of \)460.8 in this case. Similarly, the standard deviation is also scaled by the same factor, leading to a new standard deviation of $33.6. It's important to understand that scaling changes the units of the data, but it does not change the overall shape of the distribution; it's simply a change of scale. In the cattle sale example, we're changing the context from weight to sale price, providing useful insights for economic analysis but keeping the statistical characteristics relative to each other.