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The mean is the average value of a data set and offers insight into the overall trend of the data. In the case of our Dow Jones Industrial Average exercise, we are interested in finding the mean price per share for the 30 companies listed. The process begins with determining "class midpoints," which is essentially the average of the boundaries of a class interval. For example, for the price range \(0-9\), its midpoint is \((0 + 9)/2 = 4.5\). This is repeated for each price range. The next step involves calculating a "weighted mean," where each midpoint is weighted by the number of companies in that respective range. The formula used is \(\bar{x} = \frac{\sum f_i x_i}{\sum f_i}\), where \(f_i\) is the frequency and \(x_i\) is the midpoint. By multiplying each midpoint by its frequency, summing these products, and dividing by the total count of companies (30), we derive the mean of approximately \(31.22\). This mean tells us about the central tendency of the data, which is useful in understanding the typical price per share.

While the mean gives us an average, the standard deviation provides more depth by measuring how spread out the values are around the mean. It's a key metric for understanding the variability in your data. In our exercise, after computing the mean, we focus on calculating the standard deviation to gauge how much the share prices deviate from that mean value. The calculation begins with finding the deviation of each class midpoint from the mean, squared to eliminate negative values. This is represented as \((x_i - \bar{x})^2\). Each squared deviation is then multiplied by the class frequency, \(f_i\), summed across all classes, and averaged by dividing by the total frequency (30), resulting in the variance, \(\sigma^2 = 533.0\). The square root of this variance gives the standard deviation, \(\sigma \approx 23.08\). This larger standard deviation, compared to the previous value from January 2006, shows us that there is greater variability in the price of the shares over this period. A higher standard deviation suggests that the prices are more spread out.

Understanding data variability is crucial as it reveals how much individual data points differ from each other and from their mean. This aspect becomes apparent when comparing different time periods. In our exercise, the Dow Jones Industrial Average companies' stock prices show increased variability over three years. While the mean price gives an average snapshot, the variability—measured in part by the standard deviation—tells the story of consistency or fluctuation within data sets. For January 2006, a standard deviation of \(18.14\) indicated a certain level of spread around the mean of \(\\(45.83\). Fast forward three years, and the increase in standard deviation to \(23.08\) suggests more significant fluctuations in stock prices. Thus, variability helps in understanding the financial stability or volatility of these companies over time. In summary, while a decrease in mean price per share from \(\\)45.83\) to \(\$31.22\) shows an overall reduction, the increased variability implies that share prices have become less predictable, which could be a sign of changing market dynamics.