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The mean, often called the average, is a measure that summarizes the central location of a dataset. To compute the mean of the firefighter fatalities data, we sum up all the values and divide by the total number of values. Here, the dataset is given as \([20, 18, 23, 30, 20, 12, 24, 9, 25, 15, 8, 11, 15, 34]\). The sum of these numbers is \(264\), and there are \(14\) numbers in total. Thus, the mean is calculated as follows:

\( \text{Mean} = \frac{264}{14} \approx 18.86 \)

This mean tells us that, on average, about \(18.86\) firefighters lost their lives each year over this period. However, it doesn't tell us about the variability or spread of the data.

The median represents the middle value of a dataset when it is ordered from least to greatest. To find the median, we first sort the given data to get \([8, 9, 11, 12, 15, 15, 18, 20, 20, 23, 24, 25, 30, 34]\). Since there are an even number of values (14), the median is the average of the two middle values (the 7th and 8th values). Therefore, we calculate it as:

\( \text{Median} = \frac{18 + 20}{2} = 19 \)

The median in this case is \(19\), suggesting that half of the years recorded fewer than \(19\) firefighter deaths, and half recorded more. The median provides a good measure of the center that is less affected by extreme values or outliers.

The mode of a dataset is the value or values that appear most frequently. For our firefighter fatality data, when sorted \([8, 9, 11, 12, 15, 15, 18, 20, 20, 23, 24, 25, 30, 34]\), we see that the numbers \(15\) and \(20\) each appear twice. Therefore, both of these numbers are modes:

\( \text{Mode} = 15, 20 \)

In situations where there are multiple modes, the data is called bimodal. The mode can provide insights into how certain values are more common than others in the dataset. However, it doesn’t give a clear picture of the overall data distribution.

Dispersion in statistics refers to the spread of values in a dataset around its central value. Common measures of dispersion include variance and standard deviation. In our example data, measures like the mean, median, and mode do not show how the data points are spread out. For example, to calculate the variance and standard deviation, we need to measure how much each value deviates from the mean.

A measure of dispersion provides insight into the variability within the dataset. In the given data, a higher dispersion means values are spread out, while a lower dispersion indicates they are clustered close to the mean. Without considering dispersion, we miss out on understanding the consistency or reliability of the data.

Measures of center, such as the mean, median, and mode, are key statistical tools that summarize a dataset by identifying a representative central point. Each measure gives a different perspective:

• Mean: Provides the average value, useful for identifying the overall level of the data.
• Median: Indicates the middle value, resistant to outliers, offering a better measure in skewed distributions.
• Mode: Shows the most common value or values, highlighting the data's frequent occurrences.

These measures help simplify complex data into comprehensible information. However, they cannot indicate data variability or the full distribution on their own. This can lead to incomplete interpretations if the spread or dispersion is ignored.