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The first step in our exercise is to perform a summation. Summation is a fundamental concept in statistics, where we add together a series of numbers to get their total sum. Imagine counting the total number of shoe pairs each woman owns. We have six women, each with a different number of pairs. To get \(\sum x\), simply add all these values together:

• 8 pairs
• 14 pairs
• 3 pairs
• 7 pairs
• 10 pairs
• 5 pairs

So, the sum \(\sum x = 8 + 14 + 3 + 7 + 10 + 5 = 47\). Summation helps us understand the totality of data points in a given set. By seeing the sum, we can quickly grasp an overview of the collective total within that dataset. Adding up numbers like this is not only useful in statistics but also in everyday tasks like budgeting or planning.

Once you have the sum from the previous step, the next concept is squaring this sum. The square of a sum is exactly what it sounds like – you take the total sum of the values and multiply it by itself. Mathematically, it is represented as \( (\sum x )^2 \). In our example, we already found that the sum of the shoes is 47. To find \( (\sum x )^2 \), calculate:

• 47 multiplied by 47: \( 47 \times 47 = 2209 \).

The "square of a sum" quickly increases the magnitude of values, which is why it's commonly used in statistical analysis and algebraic expressions. It helps capture not just the total amount, but also gives weight to larger sums by extending their impact, which can be particularly useful in field such as physics and economics.

Our final concept is the sum of squares, another essential part of descriptive statistics. Here, instead of adding up the numbers first and then squaring them, you square each of the individual numbers and then add these squares together. This provides a different perspective on the data, highlighting how much variation or spread there is among data points.To find the sum of squares of the shoe pairs, calculate the square of each number:

• 8 squared: \(8^2 = 64\)
• 14 squared: \(14^2 = 196\)
• 3 squared: \(3^2 = 9\)
• 7 squared: \(7^2 = 49\)
• 10 squared: \(10^2 = 100\)
• 5 squared: \(5^2 = 25\)

Now, sum all these squared values: \(\sum x^{2} = 64 + 196 + 9 + 49 + 100 + 25 = 443\)."Sum of squares" is especially useful in variance and standard deviation calculations, providing a measure of data dispersion. This concept is widely applied in statistical analysis to understand the distribution and variability in data.