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To find the population standard deviation, we start by calculating the **Population Mean**, which is a fancy term for finding the average. Imagine you have two test scores: one for Math and one for Science. If you want to know your average, you'd add both scores and then divide by the number of scores you have. The formula for the population mean \( \mu \) is:

• \( \mu = \frac{x + y}{2} \)

Here, \( x \) and \( y \) are your scores. By adding them and dividing by 2 (because there are 2 scores), you get a number that tells you where the center or balance point of your scores is on the number line.

Understanding the population mean helps us see how far each score is from this "center." It's like knowing the position of the median of a see-saw to understand where each end will tilt.

Once we have the population mean, the next step is to determine how far each score lies from this mean. This is called finding the "**Sum of Squared Differences**." It's like checking how far every friend in a group is standing from a central point. For each score, \( x \) and \( y \), we calculate their differences from the mean, square those differences, and then add them.

• For score \( x \), the difference is \( x - \mu \).
• For score \( y \), the difference is \( y - \mu \).

Now square these differences:

• \((x - \mu)^2 = (x - \frac{x+y}{2})^2\)
• \((y - \mu)^2 = (y - \frac{x+y}{2})^2\)

Adding these squared differences together gives:

• \((x - \frac{x + y}{2})^{2} + (y - \frac{x + y}{2})^{2}\)

This squared step ensures that all differences are positive and emphasizes larger differences, just like squares in geometry make sizes look larger.

It's mathematical "magnifying." Sum of Squared Differences encapsulates the total deviation from the mean, showing the overall spread or variability in the dataset.

Now, the tricky part, but don't worry, it's easy! We've found the average spread of your scores from the mean with Sum of Squared Differences, but it’s not yet the standard deviation. Why? Because to get there, we have to dive into the "**Square Root Calculation**." Imagine if measuring how children can jump, but then you make that figure normal-sized by taking the square root.From the Sum of Squared Differences, we divide by the number of scores (2, in our case) to "normalize" or average their impact:

• \(\frac{(x - \frac{x + y}{2})^{2} + (y - \frac{x + y}{2})^{2}}{2}\)

Taking the square root of the result adjusts the squared scores back to their original scale:

• \(\sigma = \sqrt{\frac{(x - \frac{x + y}{2})^{2} + (y - \frac{x + y}{2})^{2}}{2}}\)

This calculation gives us the population standard deviation, a handy statistic to summarize how your scores vary from the average.

Like evening out your raised pulse after a sprint, it tells you the typical distance of each score from the mean.