Q.Mean and standard deviation of 100 items are 50 and 4, respectively. Find the sum of all the item and the sum of the squares of the items.

The mean and standard deviation formulas connect directly to the sums we need.

Step 1: Sum of all items from the mean

The mean is $\bar{x} = \frac{\sum x_i}{n}$, so:

Step 2: Sum of squares from the standard deviation

The standard deviation is $\sigma = \sqrt{\frac{\sum x_i^2}{n} - \bar{x}^2}$. Squaring both sides:

$\sigma^2 = \frac{\sum x_i^2}{n} - \bar{x}^2$ …

The mean tells us the average value of all items, while the standard deviation measures how spread out the values are from that average. When we know these summary statistics, we can reverse-engineer the raw sums that went into computing them. The key is to recall the formulas that define these measures and solve for the quantities we want.

The mean of $n$ items is defined as:

The standard deviation is the square root of the variance, and variance has the computational formula:

This second formula is particularly useful because it directly relates the sum of squares to quantities we know.

Finding the sum of all items

1. Apply the mean formula. We know $\bar{x} = 50$ and $n = 100$:

2. Solve for the sum. Multiply both sides by 100:

That's straightforward—the mean is just the total divided by the count, so the total is mean times count.

Finding the sum of squares

3. Start with the variance formula. The standard deviation is $\sigma = 4$, so the variance is: $\sigma^2 = 4^2 = 16$ …