Calculate each value of ∑𝑋2, (∑𝑋)2, ∑ (X − 2), and ∑(𝑋 − 1)2, for the following scores: 6, 2, 4, 2

Let's break down each calculation step by step:

### Given scores: 
\[ X = 6, 2, 4, 2 \]

### 1. Calculate \( \sum X^2 \)
This is the sum of the squares of each score.

\[
\sum X^2 = 6^2 + 2^2 + 4^2 + 2^2 
\]

\[
\sum X^2 = 36 + 4 + 16 + 4 = 60
\]

### 2. Calculate \( (\sum X)^2 \)
First, we find the sum of all scores and then square the result.

\[
\sum X = 6 + 2 + 4 + 2 = 14
\]

Now, square this sum:

\[
(\sum X)^2 = 14^2 = 196
\]

### 3. Calculate \( \sum (X - 2) \)
Subtract 2 from each score and then sum the results.

\[
X - 2 = 6 - 2, 2 - 2, 4 - 2, 2 - 2 
\]

\[
X - 2 = 4, 0, 2, 0
\]

\[
\sum (X - 2) = 4 + 0 + 2 + 0 = 6
\]

### 4. Calculate \( \sum (X - 1)^2 \)
Subtract 1 from each score, square the result, and then sum all the squared results.

\[
(X - 1)^2 = (6 - 1)^2, (2 - 1)^2, (4 - 1)^2, (2 - 1)^2
\]

\[
(X - 1)^2 = 5^2, 1^2, 3^2, 1^2
\]

\[
(X - 1)^2 = 25, 1, 9, 1
\]

\[
\sum (X - 1)^2 = 25 + 1 + 9 + 1 = 36
\]

### Summary of Values:

1. \( \sum X^2 = 60 \)
2. \( (\sum X)^2 = 196 \)
3. \( \sum (X - 2) = 6 \)
4. \( \sum (X - 1)^2 = 36 \)

Would you like a more detailed explanation on any of these calculations? Here are 5 questions related to this topic:

1. How would the calculations change if the scores were negative?
2. What is the difference between \( \sum X^2 \) and \( (\sum X)^2 \)?
3. Can you calculate the mean and variance of this data set?
4. How would you calculate \( \sum (X - 3) \) for the given scores?
5. What does each of these summation operations represent in statistics?

**Tip:** Always double-check the steps when working with multiple summations to ensure accuracy!

Learn how to calculate the sum of squares (∑X²), square of the sum (∑X)², sum of deviations (∑(X - 2)), and sum of squared deviations (∑(X - 1)²) for the scores 6, 2, 4, 2. This problem involves basic statistical calculations and is suitable for high school students. Get step-by-step explanations and understand the concepts of summation, squared terms, and statistical operations.

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