https://file.aiquickdraw.com/mathbot/file/1727309883394xb7b8voe.jpg

The formula in the image appears to be related to the **standard deviation** calculation, likely for a sample. The first equation shows the formula for the sample standard deviation \(\Delta X\):

\[
\Delta X = \sqrt{\frac{1}{N} \left( \frac{\sum X^2}{N-1} - \left( \frac{\sum X}{N} \right)^2 \right)}
\]

In the second part, specific numbers are substituted into the formula:

\[
\Delta X = \frac{1}{5} \sqrt{5 \cdot 1163.4 - (76.27)^2}
\]

This suggests the following steps to solve it:

### Step-by-Step Solution:
1. **Substitute values:**
   \[
   \Delta X = \frac{1}{5} \sqrt{5 \cdot 1163.4 - (76.27)^2}
   \]
   First, calculate \( (76.27)^2 \):
   \[
   76.27^2 = 5817.9529
   \]

2. **Multiply** \(5 \cdot 1163.4\):
   \[
   5 \cdot 1163.4 = 5817
   \]

3. **Subtract the squared sum:**
   \[
   5817 - 5817.9529 = -0.9529
   \]

4. **Take the square root:**
   \[
   \sqrt{-0.9529} \quad \text{(This results in an imaginary number since you cannot take the square root of a negative number)}
   \]

The result involves a negative number under the square root, meaning the result is an imaginary number. This suggests a possible issue with the input values.

Would you like help verifying the calculations or interpreting the imaginary result?

---

Here are 5 relative questions to expand understanding:

1. What is the difference between population and sample standard deviation?
2. Could you provide the full context for this data? Is it related to statistics from an experiment?
3. How do you interpret imaginary results in statistical contexts?
4. What is the significance of \(N\) in the formula? How is it determined?
5. Are the steps above clear, or should I break down any further part of the calculation?

**Tip:** Always check if the value inside the square root is non-negative when calculating standard deviations.

Step-by-step solution to calculate the sample standard deviation using the formula ΔX = sqrt[(ΣX^2/N) - (ΣX/N)^2]. Learn how to apply the standard deviation formula with specific data and the importance of variance calculations.

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