it says the range and standard deviation is wrong

Let's recheck the calculations for the **range** and **standard deviation**.

### Corrected Calculations

#### b. Range

The range is the difference between the maximum and minimum values in the dataset.

Given the values: $115, 150, 144, 126, 90, 90, 95, 110, 120, 88$

- **Maximum value:** 150
- **Minimum value:** 88

\[
\text{Range} = 150 - 88 = 62
\]

The range is **correctly calculated as $62**.

#### d. Standard Deviation

The standard deviation calculation must be precise. Let's redo it with full accuracy:

1. **Mean (as calculated before):** $ \mu = 112.8 $

2. **Squared differences from the mean:**

\[
\begin{align*}
(115 - 112.8)^2 & = 4.84 \\
(150 - 112.8)^2 & = 1377.44 \\
(144 - 112.8)^2 & = 970.56 \\
(126 - 112.8)^2 & = 173.44 \\
(90 - 112.8)^2 & = 524.16 \quad \text{(this value appears twice, so double it)} \\
(90 - 112.8)^2 & = 524.16 \\
(95 - 112.8)^2 & = 321.64 \\
(110 - 112.8)^2 & = 7.84 \\
(120 - 112.8)^2 & = 51.84 \\
(88 - 112.8)^2 & = 619.84 \\
\end{align*}
\]

3. **Sum of squared differences:**

\[
4.84 + 1377.44 + 970.56 + 173.44 + 524.16 + 524.16 + 321.64 + 7.84 + 51.84 + 619.84 = 4575.76
\]

4. **Variance (as calculated before):**

\[
\text{Variance} = \frac{4575.76}{10} = 457.576
\]

5. **Standard Deviation:**

\[
\text{Standard Deviation} = \sqrt{457.576} \approx 21.40
\]

The standard deviation, rounded to two decimal places, is **$21.40**.

### Summary:
- **Range:** Correctly calculated as **$62**
- **Standard Deviation:** Corrected to **$21.40**

Would you like further clarification or have any more questions? Here are some follow-up questions:

1. How does the standard deviation reflect the consistency of the electric bills?
2. What is the importance of using standard deviation in analyzing monthly expenses?
3. How would increasing one of the lower bills affect the standard deviation?
4. Can the standard deviation be zero? If yes, under what condition?
5. What other statistical measures could you use to analyze this data?

**Tip:** Always ensure all steps in your calculations are checked carefully, especially when dealing with squares and square roots, to avoid minor errors.

Explore how to accurately calculate and correct the range and standard deviation for a dataset. This detailed analysis covers statistical concepts and formulas, ensuring precision in statistical calculations. Suitable for advanced high school students and anyone interested in data analysis.

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