To solve the problem, let's go through each part step by step:

a. Mean Monthly Electric Bill

The mean is calculated by summing all the values and dividing by the number of values.

$\text{Mean} = \frac{\text{Sum of all bills}}{\text{Number of bills}}$

$\text{Sum of all bills} = 115 + 150 + 144 + 126 + 90 + 90 + 95 + 110 + 120 + 88 = 1128$

$\text{Number of bills} = 10$

$\text{Mean} = \frac{1128}{10} = 112.8$

So, the mean monthly electric bill is $112.80.

b. Range

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

$\text{Range} = \text{Maximum value} - \text{Minimum value}$

$\text{Maximum value} = 150, \quad \text{Minimum value} = 88$

$\text{Range} = 150 - 88 = 62$

So, the range is $62.

c. Variance

The variance is calculated by finding the average of the squared differences from the mean.

First, find each squared difference from the mean:

$\text{Variance} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2$

Where $x_i$ are the individual data points, and $\mu$ is the mean.

$\text{Variance} = \frac{1}{10} \left[ (115 - 112.8)^2 + (150 - 112.8)^2 + \dots + (88 - 112.8)^2 \right]$

Calculating each squared difference:

$(115 - 112.8)^2 = 4.84, \quad (150 - 112.8)^2 = 1397.44, \quad \dots, \quad (88 - 112.8)^2 = 619.84$

Sum these squared differences:

$\text{Sum of squared differences} = 4.84 + 1387.84 + 972.64 + 175.56 + 524.16 + 524.16 + 320.04 + 8.64 + 51.84 + 619.84 = 4688.4$

$\text{Variance} = \frac{4688.4}{10} = 468.84$

So, the variance is 468.84.

d. Standard Deviation

The standard deviation is the square root of the variance:

$\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{468.84} \approx 21.65$

So, the standard deviation is $21.65.

Summary:

• Mean: $112.80
• Range: $62
• Variance: 468.84
• Standard Deviation: $21.65

Would you like further details or have any questions? Here are some additional related questions:

1. How would the variance change if one of the bills were significantly higher?
2. What does the standard deviation tell us about the spread of the bills?
3. How would the range change if the smallest bill was $10 higher?
4. What other measures of central tendency could be useful in this context?
5. How might seasonal changes affect the variability in monthly electric bills?

Tip: When calculating variance and standard deviation, always double-check each step to ensure accuracy, as small errors can compound.