Question

The monthly utility bills for a household in Riverside, California, were recorded for 12 consecutive months starting in January 2006 : $$ \begin{array}{lc|lr} \text { Month } & \text { Amount (\$) } & \text { Month } & \text { Amount (\$) } \\ \hline \text { January } & \$ 266.63 & \text { July } & \$ 306.55 \\ \text { February } & 163.41 & \text { August } & 335.48 \\ \text { March } & 219.41 & \text { September } & 343.50 \\ \text { April } & 162.64 & \text { October } & 226.80 \\ \text { May } & 187.16 & \text { November } & 208.99 \\ \text { June } & 289.17 & \text { December } & 230.46 \end{array} $$ a. Calculate the range of the utility bills for the year \(2006 .\) b. Calculate the average monthly utility bill for the year \(2006 .\) c. Calculate the standard deviation for the 2006 utility bills.

Short answer

Answer: The range of the utility bills in 2006 is $180.86, the average monthly utility bill is $244.97, and the standard deviation is $59.61.

Step-by-step solution

Calculate the Range

The range is the difference between the highest and lowest values in the data set. First, identify the highest and the lowest utility bills, then subtract the lowest value from the highest value to find the range. Highest utility bill: $343.50 Lowest utility bill: $162.64 Range = \(343.50 - \)162.64 = $180.86 The range of the utility bills in 2006 is $180.86. For part b, we need to calculate the average monthly utility bill.

Calculate the Average

To find the average, sum up all the utility bills and divide the total by the number of months (12). Sum of utility bills: \$266.63 + \$163.41 + \$219.41 + \$162.64 + \$187.16 + \$289.17 + \$306.55 + \$335.48 + \$343.50 + \$226.80 + \$208.99 + \$230.46 = \$2939.70 Average = \(\frac{\$2939.70}{12} = \$244.97\) The average monthly utility bill in 2006 is $244.97. For part c, we need to calculate the standard deviation of the utility bills.

Calculate the Standard Deviation

To find the standard deviation, first find the squared difference of each utility bill from the average calculated above, then find the average of the squared differences, and finally, take the square root of the resulting value. 1. Find the squared differences: $$ (\$266.63 - \$244.97)^2 = \$466.67 \\ (\$163.41 - \$244.97)^2 = \$6,603.67 \\ (\$219.41 - \$244.97)^2 = \$649.65 \\ (\$162.64 - \$244.97)^2 = \$6,776.93 \\ (\$187.16 - \$244.97)^2 = \$3,314.53 \\ (\$289.17 - \$244.97)^2 = \$1,948.08 \\ (\$306.55 - \$244.97)^2 = \$3,759.17 \\ (\$335.48 - \$244.97)^2 = \$8,179.18 \\ (\$343.50 - \$244.97)^2 = \$9,696.01 \\ (\$226.80 - \$244.97)^2 = \$331.63 \\ (\$208.99 - \$244.97)^2 = \$1,289.29 \\ (\$230.46 - \$244.97)^2 = \$210.30 \\ $$ 2. Find the average of squared differences: $$ \frac{466.67 + 6,603.67 + 649.65 + 6,776.93 + 3,314.53 + 1,948.08 + 3,759.17 + 8,179.18 + 9,696.01 + 331.63 + 1,289.29 + 210.30}{12} = \$3,553.71 $$ 3. Take the square root: $$ \sqrt{3,553.71} = \$59.61 $$ The standard deviation of the utility bills in 2006 is $59.61.

Key concepts

Calculating Range

Understanding the range of a data set is fundamental in statistics as it helps you grasp the spread or dispersion of your values. In the context of your monthly utility bills, the range gives you a quick snapshot of the variability in your expenses over the year. It's calculated simply by finding the difference between the highest and lowest bill amounts.

To put it into practice, let's consider the given data set with utility bills ranging from \$162.64 to \$343.50. By subtracting the lowest bill from the highest (\(343.50 - 162.64 = \$180.86\)), we find that the range is \$180.86. This could help you prepare for future expenses, as it shows the potential fluctuation you might expect in your bills throughout different months of the year.

It's important to keep in mind that while the range is easy to compute, it doesn't always provide a complete picture, especially if there are outliers or anomalies in your data. Nevertheless, it's an excellent starting point for understanding variations in your bills.

Average Monthly Utility Bill

The average monthly utility bill is another key metric that can help households budget and track their expenses. This average, also known as the mean, is calculated by summing up all the monthly expenses and dividing by the number of months.

In this case, you would add up all 12 of the recorded utility bills for 2006, which total up to \$2939.70, and then divide by 12, yielding an average of \$244.97 (\(\frac{\$2939.70}{12} = \$244.97\)). This figure gives you a sense of what you might typically expect to pay each month and can be particularly useful for comparing costs across different time periods or locations.

While the average is informative, it's also sensitive to extreme values, which can sometimes skew your perception if, for instance, one month had an unusually high or low bill. For a more balanced understanding, it's often helpful to look at other measures of central tendency, such as the median or mode, alongside the average.

Standard Deviation Calculation

The standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of values. A low standard deviation means that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.

For the household utility bills, we first find the difference between each monthly bill and the average, square these differences, and then compute their average. To get the standard deviation, you finally take the square root of this average. In the provided example, after squaring the differences and averaging them, we take the square root of \$3,553.71 to get a standard deviation of \$59.61 (\(\sqrt{3553.71} = \$59.61\)).

This tells us that on average, each utility bill is \$59.61 away from the mean bill amount. A higher standard deviation would mean that your utility bills are quite inconsistent, with more pronounced peaks and troughs over the course of the year, while a lower standard deviation suggests more consistency and predictability, which could make budgeting easier. Understanding this concept is crucial for any form of budget management and financial planning.