Question

Consider the following list of sale prices (in \(\$ 1000 \mathrm{~s}\) ) for eight houses on a certain road: \(\$ 820, \$ 930, \$ 780,\) \(\$ 950, \$ 3540, \$ 680, \$ 920, \$ 900 .\) One of the houses is worth much more than the other seven because it is much larger, it is set well back from the road, and it is adjacent to the shore of a lake to which it has private access. (a) What is the mean price of these eight houses? (b) Is the mean a good description of the value of the houses on this block? Explain your reasoning.

Short answer

Explain your reasoning. Answer: The mean is not a good representation of the value of the houses on this block due to the presence of an extreme value that skews the mean upward. In this situation, it is best to use another measure of central tendency, such as median, to get a better description of the value of the houses in this block.

Step-by-step solution

Calculate the sum of house prices

First, we need to find the sum of all the houses' prices in the list. Add the given house prices: \(820 + 930 + 780 + 950 + 3540 + 680 + 920 + 900\).

Calculate the mean price

To find the mean price of these eight houses, we will divide the sum of the house prices by the number of houses. In this case, there are 8 houses, so we will divide the sum by 8. Mean price = \(\frac{sum \ of \ house \ prices}{number \ of \ houses}\)

Analyze if the mean is a good description of value

After calculating the mean price, discuss if this value is a fair representation of the value of houses on the block. Consider factors like one of the houses being much larger than the others, having a setback from the road, and having private access to a lake. Now let's execute these steps:

Calculate the sum of house prices

To calculate the sum of house prices, add the values together: Sum = \(820 + 930 + 780 + 950 + 3540 + 680 + 920 + 900 = 8520\)

Calculate the mean price

Now that we have the sum of the house prices, we can calculate the mean price by dividing the sum by the number of houses (8). Mean price = \(\frac{8520}{8} = 1065\) So the mean price of these eight houses is \(\$ 1065,000\).

Analyze if the mean is a good description of value

The mean price of the houses is \(\$ 1065,000\). However, it is evident from the problem statement that one of the houses has significantly higher value due to its larger size, private lake access, and setback from the road. This house's value, \(\$ 3540,000\), is significantly higher than the mean price and skews the mean upwards. The mean is sensitive to extreme values, so it doesn't always give a proper representation when there are outliers, as in this case. In this situation, it is best to use another measure of central tendency, such as median, to get a better description of the value of the houses in this block. In conclusion, the mean price of these eight houses is \(\$ 1065,000\), but the mean is not a good description of the value of the houses on this block due to the presence of an extreme value that skews the mean upward.

Key concepts

Central Tendency

Central tendency is a statistical measure that identifies a single value to represent a dataset. This value is typically at the center of the dataset and conveys an overall trend. The most common measures of central tendency are the mean, median, and mode.

The **mean** is what we often call the average. It is calculated by adding up all the numbers in a dataset and dividing by the number of values. For example, if we have sales prices for houses, the mean provides an overview of the overall price level by sharing the total sale value equally among the number of houses.

The **median** is the middle value when the numbers are arranged in order. It's particularly useful when data is skewed by outliers or extreme values, as it is not affected by them and therefore provides a more accurate central value. In contrast, the **mode** is the number that appears most frequently in a dataset and is less commonly used for numerical data but is helpful for categorical datasets, like types of houses sold.

Choosing the right measure depends on understanding the distribution and objective of your analysis. In skewed distributions like our house prices, median often shines as a robust indicator of central tendency.

Outliers in Data

Outliers are data points that significantly differ from other observations. They can highly influence the results of statistical analyses, especially measures like the mean. In the context of our house prices, the house valued at \(\$3540,000\) stands as an outlier. It is priced significantly above the rest, impacting the mean by pulling it upwards.

Recognizing outliers is crucial because they can distort insights and lead to incorrect interpretations. To spot outliers, you can use methods like:

• Visual tools such as box plots or scatter plots to easily identify values that fall outside the normal range.
• Statistical techniques like analyzing standard deviations to see how far a value strays from the rest of the data.

When dealing with datasets, always check if outliers are relevant to your analysis or if they should be removed to ensure data quality. In some cases, these points might indicate special conditions or errors that need attention rather than inclusion in the general analysis.

Median vs Mean

When analyzing data, especially with potential outliers, understanding the difference between median and mean is essential. Both are measures of central tendency but behave differently when outliers are present.

The **mean** is sensitive to extreme values because it accounts for all data points in its calculation. In our exercise, the mean sale price is skewed by the high value of one house. This means it doesn't accurately reflect the typical sale price for this road.

The **median,** on the other hand, identifies the middle value when the dataset is ordered. For datasets with outliers, the median provides a better sense of the central value as it is not swayed by extreme numbers. Using the median for the house prices would likely give a more representative figure of what typical houses on the road are worth, disregarding the exceptionally high price of the larger, unique house.

In summary, while both statistics offer valuable insights, choosing between them depends on the data's nature and whether outliers are present. In skewed data, the median often tells a tale closer to reality compared to the mean.