Question

For the data sets find the mean, the median, and the mode. Comment on the skewness or symmetry of the data. A survey by Consumer Reports looked at the reliability of cars as the cars get older. They surveyed owners of cars that were 3 years old, and recorded the average yearly repair and maintenance costs (in dollars) for 15 different models of compact cars \(^{2}\) : \(\begin{array}{rrrrrrrrr}125 & 45 & 115 & 25 & 110 & 115 & 120 & 45 \\ 40 & 125 & 55 & 90 & 105 & 110 & 80 & \end{array}\)

Short answer

Answer: The mean yearly repair and maintenance cost is 86 dollars, the median is 90 dollars, and the modes are 45, 110, and 115 dollars. The dataset is predominantly symmetrical with a minor negative skewness.

Step-by-step solution

Arrange the data in ascending order

First, we need to arrange the given data set in ascending order. This will make it easier to calculate the median and the mode: \(\{25, 40, 45, 45, 55, 80, 90, 105, 110, 110, 115, 115, 120, 125, 125\}\)

Calculate the mean

To calculate the mean, we add up all the values in the dataset and divide by the total number of values, which is 15 in this case. Using the formula: \(\text{Mean} = \frac{\sum_{i=1}^{15} x_i}{n}\), we obtain: \(\text{Mean} = \frac{25 + 40 + 45 + 45 + 55 + 80 + 90 + 105 + 110 + 110 + 115 + 115 + 120 + 125 + 125}{15} = \frac{1290}{15} = 86\)

Calculate the median

Median is the middle value when the data set is arranged in ascending order. Since the dataset has 15 values, the middle value is the 8th value. Therefore, the median is 90.

Calculate the mode

The mode is the value that occurs the most frequently in the dataset. In this dataset, the values 45, 110, and 115 each occur twice which is the most frequent. Therefore, the dataset has three modes: 45, 110, and 115.

Analyze the skewness or symmetry

The mean, median, and mode can be used to determine the skewness or symmetry of the data. In this case, we have: - Mean: 86 - Median: 90 - Modes: 45, 110, and 115 Since the mean is slightly smaller than the median and the mode values are dispersed, we can infer that the distribution is mostly symmetrical with a slight negative skewness. Based on the given data set, the mean yearly repair and maintenance cost is 86 dollars, the median is 90 dollars, and the modes are 45, 110, and 115 dollars. The dataset is predominantly symmetrical with a minor negative skewness.

Key concepts

Mean Calculation

Understanding the mean, or average, of a dataset is crucial for grasping the overall tendency of the numerical data. To calculate the mean, you sum up all the individual values in the data set and then divide by the number of values you have added.

For instance, if you're evaluating car maintenance costs, like in the exercise, you would add each year’s cost and then divide by the total number of years considered. In mathematical terms, the mean is represented as \(\text{Mean} = \frac{\sum_{i=1}^{n} x_i}{n}\), where \(x_i\) represents each value and \(n\) is the total number of values.

In the case of the Consumer Reports survey for 15 different models of compact cars, with costs given, the calculated mean tells us the average annual repair and maintenance cost is approximately 86 dollars—an insightful figure for potential buyers.

Median Calculation

The median is the middle value of an ordered dataset and provides a central point that divides the data into two equal halves. To find the median, you first need to arrange the data values in ascending order. If there is an odd number of values, the median is the middle one. If there's an even number, the median is the average of the two middle numbers.

Using the repair costs data from our example, we find the 8th value in the ordered list to be the median, which is 90 dollars. This figure is a powerful statistic, especially when compared to the mean, as it’s less influenced by outliers and can give a better idea of what a 'typical' value might be.

Mode Calculation

The mode is the value that occurs most frequently in a dataset. It is entirely possible for a dataset to have more than one mode (bi-modal, or even multi-modal), or to have no mode at all if all values are unique.

In the context of car repair costs, the mode helps to identify the most common cost experienced by car owners. The dataset from the survey shows modes at 45, 110, and 115 dollars. Indicating these are common costs for yearly repair and maintenance among different models of compact cars. The existence of multiple modes can also hint at the presence of different groups within the data.

Data Skewness Analysis

Data skewness is about measuring the asymmetry of a data distribution. If a dataset has a perfect symmetry, the mean and median will be the same. A skew to the right (positive skew) means the mean is greater than the median, while a skew to the left (negative skew) implies the opposite.

In the survey on maintenance costs, the mean of 86 dollars is slightly less than the median of 90 dollars, suggesting a slight negative skew. This can often result from a few low values dragging down the average or mean, indicating that while most car owners have slightly higher costs, some models might have substantially lower maintenance costs. Assessing skewness can be valuable as it impacts the interpretation of the mean - telling us it may not be the center of the data distribution due to the influence of outliers.