Question

Find the range, the sample variance and the sample standard deviation. 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 \(^{2}\): $$ \begin{array}{rrrrrrrr} 125 & 45 & 115 & 25 & 110 & 115 & 120 & 45 \\ 40 & 125 & 55 & 90 & 105 & 110 & 80 & \end{array} $$

Short answer

Range: 100 Sample Variance: 1842.53 Sample Standard Deviation: 42.92

Step-by-step solution

Order the data

First, we need to put the data in ascending order: $$ \begin{array}{rrrrrrrr} 25 & 40 & 45 & 45 & 55 & 80 & 90 & 105 \\ 110 & 110 & 115 & 115 & 120 & 125 & 125 & \end{array} $$

Calculate the range

Find the minimum and maximum values in the dataset, and calculate the difference between them: Range = Max - Min = 125-25 = 100

Calculate the mean

Add up all the observations and divide by the number of observations (n = 15) to find the mean: \(\bar{x} = \frac{25 + 40 + 45 + 45 + 55 + 80 + 90 + 105 + 110 + 110 + 115 + 115 + 120 + 125 + 125}{15} = 94.333\)

Calculate the squared differences

Subtract the mean from each observation and square the result: $$ \begin{array}{rrrrrrrr} (-69.333)^2 & (-54.333)^2 & (-49.333)^2 & (-49.333)^2 & (-39.333)^2 &(-14.333)^2 & (-4.333)^2 & (10.667)^2 \\ \end{array} $$ $$ \begin{array}{rrrrrrrr} (15.667)^2 & (15.667)^2 & (20.667)^2 & (20.667)^2 & (25.667)^2 & (30.667)^2 & (30.667)^2 & \end{array} $$

Calculate the sample variance

Add the squared differences, divide by (n-1), and round to two decimal places: $$ s^2 = \frac{4804.44 + 2947.56 + 2433.44 + 2433.44 + 1545.56 + 205.44 + 18.78 + 113.78 + 245.44 + 245.44 + 426.78 + 426.78 + 659.44 + 938.89 + 938.89}{14} = 1842.53 $$

Calculate the sample standard deviation

Take the square root of the sample variance and round to two decimal places: $$ s = \sqrt{1842.53} = 42.92 $$ So, the range is 100, the sample variance is 1842.53, and the sample standard deviation is 42.92.

Key concepts

Understanding Range

In descriptive statistics, the range gives you a quick snapshot of how spread out the data points in a dataset are. It's defined as the difference between the largest and smallest value. To find the range:

• Identify the minimum value in the dataset.
• Identify the maximum value in the dataset.
• Subtract the minimum value from the maximum value.

For example, in the dataset provided, the minimum value is 25, and the maximum value is 125. Therefore, the range is calculated as follows: \[\text{Range} = 125 - 25 = 100\]
The range tells you that the repair costs in this dataset span 100 dollars, giving you insight into the variability among different car models.

Exploring Sample Variance

Sample variance is a crucial measure in statistics to understand the spread of a dataset relative to its mean. It reflects the average of the squared differences between each data point and the dataset's mean.
To calculate sample variance:

• First, find the mean of the dataset.
• Subtract the mean from each data point to find the difference.
• Square each difference to eliminate negative values and emphasize larger disparities.
• Sum all squared differences.
• Divide by the number of data points minus one \((n-1)\) because we are working with a sample, not a whole population.

In the example dataset, the calculated sample variance is:\[s^2 = \frac{\text{Sum of squared differences}}{14} = 1842.53\]The variance provides a more nuanced view than the range as it considers how much each point deviates from the mean.

Decoding Sample Standard Deviation

The sample standard deviation builds on the concept of variance, providing a measure of how spread out numbers are in a dataset, but crucially, it does so in the same units as the original data.
While the variance is informative, its unit is the square of the original data's units. To make this measure intuitively useful, we take the square root of the variance to return to the original units.
Here's how you calculate it:

• Calculate the sample variance \(s^2\).
• Take the square root of the variance to obtain the standard deviation:

\[ s = \sqrt{s^2} \]
In the provided solution, the sample standard deviation is:\[s = \sqrt{1842.53} \approx 42.92\]This result means that the typical deviation of repair costs from the mean cost in this dataset is 42.92 dollars, giving you valuable insight into typical variations you might expect in the data.