Calculating Mean, Standard Deviation, And Data Distribution For Car Mileage

In this article, we will delve into the process of calculating the mean and standard deviation for a given dataset of car mileage. We will also explore how many data points fall within one standard deviation of the mean. Understanding these statistical measures is crucial for analyzing data distributions and gaining insights into the variability within a dataset. Let's consider a sample of nine different cars, where the distances driven (in miles) using 13 gallons of gasoline are as follows: 174, 201, 271, 208, 196, 340, 214, 236, and 38.

Calculating the Mean Mileage

Mean mileage serves as a central tendency measure, representing the average distance traveled by the cars in our sample. To calculate the mean, we sum up all the individual distances and divide by the total number of cars. This calculation provides a single value that summarizes the typical mileage achieved by the cars in our dataset. Understanding the mean is fundamental in statistical analysis as it gives us a baseline to compare individual data points against and to understand the overall performance of the vehicles in terms of fuel efficiency and distance traveled. The formula for the mean (often denoted as ${\bar{x}}$) is:

${\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}}$

Where:

• ${x_i}$ represents each individual data point (mileage in this case).
• ${n}$ is the number of data points (number of cars).

Applying this to our dataset:

${\sum x_i = 174 + 201 + 271 + 208 + 196 + 340 + 214 + 236 + 38 = 1878}$

${\bar{x} = \frac{1878}{9} = 208.67\text{ miles}}$

Therefore, the mean distance traveled by these cars on 13 gallons of gasoline is approximately 208.67 miles. This average provides a crucial reference point for understanding the overall performance of the vehicles in our sample. By knowing the mean, we can begin to assess how individual cars deviate from this central value, giving us insights into the variability and consistency of fuel efficiency across the fleet. This measure is especially useful in comparative analysis, where we might compare this mean against those of other car samples or against industry benchmarks to gauge performance.

Calculating the Standard Deviation

Standard deviation quantifies the spread or dispersion of data points around the mean. A higher standard deviation indicates greater variability, while a lower value suggests data points are clustered closer to the mean. To compute the standard deviation, we first calculate the variance, which is the average of the squared differences from the mean. The square root of the variance then gives us the standard deviation. This measure is crucial because it tells us not just the average performance, but also how consistent that performance is. For example, a fleet of cars might have a good mean mileage, but if the standard deviation is high, it means that some cars perform significantly better or worse than the average, which could point to issues with vehicle maintenance, driver behavior, or other factors.

The formula for the sample standard deviation (${s}$) is:

${s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}}}$

Where:

• ${x_i}$ is each individual data point.
• ${\bar{x}}$ is the sample mean.
• ${n}$ is the number of data points.

Let's calculate the standard deviation for our dataset:

First, we find the squared differences from the mean:

• ${(174 - 208.67)^2 = 1195.49}$
• ${(201 - 208.67)^2 = 58.82}$
• ${(271 - 208.67)^2 = 3905.67}$
• ${(208 - 208.67)^2 = 0.45}$
• ${(196 - 208.67)^2 = 160.53}$
• ${(340 - 208.67)^2 = 17241.45}$
• ${(214 - 208.67)^2 = 28.41}$
• ${(236 - 208.67)^2 = 746.41}$
• ${(38 - 208.67)^2 = 29144.49}$

Next, sum these squared differences:

${\sum (x_i - \bar{x})^2 = 1195.49 + 58.82 + 3905.67 + 0.45 + 160.53 + 17241.45 + 28.41 + 746.41 + 29144.49 = 52421.72}$

Now, divide by ${n-1}$ (which is 8) and take the square root:

${s = \sqrt{\frac{52421.72}{8}} = \sqrt{6552.72} \approx 80.95\text{ miles}}$

Thus, the standard deviation for the distances traveled is approximately 80.95 miles. This value indicates the degree of variability in the distances covered by the cars in our sample. A standard deviation of 80.95 miles suggests that the distances driven by the cars vary considerably around the mean of 208.67 miles. In practical terms, this means there's a wide range in how efficiently the cars are using fuel, which could be due to a variety of factors, including driving conditions, vehicle maintenance, and driver behavior. Understanding the standard deviation is critical for making informed decisions, such as identifying cars that may need servicing or adjusting driving practices to improve fuel efficiency.

Determining Data Points Within One Standard Deviation

After calculating the mean and standard deviation, a common next step is to determine how many data points fall within one standard deviation of the mean. This provides insights into the distribution of the data and helps identify potential outliers. Data points within one standard deviation are generally considered to be within the typical range of values. To find this range, we add and subtract the standard deviation from the mean. This range helps us understand the central cluster of our data. Values falling outside this range may be considered less typical and could be of particular interest for further investigation. For instance, unusually low mileage could indicate mechanical issues, while exceptionally high mileage might suggest very efficient driving habits or other factors that enhance fuel economy.

• Lower bound: ${208.67 - 80.95 = 127.72\text{ miles}}$
• Upper bound: ${208.67 + 80.95 = 289.62\text{ miles}}$

Now, let's count how many cars fall within this range (127.72 miles to 289.62 miles):

Looking at our original data (174, 201, 271, 208, 196, 340, 214, 236, and 38), the cars that fall within one standard deviation of the mean are:

• 174 miles
• 201 miles
• 271 miles
• 208 miles
• 196 miles
• 214 miles
• 236 miles

There are seven cars whose mileage falls within one standard deviation of the mean. This suggests that the majority of the cars in our sample perform relatively consistently in terms of fuel efficiency. The fact that seven out of nine cars fall within this range indicates a fairly concentrated distribution around the average mileage. The two cars that fall outside this range—one with significantly lower mileage (38 miles) and another with higher mileage (340 miles)—may warrant further examination to understand the factors contributing to these deviations. These could range from mechanical issues to differences in driving conditions or even variations in the quality of gasoline used.

Conclusion

In summary, for the given dataset of car mileages, the mean distance traveled is 208.67 miles, and the standard deviation is 80.95 miles. Seven out of the nine cars fall within one standard deviation of the mean. These statistical measures provide a comprehensive understanding of the central tendency and variability within the dataset, enabling informed analysis and decision-making. By understanding these calculations and their implications, we can better assess the performance and consistency of vehicles, identify potential issues, and make data-driven improvements in various fields, from vehicle maintenance to fuel efficiency strategies. The practical applications of such statistical analysis extend beyond the realm of automobiles, proving valuable in diverse sectors where data-driven insights are paramount.

By calculating the mean and standard deviation, we gain valuable insights into the typical mileage and the spread of data. Determining the number of cars within one standard deviation of the mean helps us understand the distribution and identify potential outliers, which may warrant further investigation.