Fiona's Biking Miles: Calculating Variance

Fiona's Biking Miles: Calculating Variance

Fiona's active lifestyle shines through her daily biking routine! Last week, she diligently recorded the number of miles she biked each day, giving us a clear picture of her cycling adventures. The data points we have are: 4, 7, 4, 10, and 5 miles. This set of numbers not only tells us how far Fiona went on different days but also provides an opportunity to delve into some interesting statistical concepts. One such concept is variance, which helps us understand how spread out the data is. We're given that the mean number of miles Fiona biked is $\mu = 6$. Now, let's explore how to represent the variance for this specific dataset. Understanding variance is crucial in many fields, from finance to science, as it quantifies the dispersion of a set of data points around their mean. It's a fundamental measure of variability, telling us whether the data points tend to be close to the average or far away from it. In Fiona's case, a low variance would mean she biked a similar number of miles each day, while a high variance would indicate significant fluctuations in her daily mileage. This article will guide you through the process of setting up the equation to calculate this variance, offering a clear and concise explanation. We'll break down the steps involved, making it easy for anyone to follow along, whether you're a student learning statistics for the first time or simply curious about understanding data better. Let's get started on uncovering the spread in Fiona's biking journey!

Understanding Variance: The Core Concept

Variance is a statistical measure that quantifies the spread or dispersion of a set of data points. In simpler terms, it tells us how much the individual data points tend to deviate from the average (the mean) of the dataset. A low variance indicates that the data points are clustered closely around the mean, suggesting consistency, while a high variance implies that the data points are spread out over a wider range of values, indicating greater variability. To calculate variance, we first need the mean of the data. Fiona's mean mileage is given as $\mu = 6$ miles. The variance is calculated by finding the average of the squared differences between each data point and the mean. This process involves several steps: first, calculate the difference between each data point and the mean; second, square each of these differences; and third, find the average of these squared differences. The squaring is done to ensure that all differences are positive, so that deviations in either direction (above or below the mean) contribute equally to the overall spread. If we didn't square the differences, the positive and negative deviations would cancel each other out, leading to a variance of zero even if there was significant spread. The formula for population variance (often denoted by $\sigma^2$) is: $\sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}$, where $x_i$ represents each individual data point, $\mu$ is the population mean, and $N$ is the total number of data points. For sample variance (often denoted by $s^2$), the denominator is $N-1$ instead of $N$, which provides a less biased estimate of the population variance when working with a sample. In Fiona's case, we are looking at the entire dataset for last week, so we can consider it a population for this specific context. Thus, we will use the population variance formula. Understanding this formula and its components is key to correctly calculating and interpreting the variance of Fiona's biking miles. It's not just about plugging numbers into a formula; it's about understanding what that number means in the context of the data.

Calculating the Variance for Fiona's Biking Data

Let's break down the calculation of variance for Fiona's biking data: 4, 7, 4, 10, 5 miles, with a given mean of $\mu = 6$. The first step in calculating variance is to find the difference between each individual data point and the mean. We have five data points, so we'll perform this calculation five times:

• For the first day: $4 - 6 = -2$
• For the second day: $7 - 6 = 1$
• For the third day: $4 - 6 = -2$
• For the fourth day: $10 - 6 = 4$
• For the fifth day: $5 - 6 = -1$

The next crucial step is to square each of these differences. Squaring ensures that all values are positive and gives more weight to larger deviations:

• $(-2)^2 = 4$
• $(1)^2 = 1$
• $(-2)^2 = 4$
• $(4)^2 = 16$
• $(-1)^2 = 1$

Now, we sum up these squared differences: $4 + 1 + 4 + 16 + 1 = 26$. This sum represents the total squared deviation from the mean for all the data points.

Finally, to find the variance, we divide this sum by the total number of data points. Fiona biked for 5 days, so $N = 5$. Therefore, the variance $(\sigma^2)$ is:

$\sigma^2 = \frac{26}{5}$

This calculation represents the equation that shows the variance for the number of miles Fiona biked last week. It's a direct application of the variance formula, taking into account each day's mileage and its deviation from the average. This value, $\frac{26}{5}$, which equals 5.2, gives us a quantitative measure of how spread out Fiona's daily biking distances were. A variance of 5.2 suggests a moderate level of variability in her mileage throughout the week. We can interpret this by saying that, on average, the squared difference between each day's mileage and the mean mileage of 6 miles is 5.2. It's important to remember that variance is expressed in squared units, so in this case, the units would be 'miles squared,' which isn't as intuitively interpretable as the standard deviation, but it's a critical step in its calculation. The process highlights how each individual measurement contributes to the overall understanding of the data's variability.

The Equation Representing Fiona's Variance

To represent the variance for the number of miles Fiona biked last week, we need to formulate an equation that encapsulates the steps we've just taken. Given the data points $x_1=4, x_2=7, x_3=4, x_4=10, x_5=5$, and the mean $\mu=6$, the variance $(\sigma^2)$ is calculated as the average of the squared differences from the mean. The general formula for population variance is $\sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}$. Applying this to Fiona's data, we substitute the values:

$\sigma^2 = \frac{(4 - 6)^2 + (7 - 6)^2 + (4 - 6)^2 + (10 - 6)^2 + (5 - 6)^2}{5}$

This equation directly shows how the variance is computed for Fiona's biking miles. Each term in the numerator represents the squared deviation of a single day's mileage from the mean. The sum of these squared deviations is then divided by the total number of days (5) to find the average squared deviation, which is the variance.

Let's simplify this equation to see the numerical result:

$\sigma^2 = \frac{(-2)^2 + (1)^2 + (-2)^2 + (4)^2 + (-1)^2}{5}$

$\sigma^2 = \frac{4 + 1 + 4 + 16 + 1}{5}$

$\sigma^2 = \frac{26}{5}$

Thus, the equation that shows the variance for Fiona's biking data is:

$\frac{(4 - 6)^2 + (7 - 6)^2 + (4 - 6)^2 + (10 - 6)^2 + (5 - 6)^2}{5}$

This expression is precisely what the question asks for: the equation that shows the variance. While $\frac{26}{5}$ or $5.2$ are the numerical values, the equation itself demonstrates the process of calculating the variance. It highlights the calculation of differences from the mean, their squaring, and their averaging. This is a fundamental concept in statistics, helping us quantify the spread of data. Understanding this equation allows us to apply the same logic to any dataset to determine its variability. Whether you're analyzing test scores, stock prices, or, in this case, daily biking distances, the principle of variance remains the same. It's a powerful tool for understanding the nature of data and making informed decisions based on its distribution. We can also derive the standard deviation from this variance by taking the square root, which would give us a measure in the original units (miles), making it easier to interpret the typical deviation from the mean.

Conclusion: The Spread of Fiona's Miles

We've successfully navigated the process of understanding and calculating the variance for Fiona's weekly biking mileage. The data, consisting of 4, 7, 4, 10, and 5 miles, with a mean of $\mu=6$ miles, led us to the variance equation: $\frac{(4 - 6)^2 + (7 - 6)^2 + (4 - 6)^2 + (10 - 6)^2 + (5 - 6)^2}{5}$. This equation is a clear representation of how variance is computed: by averaging the squared differences between each data point and the mean. The result, $\frac{26}{5}$ or $5.2$ square miles, quantifies the spread of her daily distances. A variance of 5.2 indicates a moderate level of variability, meaning Fiona's daily biking distances weren't always the same but also didn't fluctuate wildly from her average of 6 miles. This concept of variance is fundamental in statistics for understanding the consistency and spread within a dataset. For more in-depth learning about statistical concepts like variance and standard deviation, you can explore resources from trusted institutions. Consider visiting the National Center for Education Statistics (NCES) website for data and statistical information, or the American Statistical Association (ASA) for professional resources and educational materials on statistics.