Let's break down this fascinating statistical question concerning the height distribution of sixth-grade students! We're given some key information: the average height (mean) of 12-year-olds is 58 inches, with a standard deviation of 2.3 inches. This means we're dealing with a concept called the normal distribution, a fundamental idea in statistics. Think of it as a bell curve – most students cluster around the average height, with fewer and fewer students at the extreme ends (very tall or very short). To solve this, we'll use the empirical rule, also known as the 68-95-99.7 rule, which helps us understand how data spreads out in a normal distribution. This rule is our secret weapon for unlocking the answer!
Delving into the Normal Distribution
So, what exactly is this normal distribution we keep talking about? Imagine plotting the heights of all the sixth graders on a graph. If the heights are normally distributed, the graph would look like a symmetrical bell shape. The peak of the bell is right at the mean (the average height), which in our case is 58 inches. The standard deviation, 2.3 inches, tells us how spread out the data is. A smaller standard deviation means the data is clustered tightly around the mean, while a larger standard deviation means the data is more spread out. This is crucial for understanding the range within which most students' heights fall. The beauty of the normal distribution is that we can use these two numbers – the mean and the standard deviation – to make powerful statements about the data. For instance, the empirical rule lets us quickly estimate the percentage of students within certain height ranges without having to look at each individual height. It's like having a cheat sheet for understanding height patterns! Understanding this distribution is key to answering our question about the height range encompassing 68% of the students.
Now, let's discuss the significance of the empirical rule, also known as the 68-95-99.7 rule, as it's the key to solving our problem. In a normal distribution, approximately 68% of the data falls within one standard deviation of the mean. This means that if we go one standard deviation above and one standard deviation below the average height, we'll capture about 68% of the students. Then, roughly 95% of the data falls within two standard deviations, and about 99.7% falls within three standard deviations. This rule gives us a quick and easy way to understand the spread of data without doing complex calculations. For our height problem, we're interested in the 68% range. The empirical rule allows us to quickly estimate that 68% of students will have heights within a specific range calculated by adding and subtracting one standard deviation from the mean. This concept is incredibly useful in various real-world scenarios, from predicting test scores to understanding stock market fluctuations. By grasping the empirical rule, we're able to make informed judgments about data sets and their distributions.
Applying the 68-95-99.7 Rule to Student Heights
Time to put the empirical rule into action! Remember, our mean height is 58 inches, and the standard deviation is 2.3 inches. We want to find the height range that includes approximately 68% of the sixth-grade students. According to the empirical rule, 68% of the data falls within one standard deviation of the mean. So, we need to calculate two values: one standard deviation above the mean and one standard deviation below the mean. Let's start by calculating the lower bound: subtract the standard deviation from the mean (58 inches - 2.3 inches = 55.7 inches). This tells us the lower end of our range. Next, calculate the upper bound: add the standard deviation to the mean (58 inches + 2.3 inches = 60.3 inches). This is the upper end of our range. Now we know that approximately 68% of sixth-grade students will have heights between 55.7 inches and 60.3 inches. Isn't it amazing how easily we arrived at this answer using the empirical rule? This quick calculation gives us a powerful insight into the distribution of heights in this group of students. Understanding these calculations and applying them is what makes statistics so practical.
Calculating the Height Range
Okay, let's get down to the specifics. We know the mean height for 12-year-olds is 58 inches, and the standard deviation is 2.3 inches. To find the height range that includes approximately 68% of the students, we need to do two simple calculations. First, we'll subtract the standard deviation from the mean to find the lower bound of the range: 58 inches - 2.3 inches = 55.7 inches. This means that about 34% of students will be between 55.7 inches and 58 inches tall. Next, we'll add the standard deviation to the mean to find the upper bound of the range: 58 inches + 2.3 inches = 60.3 inches. This tells us another 34% of students will be between 58 inches and 60.3 inches tall. Combining these two ranges, we see that 68% of sixth-grade students will have heights between 55.7 inches and 60.3 inches. This straightforward calculation perfectly illustrates how the empirical rule works in practice. By understanding these steps, you can easily calculate similar ranges for different datasets.
The Answer: Between 55.7 inches and 60.3 inches
So, after our calculations, we've arrived at the answer! Based on the mean height of 58 inches and a standard deviation of 2.3 inches, we can confidently say that approximately 68% of sixth-grade students will have heights between 55.7 inches and 60.3 inches. This range gives us a clear picture of the typical heights within this age group. Remember, this is an estimate based on the normal distribution and the empirical rule. While individual students may fall outside this range, the majority will fall within it. It's important to understand that statistics is about probabilities and likelihoods, not absolute certainties. But this result provides a valuable insight into the distribution of heights among sixth graders. Now, you can explain this concept to others and even use it as a benchmark for understanding the growth of children in this age group!
Why This Matters: Understanding Growth and Development
Understanding height distribution isn't just a math exercise; it has real-world applications! Height is an important indicator of a child's growth and development. By knowing the average height and the typical range, we can identify children who may be growing faster or slower than their peers. This information can be valuable for parents and healthcare professionals in ensuring children are on track for healthy development. For instance, if a child's height consistently falls significantly outside the 68% range, it might be a signal to investigate further. It's crucial to remember that height is influenced by many factors, including genetics, nutrition, and overall health. Statistics like this provide a framework for understanding typical growth patterns, but they don't tell the whole story. In cases where there are concerns, a medical professional should always be consulted. This knowledge about height distribution helps us appreciate the natural variation in human growth and development and allows us to identify potential issues early on.
Beyond the Classroom: Real-World Applications of Normal Distribution
The concepts we've discussed about normal distribution and standard deviation aren't just relevant to student heights. They're used in a wide range of fields! Think about quality control in manufacturing: companies use these statistical tools to ensure their products meet certain specifications. For example, a factory producing light bulbs might use normal distribution to monitor the lifespan of the bulbs and ensure that most of them last a certain amount of time. Or consider finance: analysts use normal distribution to model stock prices and assess investment risks. Understanding how data is distributed allows them to make more informed decisions. Even in healthcare, normal distribution plays a role. For instance, doctors might use it to interpret blood pressure readings or cholesterol levels. These examples highlight how understanding statistical concepts like normal distribution is incredibly valuable in many different fields. By mastering these concepts, you're not just learning math; you're developing skills that are applicable to numerous real-world situations.
In conclusion, by applying the empirical rule to the mean and standard deviation of sixth-grade student heights, we've determined that approximately 68% of these students will have heights between 55.7 inches and 60.3 inches. This demonstrates the power of statistical tools in understanding and interpreting data, not just in the classroom but also in various real-world applications.