Calculating Mean Using Z-Score: A Step-by-Step Guide

Hey guys! Let's dive into a common statistical problem: finding the mean (μ) when you're given the z-score, a data point (x), and the standard deviation (σ). We'll use the z-score formula, which is a super handy tool in statistics. If you've ever scratched your head wondering how to tackle this, you're in the right place. We'll break it down step-by-step so it's crystal clear. So, grab your calculators, and let's get started!

Understanding the Z-Score Formula

The z-score formula is a fundamental concept in statistics. It tells us how many standard deviations a particular data point is away from the mean of the dataset. It's expressed as:

z = (x - μ) / σ

Where:

• z is the z-score.
• x is the observed value or data point.
• μ (mu) is the population mean, which is what we're often trying to find.
• σ (sigma) is the population standard deviation.

This formula is not just a bunch of symbols; it's a way to standardize data, making it easier to compare values from different datasets. Imagine you have test scores from two different classes, each with its own average and spread. Z-scores allow you to see how an individual score compares to its respective class average in a standardized way. If someone scores a z-score of 2, it means their score is two standard deviations above the mean, regardless of the actual score values.

Now, why is understanding this formula so important? Because it’s the key to solving problems where you need to find the mean, especially when you have the other pieces of the puzzle. It's like having a recipe – if you know the ingredients and the steps, you can bake a cake. In this case, if we know the z-score, the observed value, and the standard deviation, we can rearrange the formula to solve for the mean. This is a common task in statistical analysis, whether you're working with test scores, stock prices, or any other kind of data. So, let's keep this formula in our back pocket as we move forward; we'll be using it a lot!

Problem Setup: Identifying the Given Values

Okay, let's get specific. In our problem, we're given the following information:

• z-score (z) = 4.75
• Observed value (x) = 23.8
• Standard deviation (σ) = 2.8

Our mission, should we choose to accept it (and we do!), is to find the mean (μ). Think of this like a detective novel: we have clues, and we need to use them to solve the mystery. The z-score is like a witness statement, the observed value is like a piece of evidence found at the scene, and the standard deviation is like knowing the general lay of the land. Now, we need to put it all together to find the missing piece – the mean.

But before we jump into calculations, let's take a moment to appreciate what each of these values tells us. The z-score of 4.75 is pretty high, which suggests that our observed value (23.8) is significantly above the mean. The standard deviation of 2.8 gives us a sense of how spread out the data is around the mean. Knowing these pieces helps us make sense of the final answer and ensure it's reasonable. It’s like having a mental picture of what the solution should look like, so we can spot any errors along the way. So, with our clues in hand, let's move on to the next step: rearranging the z-score formula to solve for the mean. This is where we turn our detective work into mathematical action!

Rearranging the Formula to Solve for μ

Alright, now for the fun part: let's get our algebraic muscles flexed! We need to rearrange the z-score formula to isolate μ (the mean) on one side. Remember the original formula:

z = (x - μ) / σ

Our goal is to get μ by itself. Here's how we do it:

1. Multiply both sides by σ: This gets rid of the division on the right side.
   • z * σ = x - μ
2. Add μ to both sides: This moves μ to the left side.
   • z * σ + μ = x
3. Subtract z * σ from both sides: This isolates μ.
   • μ = x - z * σ

Voilà! We've successfully rearranged the formula. This is a crucial step because it transforms the formula from finding the z-score to finding the mean. It's like having a universal adapter for different types of problems. Now, instead of plugging in the mean to find the z-score, we can plug in the z-score, the observed value, and the standard deviation to find the mean. This little bit of algebra is super powerful and will be our key to unlocking the solution.

Think of it like this: we started with a map to find a location (the z-score), but we needed a map to find our starting point (the mean). By rearranging the formula, we created that new map. Now that we have our formula ready, we can move on to the next step: plugging in the values and calculating the mean. Let's do it!

Plugging in the Values and Calculating μ

Okay, the stage is set, the formula is ready, and now it's time for the main event: plugging in the values and calculating the mean. We've got our rearranged formula:

μ = x - z * σ

And we know:

• x = 23.8
• z = 4.75
• σ = 2.8

Let's substitute these values into the formula:

μ = 23.8 - 4.75 * 2.8

Now, we just need to do the math. Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). In this case, we do the multiplication first:

4. 75 * 2.8 = 13.3

Now, substitute that back into the equation:

μ = 23.8 - 13.3

Finally, we do the subtraction:

μ = 10.5

And there we have it! We've calculated the mean. It's like putting the last piece of a puzzle in place and seeing the whole picture come together. But before we celebrate too much, let's take a moment to double-check our answer and make sure it makes sense in the context of the problem. This is a crucial step in any problem-solving process – it's like proofreading your work before submitting it. So, let's move on to the next step: checking our answer for reasonableness.

Checking the Answer for Reasonableness

Great job on calculating the mean! We got μ = 10.5. But before we declare victory, let’s take a step back and ask ourselves: does this answer make sense? This is a crucial step in problem-solving, guys. It’s like being a detective who not only finds the suspect but also makes sure the evidence all adds up.

Remember, we had a z-score of 4.75, which is quite high. This tells us that the observed value (x = 23.8) is significantly above the mean. A standard deviation of 2.8 gives us the scale of how spread out the data is. If our calculated mean were, say, 50, it wouldn't make sense because 23.8 wouldn't be 4.75 standard deviations above 50.

Our calculated mean of 10.5 seems much more reasonable. If we add 4.75 times the standard deviation (2.8) to 10.5, we should get close to our observed value of 23.8. Let’s check:

10. 5 + (4.75 * 2.8) = 10.5 + 13.3 = 23.8

Bingo! It checks out perfectly. This gives us confidence that our answer is correct. It’s like having a witness confirm the detective’s theory. This step of checking for reasonableness is not just about math; it’s about critical thinking. It’s about understanding the relationships between the numbers and making sure the answer fits the story the data is telling us. So, with a reasonable answer in hand, we can confidently move on to the final step: stating the solution.

Stating the Solution

Alright, we've done the hard work, crunched the numbers, and double-checked our answer. Now, let's put a bow on it and clearly state our solution. The problem asked us to find the mean (μ) given the z-score, observed value, and standard deviation. We’ve gone through all the steps, and we've arrived at our answer:

The mean, μ, is 10.5.

It's important to state the solution clearly so that anyone reading your work can easily understand the result. Think of it like writing the final chapter of a book – you want to make sure the ending is clear and satisfying. Just writing down