Mean Deviation Calculation: Ages Of Bronchial Patients

Hey guys! Let's dive into a cool math problem. We're gonna calculate the mean deviation of the ages of patients treated for bronchial diseases in a hospital during a week. This involves understanding how spread out the ages are from the average. We'll break down the steps and interpret the results so you can totally grasp it. Ready?

Understanding the Data and Key Concepts

First off, what's mean deviation? It's a measure of how much, on average, the data points in a dataset differ from the mean (average) of the dataset. A higher mean deviation means the data points are more spread out, and a lower one means they're clustered closer to the average. Think of it like this: if everyone's age is super close to the average age of all the patients, the mean deviation is low. If ages vary widely, the mean deviation is high.

Here’s the data table we're working with:

• Edad (Age): This represents the age ranges of the patients, grouped into intervals. For instance, [10; 15[ means patients aged 10 years old or more, but less than 15 years old. The brackets and parenthesis are important here! Note how 15 is not included in the first interval but it is included in the second interval.
• fi (Frequency): This tells us how many patients fall into each age range. So, there are 3 patients aged between 10 and 15, 5 between 15 and 20, and 7 between 20 and 25. These are the number of patients.

To calculate the mean deviation, we'll need to go through a few steps. It sounds more complicated than it is, I swear. We're going to find:

1. The midpoint of each age interval.
2. The mean (average) age.
3. The absolute deviation of each midpoint from the mean.
4. The weighted average of these deviations (weighted by the frequency). This gives us the mean deviation.

Now, let's get into the nitty-gritty and calculate everything step by step. I promise, by the end of this, you’ll be able to calculate this yourself, and you’ll know how to interpret the results. Let's make this understandable and a piece of cake!

Step-by-Step Calculation of Mean Deviation

Alright, let’s crunch some numbers! We'll follow the steps mentioned earlier to get to the mean deviation. The key here is to stay organized; this helps a lot to avoid confusion and errors. Just take it one step at a time, and you'll do great! We have a series of operations to apply to our data, which we will detail below.

1. Find the Midpoint (xi) of Each Age Interval

The midpoint is the average of the lower and upper limits of each interval. It represents the central age within that range. To calculate this, we do this: Midpoint = (Lower Limit + Upper Limit) / 2. So, for each interval:

• [10; 15[: (10 + 15) / 2 = 12.5
• [15; 20[: (15 + 20) / 2 = 17.5
• [20; 25[: (20 + 25) / 2 = 22.5

Now, let's add these to our table:

2. Calculate the Mean (Average) Age (x̄)

To find the mean, we need to do a weighted average. We multiply each midpoint (xi) by its frequency (fi), sum these products, and then divide by the total number of patients. The formula is: x̄ = Σ(xi * fi) / Σfi

First, let’s calculate xi * fi for each interval:

• [10; 15[: 12.5 * 3 = 37.5
• [15; 20[: 17.5 * 5 = 87.5
• [20; 25[: 22.5 * 7 = 157.5

Now, sum these up: 37.5 + 87.5 + 157.5 = 282.5

Next, sum the frequencies to find the total number of patients: 3 + 5 + 7 = 15

Finally, calculate the mean: x̄ = 282.5 / 15 = 18.83 (rounded to two decimal places).

3. Determine the Absolute Deviation of Each Midpoint from the Mean |xi - x̄|

This tells us how far each midpoint is from the mean age. We calculate this by subtracting the mean from each midpoint and taking the absolute value (ignoring the sign). The formula is: |xi - x̄|

• [10; 15[: |12.5 - 18.83| = 6.33
• [15; 20[: |17.5 - 18.83| = 1.33
• [20; 25[: |22.5 - 18.83| = 3.67

Let’s add these to our table:

4. Compute the Mean Deviation

To do this, we multiply the absolute deviation of each midpoint by its frequency, sum those products, and divide by the total number of patients. The formula is: Mean Deviation = Σ(fi * |xi - x̄|) / Σfi

First, calculate fi * |xi - x̄| for each interval:

• [10; 15[: 3 * 6.33 = 18.99
• [15; 20[: 5 * 1.33 = 6.65
• [20; 25[: 7 * 3.67 = 25.69

Next, sum these products: 18.99 + 6.65 + 25.69 = 51.33

Finally, divide by the total number of patients: 51.33 / 15 = 3.42 (rounded to two decimal places).

Interpreting the Result: What Does it Mean?

So, we calculated a mean deviation of 3.42 years. What does this mean in practical terms? Well, it means that, on average, the ages of the patients are about 3.42 years away from the mean age of 18.83 years. This mean deviation gives us a sense of how spread out the ages are. A smaller number would have indicated that the ages were closer together, while a larger number would have suggested a wider distribution.

In our case, a mean deviation of 3.42 is a moderate spread. It tells us that while there is an average age, the patients' ages aren't all clustered tightly around that average. There's a bit of variation. The fact that the mean deviation is not too high means that the age distribution is not extremely dispersed, which can be a relief in a healthcare setting because it could indicate a more uniform health situation among the patient group, at least in terms of age. It's a key statistic to understand the dispersion of the data and gives a more comprehensive view than just looking at the average alone.

Now, let's summarize the whole thing, which is important to reinforce what we have learned.

Summary

Alright, let’s recap what we've done and why it matters:

1. Calculated Midpoints: We found the middle age for each interval.
2. Determined the Mean Age: We found the average age of the patients.
3. Calculated Absolute Deviations: We figured out how far each age range was from the average.
4. Computed Mean Deviation: We found the average of these deviations, which is our key result.

In the end, we learned how to compute and interpret the mean deviation, which is a key concept in statistics. We used this to understand the age distribution of patients with bronchial diseases. Understanding the spread of data helps in many areas, from healthcare to finance. Keep practicing, and you'll get better! That is all.

Hope this explanation helped you understand the concepts and the steps involved in calculating and interpreting the mean deviation. Keep practicing, and you’ll master it in no time. If you have any questions, feel free to ask! You got this! I know that you can do it!