Calculating Standard Deviation: Children Of School Employees
Hey guys! Today, we're diving into a cool math problem. We'll be using the frequency table you provided to calculate the standard deviation of the number of children that 80 school employees have. Understanding standard deviation is super useful because it tells us how spread out the data is. Ready to break it down?
Understanding the Frequency Table and the Goal
Okay, so the frequency table gives us a neat summary of the data. Here's a quick recap of what it shows:
| Number of Children | Absolute Frequency |
|---|---|
| 0 | 20 |
| 1 | 36 |
| 2 | 14 |
| 3 | 8 |
| 4 | 2 |
This table tells us, for example, that 20 employees have no children, 36 have one child, and so on. Our main goal is to find the standard deviation, which measures the amount of variation or dispersion in a set of values. In this case, we want to know how much the number of children varies among the school's employees. The smaller the standard deviation, the closer the data points are to the mean (average), and the larger the standard deviation, the more spread out the data is. It's like, if everyone had roughly the same number of kids, the standard deviation would be low. If some had none and others had many, it'd be higher. We need to calculate a couple of things before we can get to the standard deviation. First, we need to determine the mean, or the average number of children. Second, we'll calculate the difference between each data point (number of children) and the mean. Third, we square each of those differences, and then find the average of the squared differences. Finally, we take the square root of that average. That's our standard deviation. Let's start with calculating the mean, alright?
Step 1: Calculate the Mean (Average)
Alright, let's start with the first step! To find the mean, we'll use a weighted average, because some numbers (like '1 child') appear more often than others. Here's how we do it:
- Multiply the number of children by its frequency.
- Sum all of those results.
- Divide the sum by the total number of employees (80).
Here’s the calculation:
- (0 children * 20 employees) = 0
- (1 child * 36 employees) = 36
- (2 children * 14 employees) = 28
- (3 children * 8 employees) = 24
- (4 children * 2 employees) = 8
Now, let's add up all those products: 0 + 36 + 28 + 24 + 8 = 96
Finally, we divide this sum (96) by the total number of employees (80): 96 / 80 = 1.2.
So, the mean number of children per employee is 1.2. Got it? The mean gives us the central tendency of the data – a kind of average. Cool, right? Now we can move on to the next part, which is all about finding out how spread out the data is around this mean. Keep up the good work!
Step 2: Calculate the Deviations
Now we're moving onto a slightly more complex step, but don't worry, we'll walk through it together. We're going to figure out how much each data point (number of children) deviates from the mean (1.2). This is a crucial step in calculating the standard deviation. We do this by subtracting the mean from each number of children, then we multiply by its frequency to account for how often it appears. It’s like, how far off is each value from the average, and then, how important is that difference? Let's break it down:
| Number of Children (x) | Frequency (f) | x - Mean (1.2) | (x - Mean) * f |
|---|---|---|---|
| 0 | 20 | 0 - 1.2 = -1.2 | -1.2 * 20 = -24 |
| 1 | 36 | 1 - 1.2 = -0.2 | -0.2 * 36 = -7.2 |
| 2 | 14 | 2 - 1.2 = 0.8 | 0.8 * 14 = 11.2 |
| 3 | 8 | 3 - 1.2 = 1.8 | 1.8 * 8 = 14.4 |
| 4 | 2 | 4 - 1.2 = 2.8 | 2.8 * 2 = 5.6 |
Let’s focus on the first row, where the number of children is 0. Subtracting the mean (1.2) gives us -1.2. We then multiply this by the frequency of 20, which gives us -24. This -24 represents the total deviation from the mean for all the employees who have zero children. We do this calculation for each row in the table. The result is: -7.2, 11.2, 14.4, and 5.6. Then we add up all of these numbers for a total of -24 + -7.2 + 11.2 + 14.4 + 5.6 = 0.
Step 3: Square the Deviations and Calculate the Variance
Next up, we square each deviation. This step is super important because it gets rid of those negative numbers. Squaring the deviations makes sure that values both above and below the mean contribute positively to our calculation of the spread. Here’s what it looks like, continuing with our table:
| Number of Children (x) | Frequency (f) | x - Mean (1.2) | (x - Mean) ^2 | (x - Mean)^2 * f |
|---|---|---|---|---|
| 0 | 20 | -1.2 | 1.44 | 1.44 * 20 = 28.8 |
| 1 | 36 | -0.2 | 0.04 | 0.04 * 36 = 1.44 |
| 2 | 14 | 0.8 | 0.64 | 0.64 * 14 = 8.96 |
| 3 | 8 | 1.8 | 3.24 | 3.24 * 8 = 25.92 |
| 4 | 2 | 2.8 | 7.84 | 7.84 * 2 = 15.68 |
Now, let's look at the first row. The number of children is 0. The deviation from the mean is -1.2. We square -1.2, which gives us 1.44. Then we multiply it by the frequency of 20, which results in 28.8. We repeat this process for each row. The results are: 1.44, 8.96, 25.92, and 15.68. The sum of these values is 28.8 + 1.44 + 8.96 + 25.92 + 15.68 = 80.8.
We divide this result (80.8) by the total number of employees (80): 80.8 / 80 = 1.01. This value is known as the variance. The variance gives an average of the squared differences from the mean.
Step 4: Calculate the Standard Deviation
Alright! We’re on the final stretch! The standard deviation is simply the square root of the variance. We've already done all the hard work, so this step is pretty straightforward. Remember from the previous step that the variance is 1.01. So, we just need to find the square root of 1.01.
√1.01 ≈ 1.005
So, the standard deviation is approximately 1.005. This means that the number of children an employee has deviates, on average, by about 1.005 from the mean (1.2 children). If we had a low standard deviation, say close to zero, it would mean that most employees have almost the same number of children. But with a value around 1.005, we can conclude that there’s a noticeable variation in the number of children among the employees. The greater the standard deviation, the more spread out the data. High standard deviations mean the data points are further from the mean, and in this case, the spread shows a range in the number of children employees have.
Conclusion: Understanding the Results
Congratulations, we did it! We’ve successfully calculated the standard deviation of the number of children for the school employees, using the frequency table provided. Remember, the standard deviation is a measure of how much the values in a dataset vary or spread out. In this case, our standard deviation is approximately 1.005, which tells us that the number of children employees have varies somewhat from the average of 1.2. The standard deviation helps us understand how the data is distributed. With this kind of analysis, you can get a clearer picture of your data and draw more informed conclusions. Thanks for hanging in there, guys! I hope this helped you understand how to calculate the standard deviation using a frequency table. Keep practicing, and you'll become a pro in no time! Keep exploring and learning, and you'll find that statistics can be a lot of fun. Until next time!