Calculating Variance And Standard Deviation A Step-by-Step Guide

by Sam Evans 65 views
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Let's break down how to find the variance for the given set of numbers. Variance is a measure of how spread out a set of data is. In simpler terms, it tells us how much the numbers in a data set differ from the average (mean). It's a crucial concept in statistics, helping us understand the distribution and variability within a dataset. Now, in this guide, I will give you a very detailed approach to understanding variance, so you will never have questions about it again.

To calculate the variance, we'll follow these steps:

  1. Calculate the mean (average): This involves adding up all the numbers in the set and then dividing by the total number of values. The mean serves as the central point around which the data is distributed. It provides a baseline for understanding the typical value in the dataset. Let's begin, guys!

  2. Find the difference from the mean: For each number in the set, we subtract the mean to find the deviation. These deviations show how far each number is from the average. Some will be positive, others negative, depending on whether the number is above or below the mean.

  3. Square the differences: We square each of the differences we calculated in the previous step. Squaring the differences has two key purposes. First, it eliminates negative signs, so that values below the mean don't cancel out values above the mean. Second, it gives larger deviations more weight in the final result.

  4. Sum the squared differences: We add up all the squared differences. This sum represents the total variability in the dataset, but it's still influenced by the number of data points.

  5. Divide by the number of values (or n-1 for sample variance): This gives us the average of the squared differences, which is the variance. If we are working with the entire population, we divide by n (the number of values). If we are working with a sample, we divide by n-1 to get an unbiased estimate of the population variance. Dividing by n-1 is known as Bessel's correction, which accounts for the fact that the sample mean is likely to be closer to the sample data points than the true population mean.

Now, let's apply these steps to the given data set: (250)2+(−400)2+(−250)2+(350)2+(100)2+(−50)2=420,000(250)^2+(-400)^2+(-250)^2+(350)^2+(100)^2+(-50)^2=420,000

First, we need to figure out what the actual data set is. The equation provided seems to be calculating the sum of squares directly. It looks like the numbers in our data set are 250, -400, -250, 350, 100, and -50. These are the values we'll use to calculate the variance and standard deviation.

So, we have the following values:

  • 250
  • -400
  • -250
  • 350
  • 100
  • -50

Let's start by finding the mean (average):

To calculate the mean, we sum up all the values and divide by the number of values (which is 6 in this case). It is always the first step, guys!

Mean = (250 + (-400) + (-250) + 350 + 100 + (-50)) / 6

Mean = (250 - 400 - 250 + 350 + 100 - 50) / 6

Mean = (0) / 6

Mean = 0

Okay, the mean is 0. That simplifies things a bit for the next step!

Next, we find the difference from the mean for each value: Since the mean is 0, this step is quite straightforward. The difference from the mean for each value is simply the value itself.

  • 250 - 0 = 250
  • -400 - 0 = -400
  • -250 - 0 = -250
  • 350 - 0 = 350
  • 100 - 0 = 100
  • -50 - 0 = -50

Now, let's square each of these differences: Squaring each difference ensures that we're dealing with positive values, and it also gives more weight to larger deviations from the mean.

  • 2502250^2 = 62,500
  • (−400)2(-400)^2 = 160,000
  • (−250)2(-250)^2 = 62,500
  • 3502350^2 = 122,500
  • 1002100^2 = 10,000
  • (−50)2(-50)^2 = 2,500

Now, we sum these squared differences:

Sum of squared differences = 62,500 + 160,000 + 62,500 + 122,500 + 10,000 + 2,500

Sum of squared differences = 420,000

This result matches the initial equation provided in the problem, which confirms that we are on the right track!

Finally, we divide by the number of values minus 1 (since we're calculating the sample variance): We use n-1 (which is 6-1 = 5) instead of n when calculating the variance for a sample. This is known as Bessel's correction, and it provides a less biased estimate of the population variance.

Sample Variance = 420,000 / (6 - 1)

Sample Variance = 420,000 / 5

Sample Variance = 84,000

So, the sample variance for this dataset is 84,000. That's quite a high number, indicating that the data points are quite spread out from the mean.

Alright, let's tackle the standard deviation! The standard deviation is essentially a measure of how dispersed a set of data is. Think of it as the average distance the data points are from the mean. It's like the variance's cooler, more interpretable cousin because it's in the same units as the original data. So, if we're talking about dollar amounts, the standard deviation will also be in dollars, which makes it much easier to understand than the variance, which would be in dollars squared.

To find the standard deviation, we simply take the square root of the variance. It's that simple! The standard deviation gives us a clearer picture of the typical deviation from the mean.

Let's recap: We've already calculated the sample variance in the previous section, which was 84,000. Now, we just need to take the square root of that value.

Standard Deviation = √Variance

Standard Deviation = √84,000

Using a calculator, we find:

Standard Deviation ≈ 289.8275

The question asks us to round the standard deviation to the nearest whole number. So, we round 289.8275 to 290.

Therefore, the standard deviation, rounded to the nearest whole number, is 290.

In Summary:

  • We first calculated the sample variance, which came out to be 84,000. This value tells us how much the data points vary around the mean. A larger variance indicates greater variability.
  • Then, we found the standard deviation by taking the square root of the variance. This gave us a standard deviation of approximately 289.8275. Rounding this to the nearest whole number, we get 290. This value represents the typical amount that each data point deviates from the mean. In this case, on average, the data points are about 290 units away from the mean.

So, to recap, we've determined the following:

  • Variance: 84,000
  • Standard Deviation (rounded to the nearest whole number): 290

These two numbers give us a good sense of the distribution of our data set. The variance tells us the overall spread, while the standard deviation gives us a more interpretable measure of the typical deviation from the mean. Understanding these concepts is crucial in various fields, from finance to engineering, where analyzing data variability is key to making informed decisions. Great job, guys! You have nailed the topic now!