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

What is the variance of the set of numbers derived from the expression $(250)^2+(-400)^2+(-250)^2+(225)^2+(-125)^2+(-275)^2+(325)^2+(75)^2+(-75)^2-423,750+(650)^2+(-150)^2+(-600)^2+(250)^2+(150)^2+(-300)^2=980,000$? What is the standard deviation, rounded to the nearest whole number?

In mathematics and statistics, variance and standard deviation are crucial measures of dispersion, indicating how spread out a set of data is. To truly understand the concept presented, we need to break down the mathematical expression, compute the variance, and then derive the standard deviation. Let's dive into each step to clarify these important concepts.

Breaking Down the Mathematical Expression

At first glance, the expression might seem daunting, but it’s essentially a series of squared numbers being added and subtracted. Understanding each component helps to simplify the overall calculation.

The expression is given as:

$(250)^2+(-400)^2+(-250)^2+(225)^2+(-125)^2+(-275)^2+(325)^2+(75)^2+(-75)^2-423,750+(650)^2+(-150)^2+(-600)^2+(250)^2+(150)^2+(-300)^2=980,000$

To find the variance and standard deviation, we first need to identify the set of numbers involved. These numbers are:

250, -400, -250, 225, -125, -275, 325, 75, -75, 650, -150, -600, 250, 150, -300

We have 15 numbers in total. Now, let's proceed to calculate the mean, which is the first step in finding the variance and standard deviation.

Calculating the Mean

The mean, often referred to as the average, is a fundamental measure of central tendency. It is calculated by summing all the numbers in the dataset and then dividing by the count of numbers. This provides a single value that represents the center of the data distribution.

For our dataset, the mean (

$\mu$

) is calculated as follows:

$\mu = \frac{250 + (-400) + (-250) + 225 + (-125) + (-275) + 325 + 75 + (-75) + 650 + (-150) + (-600) + 250 + 150 + (-300)}{15}$

$\mu = \frac{250 - 400 - 250 + 225 - 125 - 275 + 325 + 75 - 75 + 650 - 150 - 600 + 250 + 150 - 300}{15}$

$\mu = \frac{-100}{15}$

$\mu ≈ -6.67$

So, the mean of our dataset is approximately -6.67. This value will be crucial in the next step, where we calculate the variance.

Determining the Variance

The variance measures how much the individual data points in a set vary from the mean. A high variance indicates that the data points are spread out over a large range, while a low variance indicates that the data points are clustered closely around the mean. The variance is calculated by averaging the squared differences between each data point and the mean.

The formula for variance (

$\sigma^2$

) is:

$\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \mu)^2}{n}$

Where:

$x_i$

represents each individual data point.

$\mu$

is the mean of the dataset.

$n$

is the number of data points.

Let’s apply this formula to our dataset:

$\sigma^2 = \frac{(250 - (-6.67))^2 + (-400 - (-6.67))^2 + (-250 - (-6.67))^2 + (225 - (-6.67))^2 + (-125 - (-6.67))^2 + (-275 - (-6.67))^2 + (325 - (-6.67))^2 + (75 - (-6.67))^2 + (-75 - (-6.67))^2 + (650 - (-6.67))^2 + (-150 - (-6.67))^2 + (-600 - (-6.67))^2 + (250 - (-6.67))^2 + (150 - (-6.67))^2 + (-300 - (-6.67))^2}{15}$

Calculating each term:

$\sigma^2 = \frac{(256.67)^2 + (-393.33)^2 + (-243.33)^2 + (231.67)^2 + (-118.33)^2 + (-268.33)^2 + (331.67)^2 + (81.67)^2 + (-68.33)^2 + (656.67)^2 + (-143.33)^2 + (-593.33)^2 + (256.67)^2 + (156.67)^2 + (-293.33)^2}{15}$

$\sigma^2 = \frac{65879.11 + 154710.89 + 59209.78 + 53670.89 + 14001.78 + 72001.78 + 110004.89 + 6670.89 + 4668.89 + 431282.89 + 20543.11 + 352040.89 + 65879.11 + 24545.78 + 86042.89}{15}$

$\sigma^2 = \frac{1461153.3}{15}$

$\sigma^2 ≈ 97410.22$

Thus, the variance of the dataset is approximately 97410.22. This value provides a measure of the data's spread, but to get a more interpretable measure, we calculate the standard deviation.

Calculating the Standard Deviation

The standard deviation is another essential measure of dispersion that provides a more intuitive understanding of data spread. It is the square root of the variance and is expressed in the same units as the original data, making it easier to interpret. A smaller standard deviation indicates that data points are closer to the mean, while a larger standard deviation indicates a wider spread.

The formula for standard deviation (

$\sigma$

) is:

$\sigma = \sqrt{\sigma^2}$

Using the variance we calculated, we find the standard deviation:

$\sigma = \sqrt{97410.22}$

$\sigma ≈ 312.11$

Rounding to the nearest whole number, the standard deviation is approximately 312.

Conclusion

In summary, we started with a complex mathematical expression and identified a set of 15 numbers. We then calculated the mean, which is approximately -6.67. Using this mean, we computed the variance to be approximately 97410.22. Finally, we found the standard deviation by taking the square root of the variance, resulting in approximately 312 when rounded to the nearest whole number.

The variance and standard deviation are vital tools for understanding the distribution and variability within a dataset. They help us make informed decisions and draw meaningful conclusions from the data. By understanding these concepts, one can effectively analyze and interpret data in various fields, from mathematics and statistics to finance and engineering. The standard deviation, in particular, gives a clear sense of how much the data deviates from the average, providing valuable insights into the data's characteristics.