Problem 25 - Algebra 2

Question

Another measure of variation is variance, which equals \(\sigma^{2} .\) Find the variance and the standard deviation of each data set. $$ 0.7 \mathrm{g} \quad 0.84 \mathrm{g} \quad 0.9 \mathrm{g} \quad 0.8 \mathrm{g} \quad 0.69 \mathrm{g} $$

Step-by-Step Solution

Answer

Variance: 0.007, Standard Deviation: 0.081

Step 1: Calculate Mean

The first step is to find the mean (average) of the data set. Add up all the data points and then divide by the number of data points: \((0.7+0.84+0.9+0.8+0.69)/5 = 0.786\)

Step 2: Find Deviation from Mean

Next, find the variance by taking each data point, subtracting the mean from it, and squaring the result. The results will be: \((0.7-0.786)^2, (0.84-0.786)^2, (0.9-0.786)^2, (0.8-0.786)^2, (0.69-0.786)^2 = 0.007396, 0.002916, 0.012996, 0.000196, 0.009166\)

Step 3: Calculate Variance

Variance is the average of these squared differences. Add up the results and divide by the number of data results: \((0.007396+ 0.002916+ 0.012996+0.000196+0.009166)/5 = 0.006534\)

Step 4: Calculate Standard Deviation

Standard deviation is the square root of the variance. So, the variance \(\sigma^2 = 0.006534\) and the standard deviation \(\sigma = \sqrt{0.006534} = 0.081\)

Step 5: Round Result

The final answer should be rounded to three decimal places to match the precision of the original measurements. So, the variance is 0.007 and the standard deviation is 0.081.

Key Concepts

Understanding Standard Deviation

Standard deviation is a key concept in statistics that measures the amount of variation or dispersion in a set of data points. It gives us an idea of how much the numbers in a data set differ from the mean, or average, value.

To find the standard deviation, you first calculate the variance. Variance is the average of the squared differences from the mean. Once you have the variance, you simply take the square root to get the standard deviation.

The formula for standard deviation is:

• Step 1: Find the mean of the data set.
• Step 2: Subtract the mean from each data point and square the result.
• Step 3: Calculate the average of these squared differences.
  
  

For our data set example:

• The variance is 0.006534, which means the standard deviation is the square root of this value.
• The standard deviation is 0.081, providing a simple and easy-to-understand measure of how spread out the weights are around the average of the sample.

Mastering Mean Calculation

Calculating the mean is one of the first steps to analyzing a data set. The mean provides a central value that can represent the data set as a whole. Computing the mean is straightforward. Add up all of the numbers in a data set and then divide by the total number of data points.

For our weight data set:

• Add all the weights: 0.7 + 0.84 + 0.9 + 0.8 + 0.69 = 3.93
• Count the number of data points: 5
• Divide the total by the number of data points to get the mean: \(\frac{3.93}{5} = 0.786\)

The mean for this set is 0.786 grams, which serves as a simple representation of the entire sample's weight. It is the first step in calculating variance and standard deviation, which lets us determine how closely related each data point is to this central value.

Insightful Data Analysis

Data analysis involves examining, cleaning, transforming, and modeling data to discover useful information. It's all about making sense of numerical data sets. Understanding how to calculate measures of central tendency, like mean, variance, or standard deviation, is crucial in this process.

These metrics help us to:

• Identify patterns and trends.
• Make comparisons across different data sets.
• Predict outcomes based on data.

In our example, the mean gave us a central value, while the variance and standard deviation provided insight into the spread of the data points. A smaller standard deviation, like 0.081 in our calculation, shows that the data points are closely clustered around the mean. This knowledge helps inform decisions by portraying how variable data is, serving as a foundation for further statistical analysis or predictive modeling.