Problem 24 - Algebra 2

Question

Another measure of variation is variance, which equals \(\sigma^{2} .\) Find the variance and the standard deviation of each data set. $$ \$ 6.99 \quad \$ 5.50 \quad \$ 7.10 \quad \$ 9.22 \quad \$ 8.99 $$

Step-by-Step Solution

Answer

The variance of the data set of prices is approximately 1.9144 and the standard deviation is approximately 1.3832.

Step 1: Calculate the Mean

Firstly, the mean (average) of the dataset needs to be calculated. This can be done by adding up all the data points and dividing by the number of data points. In this case, \( (6.99 + 5.5 + 7.1 + 9.22 + 8.99) / 5 \) equals to $7.56.

Step 2: Compute the Variance

Next, the variance needs to be calculated. Each individual data point needs to have the (mean) subtracted from it and then square the resulting number. Lastly, add these squared values together and divide by the number of data points. The calculations are as follows: \( ((6.99-7.56)^{2}+(5.5-7.56)^{2}+(7.1-7.56)^{2}+(9.22-7.56)^{2}+(8.99-7.56)^{2}) / 5 \). The variance equals to approximately 1.9144.

Step 3: Determine the Standard Deviation

Finally, to calculate the standard deviation, the square root of the previously calculated variance will be found. So, \( \sqrt{1.9144} \) will be calculated, which is approximately 1.3832.

Key Concepts

Standard Deviation

Standard deviation is a key metric that helps us understand how spread out the numbers in a dataset are. It tells us, in essence, how much the values in a dataset vary from the average (mean). A low standard deviation means that most of the numbers are close to the mean. Conversely, a high standard deviation indicates that the numbers are spread out over a wider range.
To calculate the standard deviation, you first need the variance, which is a measure of the data's spread. Once you have variance, the standard deviation is simply the square root of that variance. For our current dataset, the variance is 1.9144, so the standard deviation is approximately 1.3832.
Understanding standard deviation helps in making predictions or decisions based on data, as it gives an idea about the reliability and variability of the data points. It’s an essential tool in statistics and data science.

Mean Calculation

Calculating the mean is one of the first steps in data analysis. The mean, or average, gives us the central value of a dataset, providing a single value that summarizes the dataset efficiently.
To find the mean, simply add up all the values in your dataset and then divide by the number of values. For instance, with the dataset \( \\(6.99, \\)5.50, \\(7.10, \\)9.22, \\(8.99 \), you add them up to get \( 37.8 \) and then divide by 5, since there are 5 data points. This gives a mean of \( \\)7.56 \).

• This step is crucial because the mean is used to calculate other statistics, like variance and standard deviation.
• It helps in understanding where the center of your data points lie.

In the context of our exercise, understanding the mean helps set the stage for further calculations.

Dataset Analysis

Dataset analysis involves looking at the components of your data to draw conclusions or make informed decisions. This process typically begins with descriptive statistics like mean, variance, and standard deviation.
In our dataset of price values, analyzing each component’s deviation from the mean can tell us a lot about the overall price stability. By subtracting the mean from each data point, squaring the result, and then averaging those squared differences, we're preparing to find the variance.

• This tells us how much variability exists in the data.
• Each step in dataset analysis builds on the last, leading to deeper insights.

This structured approach is crucial in fields like finance and economics, where understanding fluctuations in data is essential.

Measure of Variation

Measures of variation, like variance and standard deviation, provide insights into the spread of a dataset. While the mean gives the central tendency, measures of variation tell us about the data's spread. In the exercise, the variance was calculated using the formula: \[ \frac{(6.99-7.56)^{2}+(5.5-7.56)^{2}+(7.1-7.56)^{2}+(9.22-7.56)^{2}+(8.99-7.56)^{2}}{5} \]This arithmetic process gives a variance of approximately 1.9144.

• Variance breaks down the differences between each data point and the mean, showing their squared distances.
• A higher variance indicates a wider spread in the dataset values.

By taking these measures, you gain a clearer picture of your dataset. This understanding helps in predicting, controlling, and making strategic decisions based on the data’s variation and reliability.