Problem 11 - Algebra 2 and Trigonometry

Question

The given values represent data for a sample. Find the variance and the standard deviation based on this sample. 12.1, 33.3, 45.5, 60.1, 94.2, 22.2

Step-by-Step Solution

Answer

Variance: 910.67, Standard Deviation: 30.17.

Step 1: Find the Mean

Calculate the mean of the sample data. The formula for the mean is: \( \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \), where \( x_i \) are the data values and \( n \) is the number of data values. For the given data (12.1, 33.3, 45.5, 60.1, 94.2, 22.2), \( n = 6 \). Thus, the mean is \( \bar{x} = \frac{12.1 + 33.3 + 45.5 + 60.1 + 94.2 + 22.2}{6} = \frac{267.4}{6} = 44.5667 \).

Step 2: Calculate the Deviations

Find the deviation of each data point from the mean. The deviations are the differences between each data point and the mean: \( x_i - \bar{x} \) for each \( x_i \).

Step 3: Square the Deviations

Square each of the deviations obtained in Step 2. This removes negative signs and ensures each term contributes positively to the variance.

Step 4: Find the Sum of Squared Deviations

Add up all the squared deviations. This sum is used in the formula for variance.

Step 5: Calculate the Variance

Use the formula for the sample variance: \( s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} \). Substitute the sum of squared deviations from Step 4 and \( n - 1 = 5 \) (because it's a sample).

Step 6: Calculate the Standard Deviation

Take the square root of the variance to find the standard deviation: \( s = \sqrt{s^2} \).

Key Concepts

Understanding Mean Calculation

Calculating the mean is an essential statistical technique that provides a central value for a data set. The mean is essentially the "average" of all your data points. To find it, you sum up all the numbers and then divide by how many numbers there are.
In this part of the process, for instance, our data points are 12.1, 33.3, 45.5, 60.1, 94.2, and 22.2. Adding them gives a total of 267.4. We then divide this sum by the number of data points, which is 6. This calculation gives us a mean of approximately 44.5667.

• This simple process helps us understand what the 'average' person or a unit might represent in a sampling scenario.
• It is crucial as it forms the basis for further calculations such as variance and standard deviation.

Mean not only gives a snapshot average but is also the starting point for understanding how data values deviate from this center.

Sample Variance Simplified

To measure the spread or variability of data points, variance comes into play. Variance tells us how much data values deviate from the mean, in squared units. It essentially informs us about the data dispersion.
Firstly, each data point must be subtracted from the mean, and the difference is then squared. Squaring these differences ensures they are positive, emphasizing the magnitude of variance irrespective of the direction.
Next, for a sample, we calculate variance using the formula: \[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} \]where \((x_i - \bar{x})\) is the difference of each data value from the mean, and \(n - 1\) accounts for the sample size.

• In this example, after calculating all squared deviations, divide the sum by 5 (since there are 6 items, \(n - 1 = 5\)).
• Variance provides insight into the consistency and reliability of data.

Consider variance as the base groundwork for exploring further statistical measures like standard deviation.

Mastering Standard Deviation Computation

Standard deviation is closely related to variance, yet it represents dispersion in the same units as original data. It’s essentially the square root of variance, providing an intuitive gauge of variability.
Once you have the variance from your computations, the next step is its square root. This is often more intuitive and interpretable as it tells you, on average, how far the data points deviate from the mean. For instance, if the variance computed was \(s^2\), the standard deviation \(s\) would be: \[ s = \sqrt{s^2} \]

• It is a preferred measure over variance because it is expressed in the same units as data.
• Useful for gauging volatility and consistency within a set of data.

You might consider standard deviation as a vital indicator of predictability and set expectations on how much variability or "risk" can be anticipated in data points.

The Art of Data Analysis

Data analysis is the overarching process that allows us to interpret data meaningfully. By summarizing data through measures like the mean, variance, and standard deviation, we can draw insights and make informed decisions.

• Conducting a thorough analysis involves calculating various statistics that highlight different aspects, such as central tendency and variability.
• Mean gives a central point, variance shows how data spread around the mean, and standard deviation offers a practical understanding of data dispersion.

In real-world scenarios, this might help in fields such as finance for risk assessment, in academia for research purposes, or in business for strategic planning. Data analysis bridges the gap between raw data and actionable knowledge.