Problem 16 - Algebra 2 and Trigonometry

Question

The given values represent data for a sample. Find the variance and the standard deviation based on this sample. \(\begin{array}{|c|c|}\hline x_{i} & {f_{i}} \\ \hline 1 & {3} \\ {2} & {3} \\\ {3} & {3} \\ {4} & {3} \\ {5} & {3} \\ {6} & {3} \\ {7} & {3} \\\ \hline\end{array}\)

Step-by-Step Solution

Answer

Sample variance is 4.2; standard deviation is approximately 2.05.

Step 1: Calculate the Mean

First, find the mean \( \bar{x} \) of the dataset. Multiply each value \( x_i \) by its frequency \( f_i \) to get the total sum, then divide by the total number of values. The sum is: \( \sum x_i \cdot f_i = 1\cdot3 + 2\cdot3 + 3\cdot3 + 4\cdot3 + 5\cdot3 + 6\cdot3 + 7\cdot3 = 84 \). Since there are 21 data points, the mean is \( \bar{x} = \frac{84}{21} = 4 \).

Step 2: Calculate Squared Deviation for Each Value

For each \( x_i \), compute \((x_i - \bar{x})^2\) and multiply it by \( f_i \). This gives: \((1-4)^2 \cdot 3 + (2-4)^2 \cdot 3 + (3-4)^2 \cdot 3 + (4-4)^2 \cdot 3 + (5-4)^2 \cdot 3 + (6-4)^2 \cdot 3 + (7-4)^2 \cdot 3 \). This simplifies to: \(27 + 12 + 3 + 0 + 3 + 12 + 27 = 84 \).

Step 3: Compute the Sample Variance

To find the sample variance \( s^2 \), divide the sum of the squared deviations by \( n-1 \) (where \( n \) is the total number of data points). Thus, \( s^2 = \frac{84}{21-1} = \frac{84}{20} = 4.2 \).

Step 4: Calculate the Standard Deviation

The standard deviation \( s \) is the square root of the variance. Therefore, \( s = \sqrt{4.2} \approx 2.05 \).

Key Concepts

Understanding Variance

Variance is a measure of how much the data points in a dataset differ from the mean. It helps us understand the spread or dispersion of the data. When working with a sample, as in this exercise, we calculate the sample variance using the formula. The variance is found by:

• First calculating the squared difference of each data point from the mean.
• Then, averaging these squared differences for all data points, adjusted by dividing by one less than the number of points (n-1).

This adjustment is important in sample data analysis to account for the fact that we are not using the entire population. A high variance indicates that the data points are spread out over a wider range of values, whereas a low variance indicates that they are closer to the mean.

Calculating the Mean

The mean, often referred to as the average, is a central value for a dataset. In simple terms, it's the sum of all values divided by the number of values. For a frequency distribution table like the one in the exercise, the mean is calculated by multiplying each data point by its frequency and then dividing by the total number of observations. In this exercise:

• Each value is multiplied by its frequency.
• The total sum derived from these products is divided by the total count of observations.

This provides the average of the dataset, offering a central reference point, which is essential when calculating variance and standard deviation.

Exploring Frequency Distribution

Frequency distribution is a summary of how often each value in a dataset occurs. It organizes data into a table, allowing you to see how frequent each result or group of results is. In the exercise, each value from 1 to 7 occurs with the same frequency, which is 3. This regularity:

• Makes calculations straightforward.
• Shows a uniform distribution, with each outcome equally probable.

Understanding frequency distribution is crucial when performing statistical analysis, as it’s used to calculate mean, variance, and other statistical measures.

Sample Data Analysis Techniques

Sample data analysis focuses on evaluating a portion of data from a larger population. This type of analysis aims to make inferences about the population as a whole. Here's how to approach sample data analysis with the given data:

• Compute basic statistical measures like mean, variance, and standard deviation.
• Use these measures to understand the overall distribution and variability of the dataset.
• Acknowledge the sample nature by employing degrees of freedom (n-1) in variance calculations, which compensates for not analyzing the entire population.

Sample analysis is widely used since collecting data from an entire population is often impractical. It enables conclusions about population traits by examining the sample characteristics.