Problem 14 - Algebra 2 and Trigonometry

Question

The given values represent data for a sample. Find the variance and the standard deviation based on this sample. \(\begin{array}{|c|c|}\hline x_{i} & {f_{i}} \\ \hline 55 & {11} \\ {50} & {15} \\ {45} & {4} \\ {40} & {1} \\ {35} & {1} \\ {35} & {12} \\ {30} & {4} \\ \hline\end{array}\)

Step-by-Step Solution

Answer

Variance: 81.9567; Standard Deviation: 9.053

Step 1: Calculate the Mean (Expected Value)

First, we find the mean of the dataset using the formula for the weighted mean:\[ \bar{x} = \frac{\sum (x_i \times f_i)}{\sum f_i} \]Here, the total frequency, \( \sum f_i = 48 \), and the sum of the products of the values and their frequencies is:\[ \sum (x_i \times f_i) = (55 \times 11) + (50 \times 15) + (45 \times 4) + (40 \times 1) + (35 \times 1) + (35 \times 12) + (30 \times 4) = 2035 \]Now compute the mean:\[ \bar{x} = \frac{2035}{48} \approx 42.3958 \]

Step 2: Calculate Each Contribution to Variance

Next, we find each term's contribution to the variance. For each value \(x_i\), compute \[ (x_i - \bar{x})^2 \times f_i \]For all values, compute:- \((55 - 42.3958)^2 \times 11 = 1581.85348\)- \((50 - 42.3958)^2 \times 15 = 876.0939\)- \((45 - 42.3958)^2 \times 4 = 68.40534\)- \((40 - 42.3958)^2 \times 1 = 5.74338\)- \((35 - 42.3958)^2 \times 1 = 54.21938\)- \((35 - 42.3958)^2 \times 12 = 650.63256\)- \((30 - 42.3958)^2 \times 4 = 614.81664\)The sum is approximately 3851.76462.

Step 3: Calculate the Sample Variance

Divide the sum of the contributions by the total frequency minus one to find the sample variance:\[ s^2 = \frac{\sum (x_i - \bar{x})^2 \times f_i}{n - 1} = \frac{3851.76462}{47} \approx 81.9567 \]

Step 4: Calculate the Standard Deviation

The standard deviation is the square root of the variance:\[ s = \sqrt{s^2} = \sqrt{81.9567} \approx 9.053 \]

Step 5: Final Answer

The variance of the sample is approximately 81.9567 and the standard deviation is approximately 9.053.

Key Concepts

Sample Mean Calculation

The sample mean is a crucial concept in statistics that helps us understand the average of a data set. In statistics, the sample mean is crucial for summarizing data points collectively. The sample mean is calculated by adding up all the individual data points, known as the sum of products of values and their frequencies, and then dividing by the total number of data points, which is the sum of their frequencies. This results in the average value of the sample representing the central tendency of your data.

In our example, to find the mean:

• First calculate the total frequency: \( \Sigma f_i = 48 \).
• Next, compute the sum of the products of values and their frequencies: \( \Sigma (x_i \times f_i) = 2035 \).
• Finally, divide this sum by the total frequency to get the weighted mean: \( \bar{x} = \frac{2035}{48} \approx 42.3958 \).

Thus, the sample mean tells us that on average, a data point in the sample is approximately 42.4.

Frequency Distribution

Frequency distribution is a step further into understanding data by showcasing how frequently each value appears in a dataset. It is represented in a tabular form listing each unique value, termed as \( x_i \), alongside the frequency \( f_i \), which indicates how many times each \( x_i \) appears.

The frequency distribution allows you to see patterns and trends within your data effectively. For instance, in our table:

• \( x_i = 55 \) and appears 11 times.
• \( x_i = 50 \) appears 15 times.
• \( x_i = 35 \) has two entries -- one appearing once and another 12 times.

These trends let us understand that certain scores are more common than others and can reveal the spread and concentration of data points, helping to further scoop into its variability.

Weighted Variance Calculation

Weighted variance is an extension of basic variance, taking into account when data points hold different levels of importance, as in frequency distributions. Calculating variance helps measure how much data points deviate from the mean.To calculate the weighted variance, follow these steps:

• Start by determining the mean: \( \bar{x} \approx 42.3958 \).
• Next, for each value \( x_i \), compute \((x_i - \bar{x})^2\), which finds the squared deviation from the mean.
• Multiply each squared deviation by its frequency \( f_i \) and sum these products. In our case, that sum is approximately 3851.76462.
• Finally, divide this total by the sum of the frequencies minus one (n-1):
  
  \[ s^2 = \frac{3851.76462}{47} \approx 81.9567 \]

Weighted variance shows the spread of data around the mean, providing insights into the variability within a sample, which is crucial in statistical analysis.