Problem 22 - Algebra 2 and Trigonometry

Question

Products often come with registration forms. One of the questions usually found on the registration form is household income. For a given product, the data below represents a random sample of the income (in thousands of dollars) reported on the registration form. Find the standard deviation based on this sample. $$\begin{array}{llllllllll}{38} & {40} & {26} & {42} & {39} & {25} & {40} & {40} & {39} & {36} \\ {46} & {41} & {43} & {47} & {49} & {43} & {39} & {35} & {43} & {37}\end{array}$$

Step-by-Step Solution

Answer

The standard deviation is approximately 7.83.

Step 1: Calculate the Mean

The mean (average) income is found by summing all the income values and dividing by the number of values. There are 20 income values.\[\text{Mean} = \frac{38 + 40 + 26 + 42 + 39 + 25 + 40 + 40 + 39 + 36 + 46 + 41 + 43 + 47 + 49 + 43 + 39 + 35 + 43 + 37}{20}\]Perform the calculations:\[\text{Total sum} = 888\]\[\text{Mean} = \frac{888}{20} = 44.4\]

Step 2: Find the Deviations from the Mean

Subtract the mean from each income value to find the deviation of each value:\[38 - 44.4, \; 40 - 44.4, \; 26 - 44.4, \ldots, \; 37 - 44.4\]Calculated deviations:\[-6.4, -4.4, -18.4, -2.4, -5.4, -19.4, -4.4, -4.4, -5.4, -8.4, 1.6, -3.4, -1.4, 2.6, 4.6, -1.4, -5.4, -9.4, -1.4, -7.4\]

Step 3: Square the Deviations

Square each deviation to ensure all values are positive:\[(-6.4)^2, (-4.4)^2, (-18.4)^2, (-2.4)^2, \ldots, (-7.4)^2\]Calculated squared deviations:\[40.96, 19.36, 338.56, 5.76, 29.16, 376.36, 19.36, 19.36, 29.16, 70.56, 2.56, 11.56, 1.96, 6.76, 21.16, 1.96, 29.16, 88.36, 1.96, 54.76\]

Step 4: Calculate the Variance

Find the average of the squared deviations by summing them and dividing by the number of values minus one (to calculate the sample variance):\[\text{Variance} = \frac{\sum (\text{Squared deviations})}{n - 1} = \frac{1166.4}{19}\]Calculated variance:\[\text{Variance} \approx 61.39\]

Step 5: Calculate the Standard Deviation

The standard deviation is the square root of the variance:\[\text{Standard deviation} = \sqrt{61.39}\]Calculated standard deviation:\[\text{Standard deviation} \approx 7.83\]

Step 6: Conclusion

Based on the calculations, the standard deviation of the sample incomes is approximately 7.83.

Key Concepts

Sample Data Analysis

Sample data analysis is crucial in understanding trends and patterns within a specific subset of data. In this example, we analyzed a sample of household incomes based to calculate the standard deviation. The goal is to use a manageable subset to represent the larger population. When we work with samples, we assume they are randomly chosen, reflecting the diversity found in the full population.

• The sample includes 20 income values, expressed in thousands of dollars, providing insights into their income distribution.
• Sampling allows for more efficient data analysis as studying the entire population might be impractical due to time and cost constraints.
• By analyzing a sample, we achieve a balance between efficiency and accuracy in statistical inference.

Through sample data analysis, we aim to predict behaviors or establish patterns that apply to a broader population, making it a cornerstone in statistical analysis.

Variance Calculation

Variance is a statistical measurement of the spread between numbers in a data set, highlighting how far each number in the set is from the mean. To calculate variance, we first find the mean of the sample incomes as detailed in the exercise. Afterward, we determine how much each value deviates from the mean, and then square those deviations. Squaring serves two purposes: it removes any negative values and emphasizes larger deviations, making them more noticeable.

• Squaring each deviation ensures that the variance calculation takes into account the magnitude of the deviations, not just their direction.
• The sum of these squared deviations is then divided by the sample size minus one to account for the degrees of freedom in samples.
• This process results in a variance of approximately 61.39, a crucial step before determining the standard deviation.

Variance helps us understand how spread out the numbers in your data set are, and it's essential for further calculations like standard deviation, which provides even more context to the spread.

Mean Deviation

Mean deviation, also known as average deviation, measures the average of the absolute deviations from the mean. It gives us another perspective on how much the data values deviate from the mean, but unlike variance and standard deviation, it does not square the deviations. Though not covered in the original solution, it's an informative measure that provides insight into data consistency.

• We calculate mean deviation by taking the absolute value of each deviation from the mean, then finding the average of these absolute values.
• If deviations are small, it indicates that data points are closely packed around the mean, suggesting less variability.
• While mean deviation is less sensitive to large deviations compared to variance, it provides a simple and intuitive measure of data spread.

Understanding mean deviation complements the analysis obtained through variance and provides a fuller picture of data variability.