Problem 8 - Algebra 2

Question

Determine the whole number of standard deviations that includes all data values. The mean price of the nonfiction books on a best-sellers list is \(\$ 25.07 ;\) the standard deviation is \(\$ 2.62 .\) \(\$ 26.95, \$ 22.95, \$ 24.00, \$ 24.95, \$ 29.95, \$ 19.95, \$ 24.95, \$ 24.00, \$ 27.95, \$ 25.00\)

Step-by-Step Solution

Answer

The entire data set ranges within 2 standard deviations from the mean.

Step 1: Identify maximum and minimum values

Identify the highest and lowest prices from the data. This gives \$29.95 as the maximum value and \$19.95 as the minimum value.

Step 2: Calculation of respective deviations from the mean

Calculate the deviations of the maximum and minimum values from the mean. Subtract the mean from each value. This gives \$29.95 - \$25.07 = \$4.88 for maximum value and \$25.07 - \$19.95 = \$5.12 for minimum value.

Step 3: Conversion of deviations into standard deviations

Divide each deviation by the standard deviation to convert them into number of standard deviations. This results in \$4.88 / \$2.62 = 1.86 for the maximum value and \$5.12 / \$2.62 = 1.95 for the minimum value.

Step 4: Rounding off to whole numbers

Since we are interested in whole number of standard deviations, round off the above results to nearest whole numbers. Thus, the number of standard deviations for maximum and minimum values are 2 and 2 respectively.

Key Concepts

Mean and Median

When we talk about data analysis, two crucial statistical measures often surface: the mean and the median. Both are measures of central tendency, which means they help to identify the center point of a data set. The mean, also known as the average, is calculated by adding up all the values in a data set and then dividing by the number of values. In the exercise, the mean price of nonfiction books was calculated to be $25.07. This gives us a sense of the overall pricing level within the list.

The median, on the other hand, is the middle value in a list of numbers. If the dataset is arranged in ascending or descending order, the median is the number that separates the higher half from the lower half. While this exercise focuses on the mean, understanding the median can be equally important, especially in datasets with outliers. Outliers can skew the mean, making the median a better measure of central tendency in such cases.

Data Analysis

Data analysis involves examining, cleaning, transforming, and modeling data to discover useful information, draw conclusions, and support decision-making. In the context of the exercise, we are analyzing the prices of nonfiction bestsellers. We start by identifying key metrics such as the mean, and we also need to spot the highest and lowest data points.

In this exercise:

• The highest book price is $29.95.
• The lowest book price is $19.95.

Once these values are identified, we proceed to calculate how far these values deviate from the average. This process helps us understand the spread of the data and how the values are dispersed around the mean. Such analysis is crucial for businesses, as understanding data spreads can inform pricing strategies and market positioning.

Statistical Deviations

Statistical deviations, primarily the standard deviation, measure the amount of variation or dispersion in a set of values. When analyzing data, the standard deviation tells us how much the typical values deviate from the mean price. In the exercise, the standard deviation was $2.62.

Here's how we applied this to our data:

• We calculated the deviation of the highest value ($29.95) from the mean, which was $4.88.
• The deviation of the lowest value ($19.95) from the mean was $5.12.

By dividing these deviations by the standard deviation, we convert them into a number of standard deviations:

• $4.88 ÷ $2.62 = 1.86
• $5.12 ÷ $2.62 = 1.95

To determine whole numbers of standard deviations, we round these results, concluding that both maximum and minimum values are approximately 2 standard deviations away from the mean. This tells us that all values lie within 2 standard deviations, showcasing a relatively consistent dataset.