Problem 11 - Thinking Mathematically

Question

In Exercises 11-16, find a. the mean; b. the deviation from the mean for each data item; and c. the sum of the deviations in part (b). \(85,95,90,85,100\)

Step-by-Step Solution

Answer

a. The mean is 91. b. The deviations from the mean are -6, 4, -1, -6, 9. c. The sum of the deviations is 0.

Step 1: Compute the Mean

To find the mean of the data set \(85,95,90,85,100\), add all the numbers together and divide by the count of numbers, in this case 5: Mean = \(\frac{85 + 95 + 90 + 85 + 100}{5} = 91\)

Step 2: Compute Deviations from the Mean

To calculate the deviation of each data item from the mean, subtract the mean from each data item. The deviations are: \(85-91 = -6, 95-91 = 4, 90-91 = -1, 85-91 = -6, 100-91 = 9\)

Step 3: Sum the Deviations

To get the sum of the deviations, simply add together all the calculated deviations: Sum = \(-6 + 4 -1 -6 + 9 = 0\)

Key Concepts

Deviation from the Mean

Understanding deviation from the mean is a crucial concept in statistics, especially when analyzing the spread of data. The mean is a measure of central tendency, providing a central value of a data set. To understand how each data point differs from this central value, we calculate the deviation for each data item.

Here's how to determine the deviation from the mean:

• Find the mean of the data set. In our example, the data set is 85, 95, 90, 85, and 100. The mean is calculated as 91.
• Subtract the mean from each individual data point:
  • 85 - 91 = -6
  • 95 - 91 = 4
  • 90 - 91 = -1
  • 85 - 91 = -6
  • 100 - 91 = 9

These deviations tell us how each data point differs from the average value, indicating whether each point is above or below the mean.

Sum of Deviations

The sum of deviations is an interesting calculation, which often holds essential information about a data set. When you add up all these individual deviations from the mean, in a balanced data set, the sum should be zero. This is often a verification step in exercises.

Let's look at the data again:

• Deviations: -6, 4, -1, -6, 9
• Sum of deviations:
  • Add the deviations: -6 + 4 - 1 - 6 + 9
  • The result is 0.

This zero sum principle applies because the positive deviations exactly balance the negative ones, confirming our mean calculation is correct.

Data Set Analysis

Data set analysis often starts with calculating some key statistical meaures. The mean is one of them, providing a glimpse into the central tendency of the data points.
This mean acts as a reference point for further exploration, such as deviations, variations, and more. Through data set analysis:

• Identify trends and patterns within the data.
• Determine the extent to which data points deviate from the mean, giving insights into variability.
• Spot potential outliers, like the data point in our exercise, where 100 is the highest value and deviates significantly from the mean, 91.

These insights inform decisions, predictions, and understanding of data spreading across different domains.

Statistics Exercises

Statistics exercises deal with applying statistical concepts to solve problems and understand data behavior. Analyzing different data sets helps hone skills in calculating means, deviations, and sums.

Engaging in statistics exercises like the one provided gives you the opportunity to:

• Enhance computational skills when it comes to calculating the mean and deviations.
• Verify understanding by ensuring that the sum of deviations equates to zero, confirming accuracy in statistical procedures.
• Differentiate between symmetrical and asymmetrical data distributions.

Practise through exercises, as they are effective methods for gaining confidence and proficiency in data set analysis and statistical thinking.