Problem 12
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). \(94,62,88,85,91\)
Step-by-Step Solution
Verified Answer
The mean of the data set is 84. The deviations from the mean are 10, -22, 4, 1, and 7. The sum of the deviations is 0.
1Step 1: Calculate the Mean
The mean is the average of a dataset and can be calculated by adding all the elements in the dataset, and then dividing by the number of elements. In this dataset, there are five elements: 94, 62, 88, 85, 91. Mean = (94 + 62 + 88 + 85 + 91) / 5 = 84.
2Step 2: Calculate the Deviation
The deviation from the mean for each data item in the set can be calculated by subtracting the mean from each item. So, deviations for each item from the mean are as follows: 94 - 84 = 10, 62 - 84 = -22, 88 - 84 = 4, 85 - 84 = 1, 91 - 84 = 7.
3Step 3: Sum of the Deviations
The part c of the question asks for the sum of the deviations calculated in the Step 2. So, sum of all the deviations would be: 10 + (-22) + 4 + 1 + 7 = 0.
Key Concepts
Mean CalculationDeviation from the MeanSum of Deviations
Mean Calculation
Understanding the concept of mean calculation is essential for students grappling with statistical analysis. The mean, commonly known as the average, is a way of expressing the central value of a data set. To calculate the mean, you simply add up all the numbers in the set and then divide by the number of items.
For example, consider the data set in the exercise: \(94, 62, 88, 85, 91\). The mean is calculated by adding these together, which gives us \(94 + 62 + 88 + 85 + 91 = 420\). This sum is then divided by the count of numbers in the set, which is 5. Thus, the mean is \(420 \div 5 = 84\). It's important that students understand this process as the mean is a fundamental statistical measure used across various disciplines.
For example, consider the data set in the exercise: \(94, 62, 88, 85, 91\). The mean is calculated by adding these together, which gives us \(94 + 62 + 88 + 85 + 91 = 420\). This sum is then divided by the count of numbers in the set, which is 5. Thus, the mean is \(420 \div 5 = 84\). It's important that students understand this process as the mean is a fundamental statistical measure used across various disciplines.
Deviation from the Mean
Once students have grasped mean calculation, they need to understand 'deviation from the mean'. This concept refers to the difference between each data point in the set and the mean. It describes how far each number is from the average and is a measure of variability.
In the context of our exercise data set \(94, 62, 88, 85, 91\), having found the mean to be 84, we calculate the deviation of each number by subtracting the mean from it. For instance:
In the context of our exercise data set \(94, 62, 88, 85, 91\), having found the mean to be 84, we calculate the deviation of each number by subtracting the mean from it. For instance:
- For the number 94, the deviation is \(94 - 84 = 10\).
- For the number 62, it is \(62 - 84 = -22\), and so on for the rest of the numbers.
Sum of Deviations
The sum of deviations is a concept that might seem somewhat counterintuitive at first, as it totals to zero when summing the individual deviations of a set from its mean. This is due to the nature of mean being the central value. For positive deviations, there will be equivalent negative deviations, balancing out to zero.
In our given set of numbers \(94, 62, 88, 85, 91\), after finding that the deviations are 10, -22, 4, 1, and 7, adding them up results in \(10 + (-22) + 4 + 1 + 7 = 0\). This sum is an important property of the mean, indicating that the deviations on either side of the mean balance each other out. This also demonstrates the reliability of the mean as a measure of central tendency.
Understanding this concept is critical for students, as it underpins more advanced statistical concepts like variance and standard deviation.
In our given set of numbers \(94, 62, 88, 85, 91\), after finding that the deviations are 10, -22, 4, 1, and 7, adding them up results in \(10 + (-22) + 4 + 1 + 7 = 0\). This sum is an important property of the mean, indicating that the deviations on either side of the mean balance each other out. This also demonstrates the reliability of the mean as a measure of central tendency.
Understanding this concept is critical for students, as it underpins more advanced statistical concepts like variance and standard deviation.
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