Problem 14 - 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). \(150,132,144,122\)

Step-by-Step Solution

Answer

The mean of the data is 137. The deviations from the mean are 13, -5, 7, and -15. The sum of deviations is 0.

Step 1: Calculate Mean

Firstly, calculate the mean. The mean (\( \overline{X} \)) is calculated as the sum of all numbers divided by the count of numbers. The list of numbers given are 150, 132, 144, and 122. The mean is therefore calculated as \( \overline{X} = \frac{(150+132+144+122)}{4} = 137 \).

Step 2: Calculate Deviation from Mean

Next, calculate the deviation from the mean for each number. Deviation is calculated as each individual number minus the mean. The deviations therefore are as follows: \( x_1 - \overline{X} = 150 - 137 = 13 \), \( x_2 - \overline{X} = 132 - 137 = -5 \), \( x_3 - \overline{X} = 144 - 137 = 7 \), \( x_4 - \overline{X} = 122 - 137 = -15 \). Thereby, the deviations are 13, -5, 7, -15.

Step 3: Calculate Sum of Deviations

Lastly, compute the sum of deviations from step 2. Thereby, it is done like so \( \sum (x_i - \overline{X}) = 13 + (-5) + 7 + (-15) = 0 \). Hence, it sums up to zero.

Key Concepts

Mean Calculation

Understanding mean calculation is fundamental in statistics, as it represents the average value in a set of numbers. To find the mean, you sum up all the values in a dataset and then divide by the number of values.

For instance, consider the set of numbers in the exercise: 150, 132, 144, and 122. Following the formula for the mean, you would add these numbers together to get the total sum, which is 548. You would then divide this sum by 4, since there are four numbers in the dataset. This gives you: \[ \overline{X} = \frac{150 + 132 + 144 + 122}{4} = \frac{548}{4} = 137 \].

This process reveals 137 to be the mean of the given dataset, allowing us to understand what value is 'central' for this particular set of numbers.

Deviation from Mean

Once you have calculated the mean, you can easily determine how much each data point deviates from that mean. The deviation is how far away a particular value is from the mean - a crucial step to measure the dispersion within a dataset.

To calculate the deviation for each data point, you subtract the mean from that data point. For example, with our numbers 150, 132, 144, and 122 and the calculated mean of 137, the deviations are computed as follows:\[ 150 - 137 = 13 \],\[ 132 - 137 = -5 \],\[ 144 - 137 = 7 \],\[ 122 - 137 = -15 \].

The result is a set of positive and negative numbers. Positive deviations indicate the data point is above the mean, while negative deviations indicate it is below the mean. This is crucial for identifying how data points are spread out in relation to the average.

Sum of Deviations

The sum of deviations is a concept used to check the accuracy of our mean calculation. Ideally, the sum of deviations from the mean should always equal zero. This is because the mean is the balance point, with deviations on either side canceling each other out.

For the given set of numbers, once we've calculated the individual deviations, we add them together to find the sum of deviations:\[ \sum (x_i - \overline{X}) = 13 + (-5) + 7 + (-15) = 0 \].

This shows that our mean is correctly computed as the balance point among data values. If the sum of deviations isn't zero, it's a sign that there might be an error in the mean calculation or in the deviation calculations. Interestingly, the concept behind the sum of deviations helps to underpin more advanced statistical measures like variance and standard deviation.