Problem 7 - Thinking Mathematically

Question

In Exercises 7-10, a group of data items and their mean are given. a. Find the deviation from the mean for each of the data item. b. Find the sum of the deviations in part \((a)\). \(3,5,7,12,18,27 ;\) Mean \(=12\)

Step-by-Step Solution

Answer

The deviations from the mean for the data items are -9, -7, -5, 0, 6, 15 and their sum is 0.

Step 1: Calculate Deviations

Calculate the deviations for each data item from the mean. For each data item \(x_i\), the deviation \(D_i\) is calculated as \(D_i = x_i - \text{mean}\). Here are the calculations for each data item: \(D_1 = 3 - 12 = -9\), \(D_2 = 5 - 12 = -7\), \(D_3 = 7 - 12 = -5\), \(D_4 = 12 - 12 = 0\), \(D_5 = 18 - 12 = 6\), \(D_6 = 27 - 12 = 15\)

Step 2: Sum the Deviations

Add up all the deviations calculated in step 1. \(\text{Sum} = D_1 + D_2 + D_3 + D_4 + D_5 + D_6 = -9 - 7 - 5 + 0 + 6 + 15 = 0\)

Key Concepts

Understanding Data Items

Data items are essentially the individual pieces of information in a data set. In our exercise, these data items are the numbers 3, 5, 7, 12, 18, and 27. Each of these items represents a specific value that we will use to perform further calculations. It is important to recognize each data item because they are the building blocks in statistical calculations, such as finding the mean and deviations.

When working with data items, remember that the mean, or average, provides a central value around which each data item is compared. In this case, the mean is given as 12. This will be our reference point for identifying how far each data item is from this central value. Data items together with their computed deviations can offer insights into the spread and pattern in the dataset.

Performing Deviation Calculation

The deviation calculation involves finding out how much each data item differs from the mean of the dataset. This process helps us understand the distribution of the data. For each data item, we follow a simple formula: the deviation \(D_i\) is calculated as the difference between the data item \(x_i\) and the mean. In symbols, \(D_i = x_i - \text{mean}\).

Let us explore this further with the provided data items:

• For \(3\), the deviation is \(D_1 = 3 - 12 = -9\).
• For \(5\), it is \(D_2 = 5 - 12 = -7\).
• For \(7\), the deviation is \(D_3 = 7 - 12 = -5\).
• At 12, the mean, the deviation naturally is \(D_4 = 12 - 12 = 0\).
• For \(18\), the deviation becomes \(D_5 = 18 - 12 = 6\).
• Finally, for \(27\), the deviation is \(D_6 = 27 - 12 = 15\).

Understanding and calculating deviations is crucial as it sets the stage for further analyses, such as evaluating the sum of all deviations.

Summing Up Deviations

The sum of deviations is the next logical step following the calculation of each individual deviation. It provides a total that represents the collective dispersion of data items around the mean. Notably, if we calculate the sum of deviations correctly, it often equals zero when summing around a mean. This characteristic is because positive deviations tend to balance out negative ones.

Let's compute the sum for our exercise:

• The deviations from the mean for our data items are: \(-9, -7, -5, 0, 6,\) and \(15\).
• Adding these results: \(\text{Sum} = -9 + (-7) + (-5) + 0 + 6 + 15 = 0\).

When the sum of deviations is zero, it confirms our calculations were conducted accurately with respect to the mean. This concept forms the basis for understanding more advanced topics in statistics such as variance and standard deviation.