Problem 13 - 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). \(146,153,155,160,161\)

Step-by-Step Solution

Answer

a. The mean is 155. b. The deviations from the mean for each data item are -9, -2, 0, 5, and 6 respectively. c. The sum of deviations from the mean is 0.

Step 1: Calculation of the Mean

To calculate the mean of a set of numbers, add all the numbers together and divide by the count of numbers. Thus, the mean of the set \(146, 153, 155, 160, 161\) is calculated as \( Mean = (146 + 153 + 155 + 160 + 161)/5 = 775/5 = 155 \)

Step 2: Calculation of Deviation from Mean for Each Data Item

Deviation of a number from the mean is calculated as the number minus the mean. The deviation from the mean is calculated as follows: \nFor 146: \(146 - 155 = -9\), \nFor 153: \(153 - 155 = -2\), \nFor 155: \(155 - 155 = 0\), \nFor 160: \(160 - 155 = 5\), \nFor 161: \(161 - 155 = 6\)

Step 3: Calculation of the Sum of Deviations

Find the sum of all the deviations calculated in the previous step. The sum of deviations is calculated as \(-9 - 2 + 0 + 5 + 6 = 0\)

Key Concepts

Mean Calculation

When it comes to understanding data, the statistical mean is an essential concept. To put it simply, the mean is the average of a set of numbers. You calculate it by adding together all the values in the set and then dividing by the number of values. In the exercise provided, you were asked to find the mean of the numbers 146, 153, 155, 160, and 161.
To do this, you would add up all the numbers to get a total sum of 775. Since there are five numbers in the set, you'd divide 775 by 5, resulting in a mean of 155. This process, by the way, is beautifully summed up in the equation
\[ Mean = \frac{146 + 153 + 155 + 160 + 161}{5} = \frac{775}{5} = 155 \].
The mean is a useful tool for summarizing a set of data with a single number, reflecting the center point of the data set.

Deviation from the Mean

Once you've calculated the mean, you might also be interested in the variation of each data point relative to that mean. This is where the concept of deviation comes into play. Deviation is the distance of each data point from the mean, which can be either positive or negative depending on whether the data point is above or below the mean.
In our exercise, the deviations for each number from the mean (155) are as follows: -9 for 146, -2 for 153, 0 for 155, 5 for 160 and 6 for 161. These values are calculated by subtracting the mean from each number in your data set. The formula looks like this: \( Deviation = Data\,Point - Mean \). Deviations help us understand the spread of the data and indicate how much each value differs from the average.

Sum of Deviations

The sum of deviations is a concept that might seem a bit counterintuitive at first because in a perfectly balanced data set, this value will always be zero. Why is that? Because the deviations above the mean cancel out the deviations below it. This balance is a fundamental property of the mean and it's what makes the mean distinct as a measure of central tendency.
In our exercise, when we added up the deviations of each number from the mean, that is, \[-9 + (-2) + 0 + 5 + 6\], the result was indeed zero. This sum checks out our earlier calculation and assures us that the mean has been computed correctly. While the sum of deviations itself isn't particularly informative, understanding why it's zero underscores the reliable nature of the mean in summarizing data.

Descriptive Statistics

Descriptive statistics are statistical processes we use to summarize and describe the main features of a data set in a quantitative manner. In addition to the mean and deviations, this includes measures such as the median, mode, standard deviation, and range. Each of these measures provides different insights into the data.
The mean tells us about the central tendency, while the standard deviation, a step further from just the mean deviation, gives us a measure of the spread of the data points around the mean. A small standard deviation indicates that the data points tend to be close to the mean, whereas a large standard deviation indicates that the data can be spread out over a large range of values. Descriptive statistics provide a powerful suite of tools for making sense of data, allowing for a comprehensive view of both the central tendency and variability within a dataset.