Problem 8 - 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 \((a)\). \(84,88,90,95,98 ;\) Mean \(=91\)

Step-by-Step Solution

Answer

The deviations from the mean are -7, -3, -1, 4, and 7. The sum of these deviations is 0.

Step 1: Compute the Deviations

First, subtract the mean from each data item to calculate the deviation. Let's perform these calculations:\n- \(84 - 91 = -7\)\n- \(88 - 91 = -3\)\n- \(90 - 91 = -1\)\n- \(95 - 91 = 4\)\n- \(98 - 91 = 7\)

Step 2: Sum of the Deviations

Secondly, sum all the deviations calculated in the Step 1. So, \(-7 + -3 + -1 + 4 + 7 = 0\)

Step 3: Interpret the Result

Last, interpret the final result. The sum of the deviations from the mean in a set of data is always zero.

Key Concepts

Data Analysis

Data analysis involves examining, cleaning, and modeling data to extract useful insights and inform decision-making. In this exercise, we are working with a set of data points: 84, 88, 90, 95, and 98. To begin understanding this data, we need to consider how each data point relates to the central tendency, or mean, which is 91 in this case. This involves breaking down and analyzing how spread out the values are from the mean.
Understanding the spread and distribution of data can help determine if the data is tightly clustered around the mean or if there are significant deviations.
Such analyses are crucial in determining the reliability and variability in different fields such as finance, science, and everyday decision-making.

Deviation Calculation

Calculating the deviation is a straightforward process that helps us understand how much each data item differs from the mean. To find each deviation, subtract the mean from the data item:

• For 84, deviation is \(84 - 91 = -7\).
• For 88, deviation is \(88 - 91 = -3\).
• For 90, deviation is \(90 - 91 = -1\).
• For 95, deviation is \(95 - 91 = 4\).
• For 98, deviation is \(98 - 91 = 7\).

By calculating these deviations, we can observe how much each data point diverges from the average. Negative deviations indicate data points below the mean, while positive deviations indicate points above it. Calculating the deviation is essential because it lays the groundwork for more complex statistical measures like variance and standard deviation.

Mean Calculation

The mean, also known as the average, is a measure of central tendency that sums up all the data points and then divides by the number of points. In this exercise, it is given as 91. Calculating the mean is fundamental when you want a single value that best represents a set of data.
To calculate the mean yourself, you would add together each of the numbers in the dataset (84, 88, 90, 95, and 98) and then divide by the total number of data points, which is 5:
\[\text{Mean} = \frac{84 + 88 + 90 + 95 + 98}{5} = 91\]
Having the mean allows you to then compute deviations, assess the spread of the data, and it often serves as the starting point for data analysis. This concept is essential because it relates various statistical measures and helps in estimating the standard performance or characteristic of data.