Question

Show that the total sum of the distances between each data and its mean, namely \(d, d_{i}=x_{i}-\bar{x}\), is zero.

Short answer

The sum of the distances between each data point and the mean is zero as shown in the step-by-step solution.

Step-by-step solution

Define the Arithmetic Mean

First, we need to understand the formula for the arithmetic mean, which is \( \bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_{i}\). This formula essentially says that the mean of a set of data is the sum of all the data points divided by the number of data points.

Express \(d_{i}\)

Next, from the task, \(d_{i}\) is defined as the distance between each data point and the mean, which can be written as \(d_{i}=x_{i}-\bar{x}\). Thus, we need to express all the \(d_{i}\) in terms of \(x_{i}\) and \(\bar{x}\), and sum them all together to get \(\sum_{i=1}^{n}d_{i}\).

Simplify the Sum

When we add all the \(d_{i}\) together, we get \(\sum_{i=1}^{n}d_{i} = \sum_{i=1}^{n} (x_{i}-\bar{x})\). Because \(\bar{x}\) is a constant, we can separate the sum into two parts: \(\sum_{i=1}^{n}x_{i} - n\bar{x}\). As we know from the definition of \(\bar{x}\), \(\sum_{i=1}^{n}x_{i} = n\bar{x}\). Hence, the sum of all the \(d_{i}\) is: \(n\bar{x} - n\bar{x} = 0\).

Key concepts

Distance between data points and mean

In statistics, understanding how individual data points deviate from the mean is crucial. The mean is the center of the data, and each data point has a certain "distance" from this center. This distance is important because it tells us how much each data point differs from the typical value in the data set.

The distance for each data point, represented as \(d_i\), is calculated using the formula \(d_i = x_i - \bar{x}\), where \(x_i\) is the value of any data point and \(\bar{x}\) is the mean of the data set.

• \(x_i\): individual data point
• \(\bar{x}\): arithmetic mean of the data set
• \(d_i\): distance of \(x_i\) from the mean

By calculating this distance for each data point, we gain insights into the distribution and spread of the data. A small distance means the point is close to the mean, indicating uniformity, whereas a larger distance represents outliers or skewness.

Sum of deviations

The sum of deviations is an essential concept in statistics that helps us understand the overall deviation of the data set from its mean. Once we have computed the distance of each data point from the mean, these distances, or "deviations," are added up. This process helps us to analyze the data more holistically.

The sum of all deviations from the mean is represented as \(\sum_{i=1}^{n}(x_i - \bar{x})\). What's interesting about this sum is that it tells us about the data set's balance around the mean.

• \(\sum\): symbol for summation, which means adding all values together
• Each term \((x_i - \bar{x})\) represents the deviation for individual data points

However, merely summing up these deviations won't always provide a clear picture of the spread unless we understand the property of zero sum, which is healthy in data sets for symmetry and balance.

Zero sum property

A fascinating aspect of the arithmetic mean and deviations in a symmetric data set is the zero sum property. This states that when you sum up all the deviations of each data point from the mean, the total is zero.

In formulaic terms, this is expressed as \(\sum_{i=1}^{n}(x_i - \bar{x}) = 0\). This indicates that the deviations on both sides of the mean balance each other out perfectly. Let's break down why:

• Deviations greater than the mean are offset by deviations smaller than the mean.
• The arithmetic mean itself is the point at which this balance occurs.

This zero sum property of deviations demonstrates the fairness and balance of the mean as a central tendency measure. It is a key reason why the arithmetic mean is widely used in data analysis. Recognizing this property helps students appreciate the "center" as an anchor point, ensuring no part of the data set consistently overshadows the others.