Measures of Central Tendency: The Arithmetic Mean

Measures of Central Tendency

The Arithmetic Mean

The arithmetic mean is one of the most commonly used measures of central tendency in statistics. It represents the average value of a set of data and is calculated by summing all the data values and dividing by the number of values.

• Definition: The arithmetic mean is the sum of all observed values divided by the total number of observations.

• Notation: The sample mean is usually denoted by \( \bar{x} \) (pronounced "x-bar").

• Formula for the Sample Mean:

$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$

• Where \( x_i \) represents each individual data value, and \( n \) is the total number of data values in the sample.

• The mean is computed using all the data values (not just a subset).

• The mean is sensitive to extreme values (outliers), which can affect its value significantly.

Example: Calculating the Sample Mean

• Suppose a sample consists of the following data: 4, 7, 9, 10, 12.

• Sum of the data: 4 + 7 + 9 + 10 + 12 = 42

• Number of data values: 5

• Sample mean: $\bar{x} = \frac{42}{5} = 8.4$

Population Mean vs. Sample Mean

It is important to distinguish between the population mean and the sample mean:

• Population Mean (\( \mu \)): The mean of all values in the entire population.

• Sample Mean (\( \bar{x} \)): The mean of values in a sample drawn from the population.

Population Mean Formula:

$\mu = \frac{1}{N} \sum_{i=1}^{N} x_i$

• Where \( N \) is the total number of values in the population.

Properties of the Arithmetic Mean

• The mean uses every value in the data set.

• The sum of the deviations of the data values from the mean is always zero:

$\sum_{i=1}^{n} (x_i - \bar{x}) = 0$

• The mean is affected by extreme values (outliers).

• The mean is a unique value for a given data set.

Applications of the Mean

• Used to summarize data with a single value representing the center of the distribution.

• Commonly used in fields such as economics, psychology, education, and the natural sciences.

Comparison Table: Sample Mean vs. Population Mean

Aspect	Sample Mean (\( \bar{x} \))	Population Mean (\( \mu \))
Symbol	\( \bar{x} \)	\( \mu \)
Formula	$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$	$\mu = \frac{1}{N} \sum_{i=1}^{N} x_i$
Data Used	Sample	Entire Population
Purpose	Estimate population mean	Describes population

Additional info: Some content and context have been inferred and expanded for clarity and completeness, as the original notes were fragmented and partially illegible.