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:

• 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:

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:

• 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:

• 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

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