In descriptive statistics, the mean (also called the average) is one of the most fundamental measures used to represent the central tendency of a dataset. It is often used to summarize a set of values with a single number that best represents the entire distribution of data.

Definition #

The mean is calculated by summing all the values in a dataset and then dividing by the number of values. It is commonly used to understand the overall level of a dataset, such as the average score of students in a class, the average income in a region, or the average temperature of a month.

Formula #

Mean Calculation for Sample and Population #

Sample Mean #

For a sample of size (n) with values (x_1, x_2, \dots, x_n), the sample mean is calculated as:

[ \text{Mean} = \frac{x_1 + x_2 + \dots + x_n}{n} ]

Where:

• (x_1, x_2, \dots, x_n) are the individual data points.
• (n) is the total number of data points.

Population Mean #

For a population, the formula is very similar, but we use (N) to denote the total number of individuals in the population. It is often referred to as the population mean:

[ \text{Population Mean} = \frac{X_1 + X_2 + \dots + X_N}{N} ]

Where:

• (X_1, X_2, \dots, X_N) are the individual data points in the population.
• (N) is the total number of data points in the population.

Example: #

If we have the following data representing the test scores of 5 students:

[ 85, 90, 95, 80, 88 ]

The mean test score is calculated as:

[ \text{Mean} = \frac{85 + 90 + 95 + 80 + 88}{5} = \frac{438}{5} = 87.6 ]

Thus, the average score is 87.6.

Key Points #

1. The mean is sensitive to extreme values (outliers). A very high or very low value can significantly alter the mean.
2. It is best used when the data is symmetrically distributed without significant outliers.
3. In cases with outliers or skewed data, the median or mode might be more appropriate measures of central tendency.

#SPSS

Last modified on 2023-12-30