Statistics: Measures of Center

Understanding the 'typical' value in a dataset is crucial. This cheat sheet covers the three main measures of center: mean, median, and mode, and how to choose the best one based on the data's characteristics.

Core Principles

• Measures of center (central tendency) provide a single number representing the typical value in a dataset.
• A parameter describes a population, while a statistic describes a sample. Greek letters represent parameters; Roman letters represent statistics.
• The mean is the average, calculated by summing all values and dividing by the count.
• The median is the middle value when data is ordered; it's resistant to extreme values.
• The mode is the most frequently occurring value in a dataset.
• The choice of measure depends on the data type (qualitative/quantitative) and distribution shape (symmetric/skewed).

Action Steps

• Identify if the data is qualitative or quantitative.
• If quantitative, determine if the data is skewed or symmetric.
• For symmetric data, use the mean as the measure of center.
• For skewed data, use the median as the measure of center.
• For qualitative data, use the mode as the measure of center.
• Always order data before calculating the median.
• Be aware that the mean is sensitive to extreme values (outliers).

Formulas

• Population Mean: $ \mu = \frac{\sum_{i=1}^{N} x_i}{N} $
• Sample Mean: $ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} $
• Median (Odd number of observations): The middle value in an ordered list.
• Median (Even number of observations): The average of the two middle values in an ordered list.
• Mode: The value(s) that appear most frequently.

Key Terms

• Parameter: A numerical characteristic of a population.
• Statistic: A numerical characteristic of a sample, used to estimate a parameter.
• Mean: The arithmetic average of a dataset.
• Median: The middle value of a dataset when ordered.
• Mode: The most frequent value in a dataset.
• Symmetric Distribution: A distribution where the left and right sides are mirror images (e.g., bell-shaped, uniform).
• Skewed Distribution: A distribution where one tail is longer than the other (left-skewed or right-skewed).
• Outlier: A data point significantly different from other observations.