11.1 Measures of Central Tendency and Dispersion

Central tendency and dispersion measures are key tools in marketing research. The mean, median, and mode help summarize data, while range, variance, and standard deviation show how spread out values are.

Understanding when to use each measure is crucial. The mean works for normal distributions, the median for skewed data, and the mode for categorical info. Outliers can skew results, so researchers must choose wisely.

Measures of Central Tendency

Mean, median, and mode differences

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• Mean represents the arithmetic average of a dataset calculated by summing all values and dividing by the number of observations $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$ sensitive to extreme values or outliers (income data)
• Median is the middle value of a dataset when arranged in ascending or descending order robust to outliers (housing prices) for an odd number of observations, the median is the middle value for an even number of observations, the median is the average of the two middle values
• Mode is the most frequently occurring value in a dataset can have no mode if no value repeats or multiple modes if multiple values repeat with the same highest frequency useful for categorical or discrete data (favorite color)
• Mean considers all values, while median and mode do not mean is sensitive to outliers, while median and mode are robust to outliers mode is the only measure applicable to categorical data

Appropriate measures for different situations

• Mean is appropriate when data is continuous and normally distributed useful when the total value of the dataset is of interest (average test score)
• Median is appropriate when data is skewed or contains outliers useful when the central value of the dataset is of interest (median income)
• Mode is appropriate for categorical or discrete data useful when the most common value is of interest (most popular product)
• Range is appropriate for a quick and simple measure of dispersion useful when the spread of the entire dataset is of interest (temperature range)
• Variance and standard deviation are appropriate when data is continuous and normally distributed useful when the average distance from the mean is of interest (variability in product quality)

Measures of Dispersion

Range, variance, and standard deviation interpretation

• Range is the difference between the maximum and minimum values in a dataset $Range = x_{max} - x_{min}$ measures the total spread of the dataset sensitive to outliers (stock price range)
• Variance is the average of the squared deviations from the mean $s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}$ measures how far each value is from the mean units are squared, making interpretation difficult (variance in product weights)
• Standard deviation is the square root of the variance $s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}}$ measures the average distance from the mean in the same units as the original data, making interpretation easier (standard deviation of heights)

Outlier effects on statistical measures

• Outliers are extreme values that are significantly different from other observations in a dataset (unusually high or low sales figures)
• Outliers can significantly pull the mean towards the direction of the outlier making the mean an inaccurate representation of the typical value in the dataset
• Median is robust to outliers and is not significantly affected by their presence
• Mode is not affected by outliers, as it only considers the most frequently occurring value
• Outliers can greatly increase the range, as they can be the minimum or maximum value in the dataset
• Outliers can significantly increase the variance and standard deviation by increasing the average distance of values from the mean