11.1 Measures of central tendency and dispersion

Central tendency measures help summarize datasets by identifying typical values. The mean calculates the average, the median finds the middle value, and the mode identifies the most frequent value. Each measure has strengths and weaknesses, making them suitable for different types of data.

Dispersion measures complement central tendency by showing how spread out data points are. The range gives the spread between extremes, while variance and standard deviation measure average distances from the mean. Together, these tools provide a comprehensive snapshot of a dataset's characteristics.

Measures of Central Tendency

Calculation of central tendency measures

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• Mean calculates the arithmetic average by summing all values and dividing by the number of values
  • Formula: $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$ where $\bar{x}$ represents the mean, $\sum_{i=1}^{n} x_i$ is the sum of all values, and $n$ is the number of values
  • Example: For the dataset {4, 7, 9, 12, 15}, the mean is (4 + 7 + 9 + 12 + 15) ÷ 5 = 9.4
• Median identifies the middle value when data is sorted in ascending or descending order
  • For an odd number of values, the median is the middle value (dataset with 5 values, median is the 3rd value)
  • For an even number of values, the median is the average of the two middle values (dataset with 6 values, median is the average of the 3rd and 4th values)
• Mode represents the most frequently occurring value in a dataset
  • A dataset can have no mode (values appear with equal frequency), one mode (unimodal), or multiple modes (bimodal for two modes, multimodal for more than two modes)
  • Example: In the dataset {4, 7, 7, 9, 12, 15}, the mode is 7

Measures of statistical dispersion

• Range calculates the difference between the largest and smallest values in a dataset
  • Formula: $Range = x_{max} - x_{min}$ where $x_{max}$ is the maximum value and $x_{min}$ is the minimum value
  • Example: For the dataset {4, 7, 9, 12, 15}, the range is 15 - 4 = 11
• Variance measures how far each value is from the mean by calculating the average of the squared differences
  • Formula: $s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}$ where $s^2$ is the variance, $x_i$ are individual values, $\bar{x}$ is the mean, and $n$ is number of values
  • Squared differences ensure positive values and give more weight to larger deviations
• Standard deviation is the square root of the variance and measures dispersion relative to the mean
  • Formula: $s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}}$ where $s$ is the standard deviation
  • Useful for comparing spread of datasets with the same unit of measurement (comparing test scores between classes)

Central tendency vs dispersion

• Central tendency measures (mean, median, mode) describe the center or typical value of a dataset
  • Provide a single representative value for the entire dataset
  • Example: A class average score of 85% represents overall performance
• Dispersion measures (range, variance, standard deviation) describe the spread or variability of a dataset
  • Provide information about how far data points are from the central tendency
  • Example: A standard deviation of 5% indicates most scores are within ±5% of the mean

Selecting appropriate central tendency measure

• For nominal data (categories with no order), the mode is the only appropriate central tendency measure
  • Example: Most common eye color in a group
• For ordinal data (categories with order), the median is most appropriate, mode can also be used
  • Example: Median Likert scale response on a survey
• For interval and ratio data (numeric values), selection depends on distribution:
  • Mean for normally distributed or symmetrical data
  • Median for skewed data or data with outliers
  • Example: Mean income for a town vs median income if few high earners skew distribution