Central tendency measures help us understand the typical value in a dataset. The mean, median, and mode each provide unique insights, with the mean considering all values, the median representing the middle, and the mode showing the most common value.

Choosing the right measure depends on the data type and distribution. The mean works well for normal distributions, the median for skewed data or outliers, and the mode for categorical data. Understanding these differences helps in selecting the most appropriate measure for analysis.

Measures of Central Tendency

Calculation of central tendency measures

• Mean
  • Calculated by adding up all the values in a dataset and dividing the sum by the total number of values
  • Formula: $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$
    • $\bar{x}$ represents the mean
    • $\sum_{i=1}^{n} x_i$ represents the sum of all values in the dataset
    • $n$ represents the total number of values in the dataset
  • Example: For the dataset {4, 7, 9, 12, 15}, the mean is calculated as (4 + 7 + 9 + 12 + 15) ÷ 5 = 9.4
• Median
  • Represents the middle value when the dataset is arranged in ascending or descending order
  • For an odd number of values, the median is the exact middle value
  • For an even number of values, the median is calculated by taking the average of the two middle values
  • Example: For the dataset {4, 7, 9, 12, 15}, the median is 9 (the middle value)
• Mode
  • Represents the most frequently occurring value or values in the dataset
  • A dataset can have no mode (when no value repeats), one mode (unimodal), or multiple modes (bimodal or multimodal)
  • Example: For the dataset {4, 7, 7, 9, 12, 15}, the mode is 7 (appears twice)

Mean, median, and mode comparisons

• Mean
  • Affected by extreme values or outliers in the dataset, which can skew the mean towards the direction of the outliers
  • Best used when the data is normally distributed (symmetric bell-shaped curve) and without significant outliers
  • Appropriate for interval and ratio scale data, where the differences between values are meaningful (temperature in ℃, height in cm)
• Median
  • Not affected by extreme values or outliers, making it a more robust measure of central tendency for skewed distributions
  • Best used when the data is skewed (asymmetric distribution) or contains significant outliers
  • Appropriate for ordinal (ranking), interval, and ratio scale data
• Mode
  • Not affected by extreme values or outliers
  • Best used for categorical (qualitative) or discrete data, where values are distinct and separate (favorite color, number of siblings)
  • Appropriate for nominal (categories), ordinal, and discrete data

Advantages vs disadvantages of measures

• Mean
  • Advantages
    • Takes into account all values in the dataset, providing a balanced measure of central tendency
    • Unique value for a given dataset, allowing for consistent comparisons
    • Useful for further statistical analysis, such as calculating variance and standard deviation
  • Disadvantages
    • Sensitive to extreme values or outliers, which can distort the mean and misrepresent the typical value
    • May not accurately represent the central tendency for skewed distributions
• Median
  • Advantages
    • Robust to extreme values or outliers, providing a more representative measure of central tendency for skewed distributions
    • Better representation of the typical value when the data is not normally distributed
  • Disadvantages
    • Does not consider all values in the dataset, as it only focuses on the middle value(s)
    • May not be unique when the dataset has an even number of values, requiring an average of the two middle values
• Mode
  • Advantages
    • Easy to determine, as it only requires identifying the most frequently occurring value(s)
    • Useful for categorical or discrete data, where the most common category or value is of interest
  • Disadvantages
    • May not exist if no value repeats in the dataset, or may not be unique if multiple values have the same highest frequency
    • Does not consider all values in the dataset, as it only focuses on the most frequent value(s)
    • Not suitable for further statistical analysis, as it does not provide information about the spread or variability of the data

Selection of appropriate central tendency

• Use the mean when:
  • The data is normally distributed (symmetric bell-shaped curve)
  • There are no significant outliers that could skew the mean
  • Further statistical analysis, such as calculating variance or standard deviation, is required
• Use the median when:
  • The data is skewed (asymmetric distribution) or contains significant outliers
  • The data is ordinal (ranking), interval, or ratio scale
  • A more robust measure of central tendency is needed to represent the typical value
• Use the mode when:
  • The data is categorical (qualitative) or discrete, such as favorite color or number of siblings
  • A quick and easy measure of the most common value or category is needed
  • The data is nominal (categories) or ordinal (ranking) scale