Central tendency measures help us understand the typical value in a dataset. The mode shows the most common value, the median represents the middle value, and the mean calculates the average. Each measure has its strengths and weaknesses.

These measures behave differently depending on the data's distribution. For symmetrical data, they're often similar. In skewed distributions, they can vary widely. Understanding when to use each measure is crucial for accurate data interpretation.

Measures of Central Tendency

Calculation of central tendency measures

• Mode represents the most frequently occurring value in a dataset
  • Datasets can have no mode (no value appears more than once), one mode (unimodal), or multiple modes (bimodal or multimodal)
  • In grouped data, the modal class is the class interval with the highest frequency (age groups, income brackets)
• Median is the middle value when the dataset is arranged in ascending or descending order
  • For an odd number of values, the median is the middle value (5, 7, 9 - median is 7)
  • For an even number of values, the median is the average of the two middle values (4, 6, 8, 10 - median is (6 + 8) / 2 = 7)
  • The median is the 50th percentile, dividing the data into two equal parts
• Mean is the arithmetic average of all values in a dataset
  • Calculated by summing all values and dividing by the number of values (10, 20, 30, 40 - mean is (10 + 20 + 30 + 40) / 4 = 25)
  • Formula: $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$, where $\bar{x}$ is the mean, $x_i$ are the individual values, and $n$ is the number of values

Comparison of central tendency measures

• Mode is best used for categorical or qualitative data to identify the most common value or category (favorite color, most popular car brand)
• Median is best used for skewed distributions or datasets with outliers and ordinal data or data with a non-numerical order (median income, median house price)
• Mean is best used for symmetrical distributions and interval or ratio data, especially when comparing different datasets (average test scores, average height)
• Mode and median are less affected by extreme values or outliers, making them more robust measures of central tendency compared to the mean

Effects of data distribution

• Extreme values (outliers) have different effects on the mode, median, and mean
  • Mode is not affected by outliers (1, 2, 3, 3, 4, 100 - mode is still 3)
  • Median is less affected by outliers compared to the mean (1, 2, 3, 4, 100 - median is 3)
  • Mean is highly sensitive to outliers and can be pulled towards the direction of the outlier (1, 2, 3, 4, 100 - mean is 22)
• In a perfectly symmetrical distribution, the mode, median, and mean are equal (bell curve)
  • For approximately symmetrical distributions, the mean is usually the best measure of central tendency
• Skewed distributions have different relationships between the mode, median, and mean
  • Right-skewed (positively skewed): Mode < Median < Mean (income distribution)
  • Left-skewed (negatively skewed): Mean < Median < Mode (age distribution in a retirement community)
  • For skewed distributions, the median is often the best measure of central tendency, as it is less affected by the skewness

Measures of Variability

• Range is the simplest measure of variability, calculated as the difference between the maximum and minimum values in a dataset
• Quartiles divide the dataset into four equal parts, with Q1 (25th percentile), Q2 (median), and Q3 (75th percentile)
• Standard deviation measures the average distance of data points from the mean, providing insight into the spread of the data
• Variance is the square of the standard deviation, representing the average squared deviation from the mean