Understanding measures of central tendency is crucial for grasping data distribution. The mean, median, and mode provide different perspectives on the typical value in a dataset, each with unique strengths and limitations.

These measures help summarize large datasets and compare different groups. Knowing when to use each measure and how they're affected by outliers or skewed data is key for accurate data interpretation and analysis.

Measures of the Center of the Data

Calculation of mean and median

• Mean
  • Represents the arithmetic average of a dataset calculated by summing all values and dividing 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 $\frac{4+7+9+12+15}{5} = 9.4$
  • Weighted average: A variation of the mean where each value is multiplied by its importance or frequency before summing
• Median
  • Represents the middle value of a dataset when 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)
  • Example: For the dataset {4, 7, 9, 12, 15, 18}, the median is $\frac{9+12}{2} = 10.5$ (average of the two middle values)
  • Less affected by outliers compared to the mean

Sample vs population means

• Population mean
  • Denoted by the Greek letter $\mu$
  • Calculated using all values in an entire population
  • Formula: $\mu = \frac{\sum_{i=1}^{N} x_i}{N}$
    • $N$ represents the total number of values in the population
  • Example: The average height of all students in a school (entire population)
• Sample mean
  • Denoted by $\bar{x}$
  • Calculated using values from a representative sample of the population
  • Used to estimate the population mean when measuring the entire population is impractical or impossible
  • Formula: $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$
    • $n$ represents the number of values in the sample
  • Example: The average height of a random sample of 100 students from a school (sample)

Mode and bimodal datasets

• Mode
  • Represents the most frequently occurring value or values in a dataset
  • A dataset can have one mode (unimodal), two modes (bimodal), more than two modes (multimodal), or no mode if no value appears more than once
  • Example: In the dataset {4, 7, 7, 9, 12, 15}, the mode is 7 (appears twice)
  • Example: In the dataset {4, 7, 9, 12, 15}, there is no mode (no value appears more than once)
• Bimodal dataset
  • A dataset with two distinct modes
  • The two modes can have equal or unequal frequencies
  • Suggests the presence of two distinct groups or clusters within the data
  • Example: A dataset of exam scores with peaks at 65 and 85, indicating two groups of students (one group performing poorly and another performing well)
  • Example: A dataset of heights with peaks at 160 cm and 180 cm, suggesting two distinct height groups (possibly males and females)

Distribution and Central Tendency

• Distribution refers to the pattern of data values in a dataset
• Central tendency measures (mean, median, and mode) describe the center of the distribution
• Skewness indicates the asymmetry of the distribution, affecting the relationship between mean, median, and mode