M09.02.03 Measures of central tendency

Measures of central tendency are statistical tools that provide a single value to describe the central or typical value in a data set. They are essential for understanding data distribution by identifying a midpoint or central value around which other data points cluster. Commonly known as “averages,” measures of central tendency include the mean, median, and mode. Each measure serves different purposes depending on the data distribution, especially when dealing with skewed data.

Types of Measures of Central Tendency

1. Mean (Arithmetic Average)

The mean is calculated by summing all the values in a dataset and dividing by the number of observations.

Formula:

Example Calculation: For the dataset 3,6,7,7,9,10,12,15,16

2. Median

• The median is the middle value in an ordered dataset, dividing it into two halves.
• For an odd number of observations, the median is the middle number; for an even number, it is the average of the two middle values.
• Example Calculation: In the same dataset 3,6,7,7,9,10,12,15,16:
  • Ordered set:
  • The median is 999, as it is the middle value.

3. Mode

• The mode is the value that occurs most frequently in a dataset.
• There can be multiple modes if two or more values have the same highest frequency.
• Example Calculation: For 3,6,7,7,9,10,12,15,16:
  • The mode is 7, as it appears most frequently.

Comparison of Mean, Median, and Mode

Data Distribution and Skewness

Data distributions can affect the choice of central tendency measure:

1. Normal Distribution:
   • Symmetrical curve where mean, median, and mode are equal.
2. Positive Skew (Right-Skewed) Distribution:
   • The tail extends to the right.
   • Characteristics:
     • Mean > Median
     • Median provides a better representation of central tendency.
3. Negative Skew (Left-Skewed) Distribution:
   • The tail extends to the left.
   • Characteristics:
     • Median > Mean
     • The median is generally a more accurate representation of central tendency.

Points to Remember

• The mean is sensitive to extreme values and may not accurately represent skewed data.
• The median is robust against outliers and better suited for skewed distributions.
• Mode is useful for categorical data or identifying the most common value in a dataset.

References

1. Black, K. (2019). Business Statistics: For Contemporary Decision Making. Wiley.
2. Gravetter, F., & Wallnau, L. (2017). Statistics for the Behavioral Sciences. Cengage Learning.
3. Triola, M. F. (2020). Essentials of Statistics. Pearson.