Measures of Central Tendency

What is Central Tendency?

Central tendency is a statistical measure that identifies a single value as representative of an entire dataset. It aims to provide an accurate description of the entire data set by finding the “center” of the data.

The three most common measures of central tendency are:

• Mean (arithmetic average)
• Median (middle value)
• Mode (most frequent value)

The Mean (Average)

The mean is the most commonly used measure of central tendency. It’s calculated by summing all values and dividing by the number of values.

Properties of the Mean

1. Uses all data points - Every value contributes to the mean
2. Sensitive to outliers - Extreme values can significantly affect the mean
3. Best for symmetric distributions - When data is evenly distributed

The Median

The median is the middle value when data is arranged in order. It divides the dataset into two equal halves.

How to Find the Median

1. Arrange data in ascending order
2. If n is odd: Median = middle value
3. If n is even: Median = average of two middle values

Properties of the Median

1. Not affected by outliers - Extreme values don’t impact the median
2. Uses only position - Not all values affect the calculation
3. Best for skewed distributions - When data has extreme values

The Mode

The mode is the value that appears most frequently in a dataset. A dataset can have:

• No mode - All values appear equally often
• One mode (unimodal)
• Two modes (bimodal)
• Multiple modes (multimodal)

Properties of the Mode

1. Works with categorical data - The only measure that works for non-numerical data
2. Can be multiple values - Unlike mean and median
3. May not exist - If all values are equally frequent

Comparing the Three Measures

Measure	Best Used When	Limitations
Mean	Data is symmetric, no outliers	Sensitive to extreme values
Median	Data is skewed, has outliers	Ignores actual values
Mode	Categorical data, finding most common	May not exist or be unique

Effect of Distribution Shape

The relationship between mean, median, and mode tells us about the shape of our distribution:

• Symmetric distribution: Mean ≈ Median ≈ Mode
• Right-skewed (positive skew): Mean > Median > Mode
• Left-skewed (negative skew): Mean < Median < Mode

When to Use Each Measure

Practice Problems

Try calculating all three measures for this dataset:

Data: 23, 29, 20, 32, 23, 21, 33, 25

Click to see the solution

Mean: (23 + 29 + 20 + 32 + 23 + 21 + 33 + 25) ÷ 8 = 206 ÷ 8 = 25.75

Median:

• Ordered: 20, 21, 23, 23, 25, 29, 32, 33
• Middle values: 23 and 25
• Median = (23 + 25) ÷ 2 = 24

Mode: 23 appears twice (most frequent) → Mode = 23

Summary

In this lesson, you learned:

• The mean is the arithmetic average, sensitive to outliers
• The median is the middle value, resistant to outliers
• The mode is the most frequent value, works with categorical data
• The relationship between these measures indicates distribution shape

Try It Yourself

Use our Mean Calculator to practice calculating averages with your own data!

Next Steps

Continue your learning with:

• Measures of Variability - Standard deviation and variance
• Data Visualization - Creating charts and graphs