Data Handling » Mean, Median and Mode

Introduction to Measures of Central Tendency

In psychology, we often collect lots of data from participants. To make sense of all these numbers, we need ways to summarise them. Measures of central tendency help us find the "typical" or "average" value in our data, giving us a single number that represents the entire data set.

Key Definitions:

• Measures of central tendency: Statistical values that represent the centre or typical value of a data set.
• Mean: The arithmetic average of all values in a data set.
• Median: The middle value when all data is arranged in order.
• Mode: The most frequently occurring value in a data set.

The Mean

The mean is what most people think of as the "average." It's calculated by adding up all the values in a data set and dividing by the number of values.

Calculating the Mean

The formula for the mean is:

Mean = Sum of all values ÷ Number of values

Example: A psychologist measures the reaction times (in milliseconds) of 7 participants in a study: 245, 231, 252, 267, 239, 241, 258

To find the mean:

1. Add all the values: 245 + 231 + 252 + 267 + 239 + 241 + 258 = 1733
2. Divide by the number of values: 1733 ÷ 7 = 247.6

The mean reaction time is 247.6 milliseconds.

👍 Strengths of the Mean

• Uses all values in the data set
• Suitable for further statistical analysis
• Most people understand what an average is
• Gives a precise value (can include decimals)

👎 Limitations of the Mean

• Affected by extreme values (outliers)
• Not suitable for skewed distributions
• Can give a value that doesn't exist in the original data
• Less useful for categorical data

The Median

The median is the middle value when all data points are arranged in order from lowest to highest. If there's an even number of values, the median is the average of the two middle values.

Finding the Median

Steps to find the median:

1. Arrange all values in ascending order (lowest to highest)
2. If there's an odd number of values, the median is the middle value
3. If there's an even number of values, the median is the average of the two middle values

Example 1 (odd number of values): Using our previous reaction time data: 231, 239, 241, 245, 252, 258, 267

There are 7 values, so the median is the 4th value: 245 milliseconds.

Example 2 (even number of values): If we add another participant with a reaction time of 249: 231, 239, 241, 245, 249, 252, 258, 267

There are 8 values, so the median is the average of the 4th and 5th values: (245 + 249) ÷ 2 = 247 milliseconds.

👍 Strengths of the Median

• Not affected by extreme values (outliers)
• Better for skewed distributions
• Useful when data contains extreme values
• Good for ordinal data (rankings)

👎 Limitations of the Median

• Doesn't use all values in calculation
• Less suitable for further statistical analysis
• Can be time-consuming to calculate with large data sets
• Less precise than the mean for normally distributed data

The Mode

The mode is the value that appears most frequently in a data set. It's the only measure of central tendency that can be used with categorical (non-numerical) data.

Finding the Mode

To find the mode, simply identify which value occurs most often in your data set.

Example 1: A psychologist records the preferred learning styles of 10 students: Visual, Auditory, Kinesthetic, Visual, Visual, Auditory, Kinesthetic, Visual, Visual, Auditory

The mode is "Visual" as it appears 5 times (more than any other value).

Example 2: In a memory test, participants recalled the following number of items: 7, 8, 9, 7, 10, 7, 8, 11, 7, 9

The mode is 7 as it appears 4 times (more than any other value).

Sometimes a data set can have more than one mode:

• Unimodal: One mode (most common)
• Bimodal: Two modes
• Multimodal: More than two modes
• No mode: When all values appear with equal frequency

👍 Strengths of the Mode

• Only measure that works with categorical data
• Easy to understand and calculate
• Not affected by extreme values
• Shows the most common experience or response

👎 Limitations of the Mode

• May not exist or may not be unique
• Can be misleading if data is widely spread
• Less useful for further statistical analysis
• Doesn't consider the full distribution of values

Choosing the Right Measure

Each measure of central tendency has its own strengths and weaknesses. The choice depends on the type of data and what you want to show.

Real-World Applications in Psychology

Understanding measures of central tendency is crucial for interpreting psychological research:

• Clinical Psychology: Comparing a client's test scores to the average (mean) of the general population to assess severity of symptoms.
• Educational Psychology: Using the median to report student performance when there are a few very high or very low scores.
• Research: Reporting the mode of responses to questionnaire items to show the most common attitudes or beliefs.
• Developmental Psychology: Using means to track average developmental milestones across different age groups.

Remember that a single measure of central tendency doesn't tell the whole story about your data. It's often useful to report multiple measures along with measures of dispersion (like range or standard deviation) to give a more complete picture.