📏 Measures of Central Tendency

What is a Measure of Central Tendency?

A measure of central tendency is a single value that describes the center or typical value of a data set. The three main measures are mean, median, and mode.

What is the Mean?

The mean (or average) is the sum of all data values divided by the number of values.

\[
Mean = Sum of all observationsNumber of observations = x_i/n

\]

What is the Range?

The range is the difference between the highest and lowest values in a data set. It measures how spread out the data is.

\[
Range = Maximum value - Minimum value

\]

Interpreting Data:

• The mean is useful when data is evenly distributed (no extreme outliers)

• A large range indicates data is spread out; a small range indicates data is clustered

• The mean can be affected by extremely high or low values (outliers)

Real-Life Applications:

• Mean: average test scores, average…

What is the Median?

The median is the middle value of a data set when arranged in ascending or descending order. It divides the data into two equal halves.

How to Find the Median:

Number of values (\(n\))Median Position
Odd (\(n\))Middle value at position \(n+1/2\)
Even (\(n\))Average of the two middle values at positions \(n/2\) and \(n/2+1\)

What is the Mode?

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

• One mode (unimodal)

• Two modes (bimodal)

• More than two modes (multimodal)

• No mode (if all values occur with same frequency)

Comparison of Measures:

MeasureBest used whenAdvantageDisadvantage
MeanData is symmetricalUses all dataAffected by outliers
MedianData has outliersNot affected by outliersDoesn't use all data
ModeData is categoricalEasy to understandMay not be unique

Choosing the Right Measure of Central Tendency:

ScenarioBest MeasureWhy?
Data with no extreme valuesMeanUses all data, good for further calculations
Data with outliersMedianNot influenced by extreme values
Data with repeated valuesModeShows most common value
Categorical data (colors, brands)ModeOnly measure that works for non-numeric data
Symmetrical distributionMean = MedianBoth work well
Skewed distributionMedianBetter represents "typical" value

Interpreting Data Distributions:

• Symmetric distribution: Mean = Median = Mode

• Positively skewed (right-skewed): Mean > Median > Mode (tail on right)

• Negatively skewed (left-skewed): Mean Practical Applications:

FieldUse of Central Tendency
EducationAverage test scores (mean), middle student performance (median)
BusinessMost common purchase (mode), average sale (mean)
HealthcareAverage patient recovery time…