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title: ‘Module I: Measures of Central Tendency’ author: ‘Saed Mohamed Ahmed’ date: “2025-12-28” output: html_document: toc: true toc_depth: 2 theme: united pdf_document: toc: true —

1. Introduction

In statistics, a Measure of Central Tendency is a summary measure that attempts to describe a whole set of data with a single value that represents the middle or center of its distribution.

It is often referred to as a “summary statistic.” The three most common measures are the Mean, Median, and Mode.

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2. The Arithmetic Mean

The mean (or average) is the sum of all observations divided by the total number of observations.

Mathematical Formula

For a Sample Mean (\(\bar{x}\)): \[\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}\]

For a Population Mean (\(\mu\)): \[\mu = \frac{\sum_{i=1}^{N} X_i}{N}\]

Where: * \(\sum\) = Summation sign * \(x_i\) = The value of each individual observation * \(n\) = Number of observations in the sample

Example in R

# Dataset: Exam scores of 10 students
scores <- c(85, 90, 78, 92, 71, 88, 76, 84, 89, 95)

# Calculating Mean
mean_score <- mean(scores)
print(paste("The mean score is:", mean_score))

## [1] "The mean score is: 84.8"

Real-Life Application

Economic Indicators: Calculating the average household income in a city to determine the general standard of living.

Pros/Cons: * Pro: Uses every value in the dataset. * Con: Highly sensitive to outliers (extreme values).

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3. The Median

The median is the middle value of a dataset when it has been arranged in ascending or descending order. It splits the data into two equal halves.

Mathematical Calculation

1. Arrange data in order.
2. If \(n\) is odd: The median is the value at position \(\frac{n+1}{2}\).
3. If \(n\) is even: The median is the average of the two middle values at positions \(\frac{n}{2}\) and \(\frac{n}{2} + 1\).

Example in R

# Dataset with an outlier
salaries <- c(30000, 32000, 34000, 31000, 250000) # One very high salary

# Calculating Median vs Mean
med_sal <- median(salaries)
avg_sal <- mean(salaries)

print(paste("Median Salary:", med_sal))

## [1] "Median Salary: 32000"

print(paste("Mean Salary:", avg_sal))

## [1] "Mean Salary: 75400"

Real-Life Application

Real Estate: House prices are usually reported as “Median Price” because a single multi-million dollar mansion (outlier) would artificially inflate the mean, giving a false impression of what a “typical” house costs.

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4. The Mode

The mode is the value that appears most frequently in a dataset.

• Unimodal: One mode.
• Bimodal: Two modes.
• Multimodal: Three or more modes.

Example in R

R does not have a built-in function for the statistical mode, so we use a custom function:

get_mode <- function(v) {
   uniqv <- unique(v)
   uniqv[which.max(tabulate(match(v, uniqv)))]
}

# Dataset: Shirt sizes sold in a day
sizes <- c("S", "M", "L", "M", "M", "S", "XL", "M", "L")

mode_size <- get_mode(sizes)
print(paste("The modal shirt size is:", mode_size))

## [1] "The modal shirt size is: M"

Real-Life Application

Inventory Management: A shoe store manager needs the Mode to know which shoe size is sold most often to ensure it is always in stock.

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5. Comparing the Measures

The relationship between Mean, Median, and Mode depends on the skewness of the distribution.

1. Symmetric (Normal) Distribution:
   • Mean = Median = Mode
2. Right-Skewed (Positively Skewed):
   • Mean > Median > Mode (Outliers are on the high end)
3. Left-Skewed (Negatively Skewed):
   • Mean < Median < Mode (Outliers are on the low end)

Visualization

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6. Summary Table

Measure	Definition	Best Used For	Sensitive to Outliers?
Mean	Numerical average	Symmetric/Continuous data	Yes
Median	Middle value	Skewed data (Income, Housing)	No
Mode	Most frequent	Categorical data (Colors, Sizes)	No

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Practice Exercise

1. Create a vector in R named age with the following values: 22, 25, 22, 30, 24, 100.
2. Calculate the Mean and Median.
3. Which measure better represents the “center” of this group? Why? ```

How to use this:

1. Install RStudio.
2. Open RStudio and go to File -> New File -> R Markdown.
3. Delete the default text and paste the code above.
4. Click the Knit button to generate a beautiful document.