Measuring central tendency help us to find the center point of a data distribution.

The three major measures of central tendency are as follows:

• Mean: This is the arithmetic average of all the values.
• Median: This is the exact middle value when data is arranged in ascending/descending order.
• Mode: This is the most frequently occurring value in the dataset.

These measures help us understand what is the trend in our dataset.

For a dataset \(X = \{x_1, x_2, ..., x_n\}\):

Mean (or Average)

\[\bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_i = \frac{x_1 + x_2 + ... + x_n}{n}\]

Median

\[\text{Median} = \begin{cases} x_{(\frac{n+1}{2})}, & \text{if } n \text{ is odd} \\ \frac{x_{(\frac{n}{2})} + x_{(\frac{n}{2}+1)}}{2}, & \text{if } n \text{ is even} \end{cases}\]

Mode

\[\text{Mode} = \text{value that appears most in } X\]

# Function to calculate the mean
meanCalculation = function(x) {
  sum(x) / length(x)
}

# Running test cases with sample
dataSet = c(12, 15, 18, 22, 30, 31, 35, 40)
cat(" Mean using our function:", meanCalculation(dataSet) ,"\n" 
    ,"Mean using function in R:", mean(dataSet))

##  Mean using our function: 25.375 
##  Mean using function in R: 25.375

# Creating sample grades
set.seed(456)
nameOfCourses = c("Computers", "Data Science", "English", "Maths", "Science")
grades = data.frame(
  Courses = rep(nameOfCourses, each = 20),
  Grades = round(runif(100, min = 60, max = 100))
)
# Calculate Measures for the course
statsForCourse = grades %>%
  group_by(Courses) %>%
  summarize(
    Mean = mean(Grades),
    Median = median(Grades)
  )
# Showing the results
statsForCourse

## # A tibble: 5 × 3
##   Courses       Mean Median
##   <chr>        <dbl>  <dbl>
## 1 Computers     79.4   76.5
## 2 Data Science  86.8   88.5
## 3 English       78.5   77  
## 4 Maths         77.0   76.5
## 5 Science       86     86

Key findings about measures of central tendency:

1. Mean
   • Mostly used for normal distribution
   • Sensitive to outliers
   • Very commonly used in statistics
2. Median
   • Mostly used for skewed distributions
   • Easily used against outliers
   • Used for the statistics like income, housing prices etc.
3. Mode
   • Mostly used for data in categories
   • Used to find the most common values
   • Can be used tp find multiple peaks in distribution

The choice of what measure to use depends on data type and distribution.