BCA PART B 2.R Program to Calculate standard deviation and variance for discrete & continuous series.

 

Explanation:

1. Discrete Series:

   • The program assumes a discrete dataset with specific values and their corresponding frequencies.
   • The mean is calculated by taking the weighted sum of the values multiplied by their frequencies, divided by the total number of observations.
   • Variance is calculated as the weighted sum of squared differences between each value and the mean, divided by the total number of observations.
   • Standard Deviation is the square root of the variance.

2. Continuous Series:

   • For continuous data, we assume class intervals (e.g., 0-10, 10-20) and their corresponding frequencies.
   • First, the midpoints of each class interval are calculated. These midpoints are used to approximate the data points in each class.
   • The mean, variance, and standard deviation are calculated using the midpoints in the same way as for the discrete series.

Program :

# Function to calculate variance and standard deviation for discrete series

calculate_discrete_sd_variance <- function(values, frequencies) {

n <- sum(frequencies) # Total number of observations

mean_val <- sum(values * frequencies) / n # Weighted mean

# Variance calculation

variance_val <- sum(frequencies * (values - mean_val)^2) / n

# Standard deviation is the square root of variance

sd_val <- sqrt(variance_val)

cat("For Discrete Series:\n")

cat("Variance: ", variance_val, "\n")

cat("Standard Deviation: ", sd_val, "\n")

}

# Function to calculate variance and standard deviation for continuous series

calculate_continuous_sd_variance <- function(class_intervals, frequencies) {

mid_points <- (class_intervals[, 1] + class_intervals[, 2]) / 2 # Calculate mid-points for each class

n <- sum(frequencies) # Total number of observations

mean_val <- sum(mid_points * frequencies) / n # Weighted mean based on mid-points

# Variance calculation

variance_val <- sum(frequencies * (mid_points - mean_val)^2) / n

# Standard deviation is the square root of variance

sd_val <- sqrt(variance_val)

cat("For Continuous Series:\n")

cat("Variance: ", variance_val, "\n")

cat("Standard Deviation: ", sd_val, "\n")

}

# Discrete series: Values and their frequencies

discrete_values <- c(2, 4, 6, 8, 10)

discrete_frequencies <- c(5, 10, 15, 7, 3)

calculate_discrete_sd_variance(discrete_values, discrete_frequencies)

# Continuous series: Class intervals and their frequencies

class_intervals <- matrix(c(0, 10, 10, 20, 20, 30, 30, 40), ncol = 2, byrow = TRUE)

continuous_frequencies <- c(7, 12, 15, 6)

calculate_continuous_sd_variance(class_intervals, continuous_frequencies)

Output :

For Discrete Series: Variance: 4.7775 Standard Deviation: 2.185749 For Continuous Series: Variance: 90 Standard Deviation: 9.486833

Explanation :

This program is designed to calculate the variance and standard deviation for both discrete and continuous series. For the discrete series, the program takes a set of distinct values and their corresponding frequencies, computes the weighted mean, and then calculates the variance by finding the average of the squared differences between the values and the mean, weighted by their frequencies. The standard deviation is obtained by taking the square root of the variance. In the case of the continuous series, the program works with class intervals and their frequencies. It first calculates the midpoints of each class, which are then treated as representative values. These midpoints are used in the same process as the discrete case to compute the weighted mean, variance, and standard deviation. The program provides flexibility to handle both types of data, offering a way to understand the spread of values in a dataset by measuring how much they deviate from the mean.