Data Science and A.I. : Measures of Dispersion : Variance of Continuous Freq. Distribution

In this post, we dive into one of the core statistical concepts used in data science and artificial intelligence—calculating the variance of a continuous frequency distribution.

Breaking down the steps

If you have data distributed across class intervals with frequencies, the goal is to find the variance. The video explains the columns in the table: class intervals, frequencies, mid-values, and how to calculate \( u_i \) and \( f_i u_i \).

Mid-Values Calculation:

Add the lower and upper limits of each class interval and divide by 2. For example:
For 20–30, mid-value = \( \frac{20 + 30}{2} = 25 \)
For 30–40, mid-value = \( \frac{30 + 40}{2} = 35 \)

Calculating \( u_i \):

The formula is \( u_i = \frac{x_i – A}{h} \), where \( A = 55 \) (assumed mean) and \( h = 10 \) (class width). For example:
– \( u_1 = \frac{25 – 55}{10} = -3 \)
– \( u_2 = \frac{35 – 55}{10} = -2 \)

Multiplying frequencies with \( u_i \) values to get \( f_i u_i \), and calculating \( u_i^2 \) for further steps. The summation \( \sum f_i u_i = 2 \) and \( \sum f_i u_i^2 = 134 \) are key values in the final formula.

Variance Formula:

The variance for a continuous frequency distribution is:
\[
\text{Var}(X) = h^2 \left( \frac{1}{N} \sum f_i u_i^2 – \left( \frac{1}{N} \sum f_i u_i \right)^2 \right)
\]
Plugging the calculated values gives the variance = 222.90.

Closing remarks, summarizing that the variance calculation process for continuous and discrete distributions is quite similar, with the only key difference being the use of class intervals in continuous data.

Key Learning Points:

1. Variance Calculation: This video simplifies how to calculate variance for continuous data using frequency distributions.
2. Formula Application: Step-by-step breakdown of the variance formula with real data and clear explanations.
3. Practical Examples: The video uses class intervals and mid-values to illustrate variance calculation, a common approach in statistics for continuous data.

Presentation  Document

var cfd ex 1

Link https://www.postnetwork.co/wp-content/uploads/2024/10/var-cfd-ex-1.pdf

Video