Varianza y desviación estándar | Introducción

Introduction to Variance and Standard Deviation

Overview of Variance

• The course begins with an introduction to variance, defined as the average of the squared deviations from the mean.

• The focus will primarily be on explaining variance, with a brief mention of standard deviation at the end, highlighting their interrelation.

Population vs. Sample Data

• Variance and standard deviation can be calculated for both population data and sample data; different formulas apply based on this distinction.

• When treating data as a population, divide by the total number of data points; when treating it as a sample, divide by one less than that number (n - 1).

Notation Clarifications

• The symbols used for variance include σ² for population variance and s² for sample variance.

• Different notations may appear in textbooks; however, they all refer to similar concepts such as mean (average).

Example Calculation of Variance

Data Set Introduction

• An example is presented using ages of five children: 5 years, two children aged 6 years, 7 years, and 8 years.

Calculating Mean

• To find the mean age: sum all ages (5 + 6 + 6 + 7 + 8 = 32), then divide by the number of children (5), resulting in a mean age of 6.4 years.

Applying Variance Formula

• The formula for calculating variance is reiterated: sum each age minus the mean squared divided by the number of observations.

• Each child's age is subtracted from the mean (6.4), squared, summed up, and then divided by five since it's treated as a population.

Final Steps in Calculation

Understanding Sample vs. Population Formulas

• A reminder that if these ages were considered a sample instead of a population, you would divide by n - 1 instead of n during calculations.

Conclusion on Variance Calculation

Understanding Variance and Standard Deviation

Calculating Variance

• The operation results in a variance of 1.04 years squared after summing the values (1.96 + 0.16 + 0.16) and dividing by 5.

• Variance is expressed in square units, hence it is noted as "years squared." This emphasizes the importance of unit representation in statistical calculations.

Relationship Between Variance and Standard Deviation

• The standard deviation is derived from the variance by taking its square root; thus, it represents a measure of dispersion around the mean.

• The symbol for standard deviation is sigma (σ), which indicates that it is not squared, unlike variance.

Practical Example with Sample Data

• A practical exercise involves calculating variance and standard deviation using sample data representing weights of three individuals.

• For this example, the average weight is calculated first (165 kg / 3 = 55 kg), which serves as a reference point for further calculations.

Steps to Calculate Variance for Sample Data

• Each data point's deviation from the mean is computed:

• First data: 52 - 55

• Second data: 55 - 55

• Third data: 58 - 55

• Squaring these deviations yields values of 9, 0, and 9. The sum divided by n - 1 (where n = number of samples, here n = 3 leading to n - 1 = 2).

Final Results

• The calculated variance for this sample dataset turns out to be 9 text kg^2.

• Consequently, the standard deviation calculated as the square root of variance equals 3 text kg.

Conclusion