Verifique que la varianza de la muestra 4,9,3,6, 4 y 7 es...

Verifique que la varianza de la muestra $4,9,3,6$, 4 y 7 es 5.1, y utilice este hecho, junto con los resultados del ejercicio 8.14, para calcular
a) la varianza de la muestra $12,27,9,18,12$ y 21 ;
b) la varianza de la muestra $9,14,8,11,9$ y 12 .

Key Concepts

Squared Deviations

Squared deviations are obtained by taking the difference between each observation and the arithmetic mean, and then squaring that difference. This squaring process ensures that both positive and negative differences contribute positively to the overall dispersion measure, and it penalizes larger deviations more than smaller ones, thus playing a crucial role in the calculation of variance.

Effects of Linear Transformations on Variance

When a dataset is subjected to a linear transformation, such as scaling (multiplying by a constant) or translation (adding a constant), the variance is affected in a predictable way. Specifically, adding a constant to every observation does not change the variance, while multiplying every observation by a constant scales the variance by the square of that constant. This property is often used to simplify calculations by transforming a data set into a more manageable form without altering its variability characteristics.

Sample Variance

The sample variance is a measure of dispersion that quantifies the average squared deviation of each observation from the sample mean. It provides an indication of the spread of the data values around the central tendency and is calculated by summing the squared differences between each observation and the mean, and then dividing by the appropriate number (typically n-1 when calculating a sample variance, where n is the number of observations).

Arithmetic Mean

The arithmetic mean, or average, is a fundamental measure of central tendency in statistics. It is calculated by summing all the observations in the data set and then dividing by the number of observations. The mean serves as the central point around which the deviations (differences between each data value and the mean) are computed for measuring variance.

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