4.6
Variance
Les écarts indiquent la dispersion des données par rapport à la moyenne. Un écart positif se produit lorsque la valeur observée dépasse la moyenne, ta…
La variance est une statistique estimant la variabilité des valeurs de l’ensemble de données par rapport à la moyenne. Il est numériquement égal au carré de l’écart-type d’un ensemble de données.
La variance est un outil statistique précieux utilisé dans l’analyse de la variance, l’estimation du risque ou la volatilité des marchés financiers.
La variance de l’échantillon est notée par le carré de l’écart-type de l’échantillon s, tandis que la variance de la population est exprimée par le carré de l’écart-type de la population sigma.
Imaginez si l’on estimait la variance de l’échantillon de poids des ours polaires dans différentes régions de l’Arctique. En divisant la population en échantillons aléatoires et en calculant les variances de l’échantillon, on observe que les valeurs sont centrées autour de la valeur constante de la variance de la population. Ainsi, la variance de l’échantillon est un estimateur impartial de la variance de la population.
Le principal inconvénient de la variance est que ses unités diffèrent considérablement des unités de l’ensemble de données. Par exemple, les unités de variance des précipitations au cours d’une année seront des millimètres carrés, ce qui n’est pas utile. Par conséquent, dans la plupart des analyses, l’écart-type est préféré à la variance.
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Q1: What is the relationship between variance and standard deviation?
Variance is numerically equal to the square of the standard deviation. While standard deviation measures spread in the same units as the original data, variance's units differ from the dataset units. For example, rainfall variance is expressed in millimeters squared, making it less intuitive. This unit difference is why standard deviation is often preferred in practical analyses.
Q2: Why do deviations need to be squared when calculating variance?
Deviations show how spread out data are about the mean. When deviations are added together, the sum is always zero because positive and negative deviations cancel out. By squaring the deviations, all numbers become positive, so their sum is also positive. This allows variance to accurately represent data spread.
Q3: How does sample variance differ from population variance?
Sample variance is denoted as the square of sample standard deviation s, while population variance is the square of population standard deviation sigma. Sample variance is an unbiased estimator of population variance. When calculating sample variance, dividing by n – 1 instead of n provides a better estimate of the population variance based on theoretical mathematics.
Q4: What are the practical applications of variance in statistics?
Variance is a valuable statistical tool used in analysis of variance, estimation of risk, and measurement of volatility in financial markets. It quantifies how dataset values vary from the mean, providing insight into data dispersion. However, its unintuitive units often make standard deviation the preferred choice for most practical analyses and interpretations.
Q5: Why is dividing by n – 1 important when calculating sample variance?
Dividing by n – 1 instead of n when calculating sample variance produces a better estimate of the population variance. This adjustment accounts for the fact that sample data is used to estimate population parameters. The theoretical mathematics behind this calculation shows that n – 1 provides an unbiased estimator, making sample variance a reliable tool for inference.
Q6: What is the main disadvantage of using variance in data analysis?
The major disadvantage of variance is that its units vastly differ from the dataset units, making interpretation difficult. For instance, variance of rainfall measured in millimeters squared is unhelpful for practical understanding. This unit incompatibility is why standard deviation, which maintains the original data units, is preferred in most statistical analyses and real-world applications.
Q7: How can you verify that sample variance is an unbiased estimator?
When a population is divided into random samples and sample variances are calculated, the values center around the constant population variance value. This clustering demonstrates that sample variance is an impartial estimator of population variance. This property makes sample variance reliable for drawing inferences about population characteristics from sample data.