variance

Examples of variance in the following topics:

• Variance Sum Law II

  • Compute the variance of the sum of two variables if the variance of each and their correlation is known
  • Compute the variance of the difference between two variables if the variance of each and their correlation is known
  • which is read: "The variance of X plus or minus Y is equal to the variance of X plus the variance of Y."
  • The variance of the difference is:
  • If the variances and the correlation are computed in a sample, then the following notation is used to express the variance sum law:

• Variance

  • The variance of a data set measures the average square of these deviations.
  • Calculating the variance begins with finding the mean.
  • Once the mean is known, the variance can be calculated.
  • The variance for the above set of numbers is:
  • The population variance can be very helpful in analyzing data of various wildlife populations.

• Variance Sum Law I

  • The question is, "What is the variance of this sum?"
  • where the first term is the variance of the sum, the second term is the variance of the males and the third term is the variance of the females.
  • Therefore, if the variances on the memory span test for the males and females respectively were 0.9 and 0.8, respectively, then the variance of the sum would be 1.7.
  • More generally, the variance sum law can be written as follows:
  • which is read: "The variance of X plus or minus Y is equal to the variance of X plus the variance of Y."

• Test of Two Variances

  • Another of the uses of the F distribution is testing two variances.
  • Let $\sigma ^2_1$ and $\sigma ^2_2$ be the population variances and $s^2_1$ and $s^2_2$ be the sample variances.
  • It depends on Ha and on which sample variance is larger.
  • The first instructor's grades have a variance of 52.3.
  • The second instructor's grades have a variance of 89.9.

• Variance Estimates

  • The $F$-test can be used to test the hypothesis that the variances of two populations are equal.
  • Notionally, any $F$-test can be regarded as a comparison of two variances, but the specific case being discussed here is that of two populations, where the test statistic used is the ratio of two sample variances.
  • The expected values for the two populations can be different, and the hypothesis to be tested is that the variances are equal.
  • It has an $F$-distribution with $n-1$ and $m-1$ degrees of freedom if the null hypothesis of equality of variances is true.
  • Discuss the $F$-test for equality of variances, its method, and its properties.

• Comparing Two Population Variances

  • In order to compare two variances, we must use the $F$ distribution.
  • Let $\sigma_1^2$ and $\sigma_2^2$ be the population variances and $s_1^2$ and $s_2^2$ be the sample variances.
  • A test of two variances may be left, right, or two-tailed.
  • The first instructor's grades have a variance of 52.3.
  • The second instructor's grades have a variance of 89.9.

• Mean, Variance, and Standard Deviation of the Binomial Distribution

  • In this section, we'll examine the mean, variance, and standard deviation of the binomial distribution.
  • The mean, variance, and standard deviation are three of the most useful and informative properties to explore.
  • The easiest way to understand the mean, variance, and standard deviation of the binomial distribution is to use a real life example.
  • $s^2 = Np(1-p)$, where $s^2$ is the variance of the binomial distribution.
  • Naturally, the standard deviation ($s$) is the square root of the variance ($s^2$).

• Variance and standard deviation

  • Here, we introduce two measures of variability: the variance and the standard deviation.
  • The standard deviation is defined as the square root of the variance:
  • The variance is roughly the average squared distance from the mean.
  • The standard deviation is the square root of the variance.
  • The σ2 population variance and for the standard deviation.

• Test of a Single Variance

  • A test of a single variance assumes that the underlying distribution is normal.
  • A test of a single variance may be right-tailed, left-tailed, or two-tailed.
  • The null and alternate hypotheses contain statements about the population variance.
  • To many instructors, the variance (or standard deviation) may be more important than the average.
  • The parameter is the population variance, σ2, or the population standard deviation, σ.

• Summary

  • The populations are assumed to have equal standard deviations (or variances)
  • A Test of Two Variances hypothesis test determines if two variances are the same.