Variance is the squared deviation from the mean. (X bar - X data)^2
No, the variance is not defined as the mean of the sum of the squared deviations from the median; rather, it is the mean of the squared deviations from the mean of the dataset. Variance measures how much the data points differ from the mean, while the median is a measure of central tendency that may not accurately reflect the spread of the data in the same way. Though both concepts involve deviations, they use different points of reference for their calculations.
You want some measure of how the observations are spread about the mean. If you used the deviations their sum would be zero which would provide no useful information. You could use absolute deviations instead. The sum of squared deviations turns out to have some useful statistical properties including a relatively simple way of calculating it. For example, the Gaussian (or Normal) distribution is completely defined by its mean and variance.
Averaging the deviations of individual data values from their mean would always result in zero, since the mean is the point at which the sum of deviations is balanced. This occurs because positive and negative deviations cancel each other out. Instead, measures like variance and standard deviation are used, which square the deviations to ensure all values contribute positively, providing a meaningful representation of spread around the mean.
because of two things- a) both positive and negative deviations mean something about the general variability of the data to the analyst, if you added them they'd cancel out, but squaring them results in positive numbers that add up. b) a few larger deviations are much more significant than the many little ones, and squaring them gives them more weight. Sigma, the square root of the variance, is a good pointer to how far away from the mean you are likely to be if you choose a datum at random. the probability of being such a number of sigmas away is easily looked up.
Given a set of n scores, the variance is sum of the squared deviation divided by n or n-1. We divide by n for the population and n-1 for the sample.
The sum of total deviations about the mean is the total variance. * * * * * No it is not - that is the sum of their SQUARES. The sum of the deviations is always zero.
No, a standard deviation or variance does not have a negative sign. The reason for this is that the deviations from the mean are squared in the formula. Deviations are squared to get rid of signs. In Absolute mean deviation, sum of the deviations is taken ignoring the signs, but there is no justification for doing so. (deviations are not squared here)
No, the variance is not defined as the mean of the sum of the squared deviations from the median; rather, it is the mean of the squared deviations from the mean of the dataset. Variance measures how much the data points differ from the mean, while the median is a measure of central tendency that may not accurately reflect the spread of the data in the same way. Though both concepts involve deviations, they use different points of reference for their calculations.
You want some measure of how the observations are spread about the mean. If you used the deviations their sum would be zero which would provide no useful information. You could use absolute deviations instead. The sum of squared deviations turns out to have some useful statistical properties including a relatively simple way of calculating it. For example, the Gaussian (or Normal) distribution is completely defined by its mean and variance.
sum of scores: 24 mean of scores : 24/4 = 6 squared deviations from the mean: 9, 4,4,9 sum of these: 26 sample variance: 26/4 = 6.5
First mean is calculated.Then calculate deviations from the mean.Then the deviations are squared.Then the squared deviations are summed up.Finally this sum is divided by number of items for which the variance is being calculated. For a population, it is by the number of values, in this case 12. If it is a sample, then we divide by one less, which is 11,For these figures, the variance for the population is 11069.24306. If it is a sample, it is 12075.53788 as the result.
Averaging the deviations of individual data values from their mean would always result in zero, since the mean is the point at which the sum of deviations is balanced. This occurs because positive and negative deviations cancel each other out. Instead, measures like variance and standard deviation are used, which square the deviations to ensure all values contribute positively, providing a meaningful representation of spread around the mean.
because of two things- a) both positive and negative deviations mean something about the general variability of the data to the analyst, if you added them they'd cancel out, but squaring them results in positive numbers that add up. b) a few larger deviations are much more significant than the many little ones, and squaring them gives them more weight. Sigma, the square root of the variance, is a good pointer to how far away from the mean you are likely to be if you choose a datum at random. the probability of being such a number of sigmas away is easily looked up.
Usually the sum of squared deviations from the mean is divided by n-1, where n is the number of observations in the sample.
Given a set of n scores, the variance is sum of the squared deviation divided by n or n-1. We divide by n for the population and n-1 for the sample.
The sum of standard deviations from the mean is the error.
When throwing a single unbiased six-sided die, the mean (expected value) is calculated as the average of the outcomes: (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5. The variance measures the spread of the outcomes around the mean, which is calculated as the average of the squared deviations from the mean: the variance for a single die is 2.9167 (or 35/12). For multiple dice, the mean is the number of dice times 3.5, and the variance is the number of dice times 2.9167.