For a symmetrical distribution, the variance is equal to the standard deviation. true or false?
Answers
Answer:
True
Step-by-step explanation:
In a symmetrical distribution, where the data is evenly spread around the mean, the variance is equal to the standard deviation. Both measures quantify the dispersion or spread of the data points. The standard deviation is simply the square root of the variance, and since squaring and taking the square root are symmetric operations, they yield the same result in a symmetrical distribution.
Final answer:
The statement that in a symmetrical distribution the variance is equal to the standard deviation is false. While they are related, with the standard deviation being the square root of the variance, they are not equivalent. The properties of symmetrical distributions include the same mean and median, and a certain percentage of data is within one, two, or three standard deviations of the mean (known as the Empirical Rule).
Explanation:
The statement in your question is actually false. In a symmetrical distribution, the variance is not equal to the standard deviation. The variance is calculated as the average of the squares of the difference between each value and the mean, while the standard deviation is the square root of the variance. Hence, variance and standard deviation are related, but not equal.
For example, imagine you have a data set with values: 1, 2, and 3. The mean then is 2. The variance is calculated as [(1-2)^2 + (2-2)^2 + (3-2)^2]/3 = 2/3, and the standard deviation is the square root of this, approximately 0.82, so they are not equal, even though this distribution is symmetric.
The properties of symmetrical distributions are that the mean and median are the same and about 68 percent of the data is within one standard deviation of the mean, 95 percent within two standard deviations, and over 99 percent within three standard deviations (also known as the Empirical Rule). Remember that these properties apply only when the shape of the distribution is bell-shaped and symmetric.
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Answer:
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Answer:
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Step-by-step explanation:
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