For which measures of central location will the sum of the deviations of each value from the datas average will always be zero?

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Video Transcript
Does standard deviation and mean deviation measure dispersion the same?
Why is the sum of the deviations always zero?
What do you call the measure of central tendency which is the sum of the values of data divided by the total number of values?
What measure of central tendency do you use with standard deviation?
Which is used to measure the location of central tendency?

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Video Transcript

in this problem, we have been asked for which measure of central tendency will the some of the deviations of each value from the average always be zero? The options are the median, the mean, the mode, the geometric mean or all of them. So the answer to this question will be the mean. Let us prove that. So let us consider that we have the values X one, X 2 up tricks and and let us find the mean of these values by adding all of them and dividing it by the number of data points, which is enh this is the mean, let's assume that to be X bar. So from here we can obtain that X one plus X two up to xN, this will be equal to n times X bar. Now we want to show that the sum of deviations of each value from the mean is zero. So let us consider the deviations. So the first deviation is X one minus X bar. The deviation of the second value from the mean is X two minus x bar, and we can continue like this until we get x n minus x bar. So if we add all of these then we will have the sum of deviations and we would like to show that this is equal to zero. So let us write this as X one plus X two up to xn and here we have minus X bar plus X bar and another X bar over here, so we just showed that X one plus X two X two, X n, that is N X bar. So let's write that over here and then we have X bar plus X bar plus X bar. How many times it's end times because we get one X bar from here to X bar, this is two X bars up to this point, and up to here we get the N F X bar. So that means this is an X bar, this will be an X bar. So we end up with an X bar minus N X bar and that is equal to zero. So we have shown that the sum of the deviations from the mean is zero. And we also need to show that the sum of deviations from the other ones are not zero, so that we can Strike out the last option and know that that one is incorrect. So for that let us consider a counter example, let us consider a dataset one, So let us find the median, the median will be the arithmetic mean of these middle two values. So we have two plus two divided by two, which is two. Then let us consider the mode, that is the value which appears the most frequently. We can see that too appears the most frequently the mode is too. And then let us find the geometric mean. So we have four values that will be the fourth root of the product of all of them. So we have one times two times two times four and we need to consider the fourth root of it. So we have the fourth root of 16 and that is just equal to two. So you can see that the median, the mode and the geometric mineral too. And if we consider the status of the deviations from this number two, we end up with one minus two plus two minus two, plus two minus two, plus four minus two, so this is negative one plus zero plus zero plus two. So we have two minus one, which is one, and that is not equal to zero. So you can see that the deviations from two is not zero. So that means that some of the deviations from the median, the mode and the geometric mean this will all be non zero, because they're all too. So we have shown using this counter example that the sum of deviations from the median mode and geometric mean, they are not necessarily zero. And we have proved over here that the sum of deviations from the mean will always be zero. So that means that the correct answer will be the mean

Does standard deviation and mean deviation measure dispersion the same?

No. The average of the deviations, or mean deviation, will always be zero. The standard deviation is the average squared deviation which is usually non-zero.

Why is the sum of the deviations always zero?

The sum of the deviations from the mean is zero. This will always be the case as it is a property of the sample mean, i.e., the sum of the deviations below the mean will always equal the sum of the deviations above the mean.

What do you call the measure of central tendency which is the sum of the values of data divided by the total number of values?

The mean, often called the average, of a numerical set of data, is simply the sum of the data values divided by the number of values. This is also referred to as the arithmetic mean.

What measure of central tendency do you use with standard deviation?

Standard deviation - as the name suggests is a measure of the deviation. Deviation means change or distance. But change is always followed by the word 'from'. Hence standard deviation is a measure of change or the distance from a measure of central tendency - which is normally the mean.

Which is used to measure the location of central tendency?

The median is less affected by outliers and skewed data than the mean, and is usually the preferred measure of central tendency when the distribution is not symmetrical.