# Standard deviation and percentage form

• nobahar
In summary, the coefficient of variation is a measure of how spread out a set of data is around its mean, and it's influenced by the standard deviation.
nobahar
Hello!
Quick question:
Is this valid:
$$(\frac{standard \left \left deviation}{average} * 100)$$?
The standard deviation is just a measure of the spread of the data from the average, so by doing the above, is it retaining its 'representation' as a measure of spread, just as a percentage of the average form?

You have to be careful with the word "average" in math. I take it you mean the mean.

No. I can't see that dividing the standard deviation by the mean can be meaningful. This is because the standard deviation is a quantity having to do with the shape of the distribution while the mean has only to do with position of the distribution along the number line. The two quantities are, in a sense, independent of each other. Or rather, they convey distinct statistical information.

For instance, suppose you have a set of data from which you calculate the mean and SD. Now add 10 to every value in the set. The new set will have a mean that is the mean of original set plus 10. But the SD will be the same! By adding 10 to every value you have just picked up the whole distribution curve and shifted it along the number line 10 units. But the shape of the graph is the same.

nobahar said:
Hello!
Quick question:
Is this valid:
$$(\frac{standard \left \left deviation}{average} * 100)$$?
The standard deviation is just a measure of the spread of the data from the average, so by doing the above, is it retaining its 'representation' as a measure of spread, just as a percentage of the average form?

this is the definition of the coefficient of variation, a measure of relative dispersion. Technically it gives the standard deviation as a percentage of the mean, but the intuitive use is this: the larger this value is, the greater the spread of the numbers around their mean.

It springs from normal distributions. consider these two situations.

a) A population with incomes that are normally distributed with mean $50,000 and standard deviation$5000. If I pick the +- 2 standard deviation spread, it is $40000 to$60000, a
range of ten thousand. It is a simple matter to find the percentage of incomes in this range. Here the coefficient of variation is 10% (5000 is 10% of 50000)

2) Now consider another group of incomes, normally distributed, with mean of $2,000,000 (two million dollars) and standard deviation$200,000. The +- two standard deviation spread is $1,600,000 to$2,400,000 - eight hundred thousand dollars. On an absolute scale (purely in dollars) income is much more widely spread here than in 'a' - that's the influence of the larger standard deviation. Notice, however, that the percentage of incomes in this range will be the same as it was in 'a'. Here also, the coefficient of variation is 10%: relatively speaking, the incomes in this group are spread around the mean in the same pattern as those in 'a'

The intuitive use of this quantity is to compare relative variability among several groups of data. The larger this coefficient, the greater the variability. the only restrictions is that it applies only when the measurements are positive.

this is the definition of the coefficient of variation, a measure of relative dispersion.

Wow. How about that. I guess I'd better leave stats to my math betters. :-)

## 1. What is standard deviation?

Standard deviation is a measure of how spread out a set of data is from the average or mean. It tells us how much the data varies from the average value. A high standard deviation means that the data is spread out over a larger range, while a low standard deviation means the data is closer to the average value.

## 2. How is standard deviation calculated?

Standard deviation is calculated by taking the square root of the variance. The variance is calculated by finding the difference between each data point and the mean, squaring those differences, and then taking the average. This value is then squared to get the standard deviation.

## 3. What does standard deviation tell us about a data set?

Standard deviation provides valuable information about the spread or variability of a data set. It allows us to compare the spread of different data sets and identify outliers. It can also be used to determine the probability of a data point falling within a certain range of values.

## 4. How is standard deviation used in statistics?

Standard deviation is used in statistics to measure the spread of a data set and to describe the variability of the data. It is an important tool for analyzing and interpreting data, and is used in various statistical tests and models to assess the significance of results.

## 5. What is percentage form in relation to standard deviation?

Percentage form in relation to standard deviation is a way of expressing the standard deviation as a percentage of the mean. This allows for easier comparison of the variability of different data sets, as the percentage form accounts for the size and scale of the data. It is also useful for interpreting the significance of results in statistical analyses.

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