# What is the geometric standard deviation used for?

## Harmonic and geometric mean

Descriptive Statistics> Measures of Location

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Harmonic and geometric mean.

The arithmetic mean is not always meaningful to use as an average value.

In the following two exceptional cases, the "harmonic and geometric" means are better suited for calculating an "average value".

The harmonic mean is often used to calculate average speeds.

This is also the case in the following example: There are 2 waiters in our restaurant. One takes 2 minutes to serve a guest, the other is slower and takes 4 minutes to serve a guest.

What is the average service time per guest?

The arithmetic mean would be right in between, i.e. 3 minutes.

Is that the right solution now? No, because the fast waiter serves more guests than the slow waiter in the same time.

But when the fast waiter serves more guests, he has a stronger influence on the average speed of both waiters. The value of the average service speed per guest must therefore be faster, i.e. less than 3 minutes.

Using this formula, we can calculate the harmonic mean.

"n" is the number of guests served. In our example it was "2", because the operating speeds were given "per guest". It took the fast waiter 2 minutes and the slow waiter 4 minutes to serve a guest. The result is an average speed of 2.66 minutes per guest.

Remember: When calculating the average of speeds, it can be useful to use the harmonic mean formula. The key word for using the "geometric mean" is "average growth".

This table shows us the sales and the sales growth of our annual restaurant sales. For example, in 2006 restaurant sales were 13.1 percent higher than in 2005. How do we form an average value from these 4 growth rates? The use of the arithmetic mean would be conceivable here, but it does not make sense to calculate an average growth rate.

Why? Because the growth rates of the individual years are interdependent and we cannot simply "add" them like independent values, we have to "multiply" them using the following formula.

We have 4 values, so "n" is 4 and we therefore calculate "4. Root of the product of the individual growth rates.

This is the geometric mean and we now know the average annual growth rate of our restaurant sales.

It is easy to remember: If the average value is to be formed from growth rates, the formula of the geometric mean must be used.

The arithmetic mean is not always meaningful to use as an average value.

In the following two exceptional cases, the "harmonic and geometric" means are better suited for calculating an "average value".

The harmonic mean is often used to calculate average speeds.

This is also the case in the following example: There are 2 waiters in our restaurant. One takes 2 minutes to serve a guest, the other is slower and takes 4 minutes to serve a guest.

What is the average service time per guest?

The arithmetic mean would be right in between, i.e. 3 minutes.

Is that the right solution now? No, because the fast waiter serves more guests than the slow waiter in the same time.

But when the fast waiter serves more guests, he has a stronger influence on the average speed of both waiters. The value of the average service speed per guest must therefore be faster, i.e. less than 3 minutes.

Using this formula, we can calculate the harmonic mean.

"n" is the number of guests served. In our example it was "2", because the operating speeds were given "per guest". It took the fast waiter 2 minutes and the slow waiter 4 minutes to serve a guest. The result is an average speed of 2.66 minutes per guest.

Remember: When calculating the average of speeds, it can be useful to use the harmonic mean formula. The key word for using the "geometric mean" is "average growth".

This table shows us the sales and the sales growth of our annual restaurant sales. For example, in 2006 restaurant sales were 13.1 percent higher than in 2005. How do we form an average value from these 4 growth rates? The use of the arithmetic mean would be conceivable here, but it does not make sense to calculate an average growth rate.

Why? Because the growth rates of the individual years are interdependent and we cannot simply "add" them like independent values, we have to "multiply" them using the following formula.

We have 4 values, so "n" is 4 and we therefore calculate "4. Root of the product of the individual growth rates.

This is the geometric mean and we now know the average annual growth rate of our restaurant sales.

It is easy to remember: If the average value is to be formed from growth rates, the formula of the geometric mean must be used.

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