Weighted Harmonic Mean

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Description:



Example:



Sources:

http://en.wikipedia.org/wiki/Weighted_harmonic_mean

http://books.google.ca/books?id=CHz8dIDswBEC&pg=PT89&lpg=PT89&dq=weighted+harmonic+mean&source=web&ots=mdz9ZEcGz5&sig=8MTvSAWh_ChWtT2vZpZSJfSGJ-I&hl=en

Harmonic Mean

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Description:

The harmonic mean is the number of variables divided by the sum of the reciprocals of the variables. Typically, it is appropriate for situations when the average of rates is desired.


Example:

For instance, if for half the distance of a trip you travel at 40 kilometres per hour and for the other half of the distance you travel at 60 kilometres per hour, then your average speed for the trip is given by the harmonic mean of 40 and 60, which is 48; that is, the total amount of time for the trip is the same as if you travelled the entire trip at 48 kilometres per hour. If you had travelled for half the time at one speed and the other half at another, the arithmetic mean, in this case 50 kilometres per hour, would provide the correct average.

In finance, the harmonic mean is used to calculate the average cost of shares purchased over a period of time. For example, an investor purchases $1000 worth of stock every month for three months and the prices paid per share each month were $8, $9, and $10, then the average price the investor paid is $8.926 per share. However, if the investor purchased 1000 shares per month, the arithmetic mean (which turns out to be $9.00) would be used. Note that in this example, the investor buying $1000 worth of the stock each month means buying 125 shares at $8 the first month, 111.11 shares at $9 the second month, and 100 shares at $10 in the third month. Fewer shares are purchased at higher prices while more shares are purchased at lower prices. Thus more weight is given to the lower prices than the higher prices in the calculation of the average cost per share ($8.926). If the investor had instead purchased 1000 shares each month then equal weight would be given to high and low purchase prices, leading to an average cost per share of $9.00. This explains why the harmonic mean is less than the arithmetic mean.

Sources:

http://en.wikipedia.org/wiki/Harmonic_mean

Weighted Geometric Mean

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Description:

I found very little information about this type of mean. So far, I've only seen it used with weights that add up to 1.

Example:

none yet

Geometric Mean

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Description:

The geometric mean only applies to positive numbers. It is also often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as data on the growth of the human population or interest rates of a financial investment.

Any time you have a number of factors contributing to a product, and you want to find the "average" factor, the answer is the geometric mean. The example of interest rates is probably the application most used in everyday life.

Example:

Suppose that I’m 30% richer than last year, but last year I was 20% richer than the year before… what is the average growth? Well, my current wealth is 1.3 * 1.2 * w if w is my wealth two years ago. I can expect that if t is the average growth factor over the last two years, then my current wealth is t * t * w. Setting t = 1.25 is the wrong answer. In such a case, choosing t = sqrt(1.3 * 1.2) solves the problem.

Sources:

http://www.daniel-lemire.com/blog/archives/2006/04/21/when-use-the-geometric-mean/

http://en.wikipedia.org/wiki/Geometric_mean

http://www.math.toronto.edu/mathnet/questionCorner/geomean.html

Weighted Arithmetic Mean

Formula:



Where w are the weights

Description:

Data elements with a high value for the weight contribute more to the weighted mean than do elements with a low value for the weight. The weights must not be negative.

Example:

I buy 20 red balls and 30 blue balls. Red balls cost one dollar each, blue balls cost two dollars each. What is the average price of the balls I purchased?

20*1+30*2/50=80/50=$1.60

Another way using proportions: red balls represent 40%, blue balls 60%.

0.4*1+0.6*2/1=$1.60

Arithmetic Mean

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Description:

This is usually what people refer to when they talk about a mean or average. It is the sum of the items, divided by the number of items.

Example:

There are 4 students in a class. Their final grades are 50, 60, 75 and 90. What is the average for the class?

Average for the class = (50+60+75+90)/4 = 68.75

Statistical and Mathematical Formulas and Equations, and Their Uses

The next few posts may seem very dry, and the link to performance measurement might not be immediately clear.

They will consist of different statistical and mathematical formulas which I may or may not refer to in future posts. Without going into too much detail, let me just say that “how” you measure something doesn’t just refer to what indicators you will use to try to measure an outcome, “how” can also refer to how your indicators or measures are calculated. That’s just one of the challenges of the practical side of performance measurement, as opposed to the more theoretical side of defining outcomes, logic models and frameworks.

Two Interesting Papers on the Basics

Mark Schacter has written some interesting papers on performance measurement.

They are fairly theoretical however, but give a good overview of some of the basics, in particular of logic models, frameworks and some advice on how to choose indicators.

I would recommend reading:

Not a Toolkit. Practitioner's Guide to Measuring the Performance of Public Programs

and

What Will Be, Will Be. The Challenge of Applying Results-based Thinking to Policy


Good reading!

source: http://www.schacterconsulting.com/