Statistics Definitions > Weighted Mean

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## What is a Weighted Mean?

A weighted mean is a kind of average. Instead of each data point contributing equally to the final mean, some data points contribute more “weight” than others. If all the weights are equal, then the weighted mean equals the arithmetic mean (the regular “average” you’re used to). Weighted means are very common in statistics, especially when studying populations.

## The Arithmetic Mean.

When you find a mean for a set of numbers, all the numbers carry an equal weight. For example, if you want to find the arithmetic mean of 1, 3, 5, 7, and 10:

- Add up your data points: 1 + 3 + 5 + 7 + 10 = 26.
- Divide by the number of items in the set: 26 / 5 = 5.2.

What do we mean by “equal weight”? The first sentence in some tests (like this one) is sometimes “All questions carry an equal weight”). It’s saying that all the questions in the exam are worth the same number of points. If you have a 100 point exam and 10 questions, each question is worth 1/10th of the points. In the above question, you have of a set of five numbers. You can think of each number contributing 1/5 to the total mean (as there are 5 numbers in the set).

## The Weighted Mean.

In some cases, you might want a number to have more weight. In that case, you’ll want to find the **weighted mean**. To find the weighted mean:

- Multiply the numbers in your data set by the weights.
- Add the results up.

For that set of number above with equal weights (1/5 for each number), the math to find the weighted mean would be:

1(*1/5) + 3(*1/5) + 5(*1/5) + 7(*1/5) + 10(*1/5) = 5.2.

**Sample problem**: You take three 100-point exams in your statistics class and score 80, 80 and 95. The last exam is *much* easier than the first two, so your professor has given it less weight. The weights for the three exams are:

- Exam 1: 40 % of your grade. (Note: 40% as a decimal is .4.)
- Exam 2: 40 % of your grade.
- Exam 3: 20 % of your grade.

What is your final weighted average for the class?

- Multiply the numbers in your data set by the weights:

.4(80) = 32

.4(80) = 32

.2(95) = 19 - Add the numbers up. 32 + 32 + 19 = 83.

The percent weight given to each exam is called a weighting factor.

## Weighted Mean Formula

The weighted mean is relatively easy to find. But in *some* cases the weights might not add up to 1. In those cases, you’ll need to use the weighted mean formula. The only difference between the formula and the steps above is that you divide by the sum of all the weights.

The image above is the technical formula for the weighted mean. In simple terms, the formula can be written as:

**Weighted mean = Σwx/Σw**

Σ = the sum of (in other words…add them up!).

w = the weights.

x = the value.

To use the formula:

- Multiply the numbers in your data set by the weights.
- Add the numbers in Step 1 up. Set this number aside for a moment.
- Add up all of the weights.
- Divide the numbers you found in Step 2 by the number you found in Step 3.

In the sample grades problem above, all of the weights add up to 1 (.4 + .4 + .2) so you would divide your answer (83) by 1:

83 / 1 = 83.

However, let’s say your weighted means added up to 1.2 instead of 1. You’d divide 83 by 1.2 to get:

83 / 1.2 = 69.17.

**Warning:** The weighted mean can be easily influenced by outliers in your data. If you have very high or very low values in your data set, the weighted mean may not be a good statistic to rely on.

If you prefer an online interactive environment to learn R and statistics, this *free R Tutorial by Datacamp* is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try *this Statistics with R track*.

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well done thanks

Nice one ! Thanks ! It helps a lot !

Very helpful, it has a lot to learn and repeat in a short time.

This is so easy to understand and interprete thanx

very helpful.Thank you.

thanx dats so useful

Very simple to understand,thanks

Really helpful in understanding the basic difference !!

Thank u :)

thanx sir

Great….easy to understand. Thanks much. Itox JK – 14x

Really helpful , thanks a lot ….

how did you calculate the weight of 0.4 is becoming 80

can you please explain , i dont know how to calculate Weight of the score.

pl explain some easy understanding of examples.

Regards,

R.Saran

9943041432

Saran, do you mean you’re having trouble following the weighted mean example?

0.4 is the same as 40%. The 80 is the number of points scored in the exam (from the question). You just calculate the two.

You have made it very easy to understtad! thanks a lot

Thanks explained perfectly

it’s good but can you discribe weighted arithmetic mean by more examples & differnt tyeps of examples.

Thank you very much

This problem is very clearly understanding

Superb! Thank you so much!

Teach me how to compute weighted mean

Not 19. That’s 17

Hi, Steve,

Where are you seeing this? The only place is I see a 19 is .2(95). Google tells me that’s 19 :)

Yes. Thanks for spotting that. It’s corrected.

Hey,

The initial scores per the question statement were 80, 80 and 85.

“Sample problem: You take three 100-point exams in your statistics class and score 80, 80 and 85. ”

Why, then, is the calculation of 20% made on a score of 95 ?

Thanks for spotting that, Shruti. I fixed the typo in the article and I’ll fix the video later on (hopefully I’ll get to it today :) ).

my problem solved ……. thanks a lot

You can only use weights till 100% right? then the total weights must always add upto 1.0….please help me understand this…..i mean u used (40%, 40%, 20%) right? but whatever u use it must always add upto 100%..if i am mistaken plz help me understand……

Weights can add up to more than 1. If they do, use the formula.

As a simple example of why, think of GPAs. Technically, a GPA can only got up to 4.0 (which would equal getting 100% in all classes). But sometimes classes (like AP classes) are given more weight — say 5.0 instead of 4.0.

Thanks, but, how about : The grades of a student and the credits that he received for the courses that he or she got the grades in? Like this= English- 90, Math – 89, Science- 86 and Spanish- 89. The credits will be 3, 4, 3 and 4 respectively.

I’m assuming you want the weighted mean for the grades. Use the formula, figuring out the weights like this:

90(.3)

89(.4)

86(.3)

89 (.4)

Then divide by 1.4 (the total weight)

Wow! I really love that.Okay but a little difficult to understand

Why is the total weight 1.4

Where are you seeing the total weight of 1.4?