Weighted Mean Formula and Solved Examples

Weighted mean is the average of a data set. It's a weighted average derived by having some of the individual numbers of various weights. The weighted mean is the same as the arithmetic mean if all the values are the same. When data is presented in different ways than the arithmetic mean or sample mean, it is computed. Although the weighted mean acts similarly to the average mean in most cases, it does have certain inconsistencies. High-weighted data values contribute more to the weighted mean than low-weighted data values. Negative weights aren't feasible; some may be zero, but not all, because division by zero isn't permitted. In this article, we will look at the weighted mean formula and some solved examples.

Key takeaways: Arithmetic mean, Weighted mean, data set, sum, average, numerator, denominator, deviance, median

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What is Weighted Mean?

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A weighted average is the sum of all values that are ranked in order of importance. The sum of the weight times values divided by the sum of the weights is the weighted average of values.

Weghted mean

The ordinary arithmetic mean, which is the most often used average, is the same as the weighted arithmetic mean, except that instead of each data point contributing equally to the final average, certain data points represent more than others. The weighted mean concept is important in descriptive statistics and various other fields of mathematics. If all of the supplied weights are equal, the weighted arithmetic mean equals the conventional arithmetic mean. While weighted means function similarly to arithmetic means in most cases, they do have some contradictary features.

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Weighted Mean Formula

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The weighted mean formula below can be used to calculate the weighted mean for a non-negative set of data with non-negative weights:

\(\bar{x} =\frac{ w1x1 + w2x2 + .... + wnxn}{w1 + w2 + .... + wn}\)

Here,

x stands for the repeating value

w denotes the number of y weight occurrences

and x stands for the weighted mean

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Solved Examples

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Ques. Assume a marketing firm performs a study of 1,000 households in order to ascertain the average number of televisions owned by each family. According to the data, a high percentage of households have two or three televisions, while a lesser number have one or four. There is at least one television in every family in the sample, and no household has more than four. Calculate the average number of televisions per home.

Solution: As there are many repeated values in the dataset, we can calculate the sample mean by computing it as a weighted mean. To calculate the weighted arithmetic mean, follow these steps:

Each value in the dataset should be given a weight:

w1 = 73, w2 = 378, w3 = 459, w4 = 90

Calculate the weighted mean formula's numerator.

Add the items together after multiplying each sample by its weight:

=w1x1 × w2x2 × w3x3 × x w4x4

=173+2378+3459+490

= 2566

Add the weights together to get the denominator of the weighted mean calculation.

=73+378+459+90

=1000

Divide the numerator by the denominator

=2566/1000

=2.566

In this survey, the average number of TVs per household is 2.566.

Ques. The commerce class at a school is separated into two sections: A and B. Section A has 25 students, with an average grade of 75, and section B has 30 students, with an average grade of 60. Calculate the class's overall average grade.

Solution: To answer this question, we must first determine x1w1 and x2w2.

Here, w1 = 25 and w2 = 30

So, w1x1= 25 x 75 = 1875

w2x2 = 30 x 60 = 1800

Weighted mean = (1875 + 1800)/ 55

= 66.81

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Simpson’s Paradox

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Simpson's paradox is a probability and statistics phenomenon in which a trend appears in several different sets of data but vanishes or reverses when the data sets are combined. This is a common occurrence in social-science and medical-science statistics, and it's especially problematic when frequency data is assigned causal interpretations that aren't warranted. When confounding variables and causal relationships are properly addressed in statistical modeling, the dilemma can be resolved. Simpson's paradox has been used to demonstrate the type of erroneous conclusions that can be produced when statistics are misused.

In a technical study published in 1951, Edward H. Simpson originally identified this phenomenon, however statisticians Karl Pearson et al. in 1899 and Udny Yule in 1903 had previously mentioned comparable effects. Colin R. Blyth coined the term "Simpson's paradox" in 1972. Simpson's reversal, the Yule–Simpson effect, the amalgamation paradox, and the reverse paradox are all terms used to describe this phenomenon.

Simpson’s Paradox

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Uses of Weighted Mean

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Weighted means come in handy in a variety of situations. A student might use a weighted mean to calculate his or her % grade in a course.

For example, a student would multiply the weighted average of all assessment items in the course (e.g., assignments, examinations, projects, etc.) by the grade earned in each of the categories. Consider the following grades for a student:

Uses of Weighted Mean

We can get the weighted mean in the example above by multiplying the weights associated with each assessment item by the student's grade on each of the items. After that, we may add up the items to get the student's final grade.

We can see that the student is able to achieve a higher grade than expected by performing well on the course's most heavily weighted component: the final. Students can better plan their study time if they understand how each assessment aspect in the course is weighted.

Students will be better able to weigh a particular assessment item against other time-consuming activities (e.g., social life, personal interests, other courses, etc.) and make judgments that fit their personal utility function if they take a step back.

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Arithmetic Mean

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The arithmetic meanor simply the mean or the average is the sum of a set of numbers divided by the number of numbers in the set in mathematics and statistics. A set of results from an experiment or observational study, or a set of survey results, is typically included in the collection. In some cases in mathematics and statistics, the name "arithmetic mean" is favored since it distinguishes it from other means like geometric and harmonic means.

The arithmetic mean is utilized in practically every academic subject to some level, and it is widely employed in many distinct fields such as economics, anthropology, and history. The arithmetic average income of a country's population, for example, is called per capita income.

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Things to Remember

• Weighted mean is the average of a data set. It's a weighted average derived by giving some of the individual numbers of various weights.
• The weighted mean is the same as the arithmetic mean if all the values are the same.
• A weighted average is the sum of all values that are ranked in order of importance. The sum of the weight times values divided by the sum of the weights is the weighted average of values.
• The ordinary arithmetic mean, which is the most often used average, is the same as the weighted arithmetic mean, except that instead of each data point contributing equally to the final average, certain data points represent more than others.
• Simpson's paradox is a probability and statistics phenomenon in which a trend shows in many sets of data but disappears or reverses when the groups are combined.
• This is a common occurrence in social-science and medical-science statistics, and it's especially problematic when frequency data is assigned causal interpretations that aren't warranted.

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