Step-by-Step Guide: How to Calculate Weighted Mean in Maths
In Mathematics, the weighted mean is used to calculate the average value of the data. In the weighted mean calculation, the average value can be calculated by providing different weights to some of the individual values. We need to calculate the weighted mean when data is given in a different way compared to the arithmetic mean or sample mean. Different types of means are used to calculate the average of the data values. Let’s understand what is weighted mean and how to define weighted mean along with solved examples.
Weight Definition
Weight is defined as the measure of how heavy an object is. The weights cannot be negative. Some weight can be zero, but not all of them, since division by zero is not allowed.
The data elements which have a high weight will contribute more to the weighted mean as compared to the elements with a low weight.
What is Weighted Mean?
To calculate the weighted mean of certain data, we need to multiply the weight associated with a particular event or outcome with its associated outcome and finally sum up all the products together. It is very useful in calculating a theoretically expected outcome. Apart from weighted mean and arithmetic mean, there are various types of means such as harmonic mean, geometric mean, and so on.
Define Weighted Mean
The weighted mean is defined as an average computed by giving different weights to some of the individual values. When all the weights are equal, then the weighted mean is similar to the arithmetic mean. A free online tool called the weighted mean calculator is used to calculate the weighted mean for the given range of values.
Weighted Mean Formula
To calculate the weighted mean for a given set of non-negative data x1,x2,x3,...xn with non-negative weights w1,w2,w3,..., we use the formula given below.
\[WeightedMean\overline{( W )} = \frac {\sum_{i=1}^{n} w_{i}x_{i}}{\sum_{i=1}^{n} w_i}\]
Where \[\overline {W}\] is the weighted mean,
x is the repeating value,
n is the number of terms whose mean is to be calculated, and
w is the individual weights.
Uses of Weighted Means
Weighted means are useful in a wide variety of scenarios in our daily life. For example, a student uses a weighted mean in order to calculate their percentage grade in a course. In such a case, the student has to multiply the weighing of all assessment items in the course (e.g., assignments, exams, projects, etc.) by the respective grade that was obtained in each of the categories.
It is used in descriptive statistical analysis, such as index numbers calculation. For example, stock market indices such as Nifty or BSE Sensex are computed using the weighted average method. It can also be applied in physics to find the center of mass and moments of inertia of an object.
It is also useful for businessmen to evaluate the average prices of goods purchased from different vendors where the purchased quantity is considered as the weight. It gives a better understanding of his expenses.
A customer's decision on whether to buy a product or not depends on the quality of the product, knowledge of the product, cost of the product, and service by the franchise. The customer allocates weight to each criterion and calculates the weighted average. This will help him to make a better decision on buying the product.
To recruit a person for a job, the interviewer looks at the personality, working capabilities, educational qualification, and team working skills. Based on the profile, different levels of importance (weights) are given, and then the final selection is made.
Important Notes
The weights can be in the form of quantities, decimals, whole numbers, fractions, or percentages.
If the weights are given in percentage, then the sum of the percentage will be 100%.
Weighted average for quantities (x)i having weights in percentage (P)i% is: Weighted average = ∑ (P)i% × (x)i
Solved Example of Weighted Mean
Suppose a marketing firm conducted a survey of 1,000 households to determine the average number of TVs each household owns. The data shows that there are more households with two or three TVs and a few numbers with one of four. Every household in the sample has at least one TV and not a household has more than four. Calculate the mean number of TVs per household.
Solution: Here most of the values in this data set are repeated multiple times, we can easily compute the sample mean as a weighted mean. Following are steps to calculate the weighted arithmetic mean.
Step 1: First assign a weight to each value in the dataset.
x1=1, w1=73
x2=2, w2=378
x3=3, w3=459
x4=4, w4=90
Step 2: Now compute the numerator of the weighted mean formula.
To calculate it, multiply each sample by its weight and then add the products together to get the final value
\[\displaystyle\sum\limits_{i=1}^4 i\] wixi=w1x1+w2x2+w3x3+w4x4
= 1 x 73 + 2 x 378 + 3 x 459 + 4 x 90
= 73 + 756 + 1377 + 360
= 2566
Step 3: Now, compute the denominator of the weighted mean formula by adding their weights together.
\[\displaystyle\sum\limits_{i=1}^4 i\] wi=w1+w2+w3+w4
= 73 + 378 + 459 + 90
= 1000
Step 4: Finally divide the numerator value by the denominator value.
\[\frac{\displaystyle\sum\limits_{i=1}^4w_{i}x_{i}}{\displaystyle\sum\limits_{i=1}^4 w_{i}}\]
=\[\frac {2566}{1000}\]
=2.566
Hence, the mean number of TVs per household in this sample is 2.566.
Note: The weighted mean can be easily influenced by an outlier in our data. If we have very high or very low values in our data set, then we cannot rely on the weighted mean.
Conclusion
Weighted Mean is a mean where some of the values contribute more than others. It represents the average of a given data. The Weighted mean is similar to the arithmetic mean or sample mean. Sometimes it is also known as the weighted average.
When the weights add to 1, we just have to multiply each weight by the matching value and sum it all up.
Otherwise, we have to multiply each weight w by its matching value x, the sum that all up, and divide it by the sum of weight.
FAQs on Weighted Mean Explained: Key Concepts & Real-Life Uses
1. What is a weighted mean in simple terms?
A weighted mean is a type of average where not all values are treated equally. Instead, each value is given a specific 'weight' or 'importance'. Values with a higher weight contribute more to the final average than values with a lower weight. It provides a more accurate representation when the data points have varying levels of significance.
2. How is a weighted mean different from a simple arithmetic mean?
The key difference lies in the 'importance' assigned to data points. In a simple arithmetic mean, every value has equal importance. You just add them all up and divide by the count. In a weighted mean, each value is multiplied by its assigned weight before being summed, which gives a more balanced result when some data points are more significant than others.
3. What are some real-life examples of using a weighted mean?
Weighted mean is used in many everyday situations. Here are a few common examples:
Calculating Student Grades: A final exam (worth 50%) has more weight than a unit test (worth 20%) or a quiz (worth 5%). Your final grade is a weighted mean of your scores.
Stock Market Indices: Indices like the Nifty or Sensex use a weighted mean, where companies with larger market capitalisation have a greater impact on the index's value.
Product Ratings: A review from a 'verified purchaser' might be given more weight than an anonymous review when calculating a product's average rating.
4. Why is it important to use a weighted mean for certain calculations?
Using a weighted mean is important because it provides a fairer and more accurate reflection of the data when some points are inherently more significant. If you used a simple average to calculate a student's grade, a low score on a minor quiz would have the same impact as a low score on a major final exam, which would be misleading. The weighted mean corrects for this imbalance.
5. How is the weighted mean calculated?
To calculate the weighted mean, you follow these clear steps:
Step 1: Multiply each data value (x) by its corresponding weight (w).
Step 2: Add up all the products (the results from Step 1) to get the sum of weighted values (Σxw).
Step 3: Add up all the weights to get the total weight (Σw).
Step 4: Divide the sum of the weighted values (from Step 2) by the sum of the weights (from Step 3). The formula is: Weighted Mean = Σ(xᵢwᵢ) / Σ(wᵢ).
6. In the context of the NCERT syllabus, how does the concept of a weighted mean appear?
In the NCERT syllabus for statistics, particularly when calculating the mean of a grouped frequency distribution, you are essentially using a weighted mean. Here, the data values are the class marks (mid-points of class intervals), and their 'weights' are the corresponding frequencies (the number of times values occur in that class). You multiply each class mark by its frequency, sum them up, and divide by the total frequency.
7. What do the 'weights' in the weighted mean formula actually signify?
The 'weights' signify the relative importance, frequency, or contribution of each data point to the overall average. A higher weight means the data point has a greater influence on the result. For instance, in a survey, a weight might be applied to a demographic group to ensure its representation in the final result matches its proportion in the general population.
8. What happens to the weighted mean if all the weights are equal?
If all the weights assigned to the data values are equal, the weighted mean will be exactly the same as the simple arithmetic mean. This is because the equal weights cancel each other out during the calculation, effectively reducing the formula to the standard 'sum of values divided by the number of values'. This shows that the arithmetic mean is just a special case of the weighted mean.
