Weighted Average Real Life Examples

Introduction to Weighted Averages

The multipliers or sets of numbers that express more or less relative importance of various observations (data points) in a data set are called weights.

The weighted arithmetic mean (simply called weighted average or weighted mean) is similar to an ordinary arithmetic mean except that instead of each data point contributing equally to the final average, some data points contribute more than others. Weighted means are useful in a wide variety of scenarios. Weighted averages are used when there are a bunch of values, but some of those values are more important or contribute more to the overall result.

Example of Weighted Average

For example, a student may use a weighted mean to calculate his/her percentage grade in a course. In such an example, the student would multiply the weight of all assessment items in the course (e.g., assignments, exams, sessionals, quizzes, projects, etc.) by the respective grade that was obtained in each of the categories.

As an example, suppose in a course there are a total of 60 marks, while the distribution of marks is as follows, Assignment-1 has a weightage of 10%, Assignment-2 has a weightage of 10%, the mid-term examination has a weightage of 30% and the final term examination have the weightage of 50%. The scenario is described in the table below:

Weighted Average Formula

Mathematically, the weighted average forma is given as

$$\overline{x}_w = \frac{\sum\limits_{i=1}^n w_i x_i}{\sum\limits_{i=1}^n w_i}$$

Another Example

Consider another example: Suppose we have monthly expenditures of a family on different items with their quantity

The average expenses will be: $AM = \frac{590}{5} = 118$.

However, the weighted average of the scenario will be $\overline{x}_w = \frac{\sum\limits_{i=1}^n w_i x_i}{\sum\limits_{i=1}^n w_i} = \frac{2539.5}{12.5}=203.16$

Keeping in mind the importance of weight, the average monthly expenses of a family was 203.16, not 118.

Note that in a frequency distribution, the computation of relative frequency (rf) is also related to the concept of weighted averages.

Some Real-World Examples of Weighted Averages

• Calculating class grade: Different assignments might have different weights (e.g., exams worth more than quizzes). A weighted mean considers these weights to determine the overall grade.
• Stock market performance: A stock index might use a weighted average to reflect the influence of large companies compared to smaller ones.
• Customer Satisfaction: Finding the average customer satisfaction score when some customers’ feedback might hold more weight (e.g., frequent buyers).
• Average Customer Spending: if some customers buy more frequently.
• Expected Value: Determining the expected value of outcomes with different probabilities.

The following are some important questions. What is the importance of weighted mean? Describe its advantages and disadvantages. What is an average? What are the qualities of a good average? What does Arithmetic mean? Describe the advantages and disadvantages of Arithmetic mean. In which situations do we apply arithmetic mean?

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Summation Operator Properties and Examples (2024)

The summation operator is denoted by $\Sigma$. The summation operator is a mathematical notation used to represent the sum of numbers or terms. The summation is the total of all the terms added according to the specified range of values for the index.

Suppose, we have information about the height of students, such as 54, 55, 58, 60, 61, 45, 53.
Using variable and value notation one can denote the height of the students like

• First height in the information $X_1$, that is $X_1=54$
• Second height in the information $X_2$, that is $X_2=55$
• Last or nth information $X_n$, that is $X_n=53$.

In general, the variable and its values can be denoted by $X_i$, where $i=1,2,3, \cdots, n$.

The sum of all numeric information (values of the variable $X_1, X_2, \cdots, X_n$) can be totaled by $X_1+X_2+\cdots+X_n$. The short and useful summation for the set of values is $\sum\limits_{i=1}^n X_i$, where the symbol $\Sigma$ is a Greek letter and denotes the sum of all values ranging from $i=1$ (start) to $n$ (last) value.

The number written on top of $\Sigma$ is called the upper limit (Upper Bound) of the sum, below $\Sigma$, there are two additional components: the index and the lower bound (lower limit). On the right of $\Sigma$, there is the sum term for all the indexes.

Summation Operator

Consider the following example for the use of summing values using the Summation operator.

\begin{align*}
X_1 + X_2 + X_3 + \cdots X_n &= \sum\limits_{i=1}^{n} X_i\\
X_1Y_1 + X_2Y_2 + X_3Y_3 + \cdots X_nY_n &= \sum\limits_{i=1}^{n} X_iY_i\\
X_1^2 + X_2^2 + \cdots + X_3^2 + \cdots X_n^2 &= \sum\limits_{i=1}^n X_i^2\\
(X_1 + X_2 + X_3 + \cdots X_n)^2 &= \left( \sum\limits_{i=1}^{n} X_i \right)^2
\end{align*}

The following examples make use of the summation operator, when a number (constant) and values of the variable are involved.

\begin{align}
a+a+a+ \cdots + a = na&=\sum\limits_{i=1}^{n}a\\
aX_1 + aX_2 + aX_3 \cdots + aX_n &= a \sum\limits_{i=1}^n X_i\\
(X_1-a)+(X_2-a)+\cdots + (X_n-a) &= \sum\limits_{i=1}^n (X_i-a)\\
(X_1-a)^2+(X_2-a)^2+\cdots + (X_n-a)^2 &= \sum\limits_{i=1}^n (X_i-a)^2\\
[(X_1-a)+(X_2-a)+\cdots + (X_n-a)]^2 &= \left[\sum\limits_{i=1}^n (X_i-a)\right]^2
\end{align}

Properties of Summation Operator

The summation operator is denoted by the $\Sigma$ symbol. It is a mathematical notation used to represent the sum of a collection of (data) values. The following useful properties for the manipulation of the sum operator are:

1) Multiplying a sum by a constant
$$c\sum\limits_{i=1}^n x_i = \sum\limits_{i=1}^n cx_i$$

2) Linearity: The summation operator is linear meaning that it satisfies the following properties for constant $a$ and $b$, and sequence $x_n$ and $y_n$.
$$\sum\limits_{i=1}^N(ax_i + by_i) = a \sum_{i=1}^N x_n + b\sum\limits_{i=1}^N y_i$$

3) Splitting a sum into two sums
$$\sum\limits_{i=a}^n x_i = \sum\limits_{i=a}^{c}x_i + \sum_{i=c+1}^n x_i$$

4) Combining Summations: Multiple summations can be combined into a single summation:
$$\sum\limits_{i=1}^b x_n + \sum\limits_{i=b+1}^c x_i = \sum\limits_{i=1}^c x_i$$

5) Changing the order of individual sums in multiple sum expressions
$$\sum\limits_{i=1}^{m} \sum\limits_{j=1}^{n} a_{ij} = \sum\limits_{j=1}^{n}\sum\limits_{i=1}^{m} a_{ij}$$

6) Distributivity over Scalar Multiplication: The summation operator distributes over scalar multiplication
$$c\sum\limits_{i=1}^b x_i = \sum_{i=1}^b (cx_i)$$

7) Adding or Subtracting Sums
$$\sum\limits_{i=1}^a x_i \pm \sum_{i=1}^a y_i = \sum\limits_{i=1}^a (x_i \pm y_i)$$

8) Multiplying the Sums:
$$\sum\limits_{i_1=a_1}^{n_1} x_{i_1} \times \cdots \times \sum\limits_{i_n=a_n}^{n_n} x_{i_n} = \sum\limits_{i_1=a_1}^{n_1} \times \cdots \times \sum\limits_{i_1=a_1}^{n_n}x_{i_1}\times \cdots \times x_{i_n}$$

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MCQ Level of Measurement 13 (2024)

The post is about the MCQ Level of measurement and covers the concepts related to statistical data and variables. The understanding of these important concepts helps in understanding the important aspects of data from different fields of study and their statistical analysis.

The quiz MCQ Level of Measurement is designed to test your knowledge of data and variables in statistics.

Online MCQs about Statistics Data and Variables with Answers.

1. The collection of observations for all variables related to some research or findings is classified as

2. A data analyst is working on a project about the global supply chain. They have a dataset with lots of relevant data from Europe and Asia. However, they decided to generate new data that represents all continents. What type of insufficient data does this scenario describe?

3. The data measurement which arises from a specific process of counting is classified as a

4. Temperature on a centigrade scale (no absolute zero point) is a measurement scale of:

5. The data in which we study Regions is called

6. Participants in an experiment are asked to wear headphones. Across a four-minute long interval, the experimenter presents audio clips of different instruments. Participants are asked to raise their hands every time they hear a new instrument. Their total score is the number of correct responses. The measurement scale is

7. Classifying elementary school children as nonreaders (0), starting readers (1), or advanced readers (2) to place each child in a reading group is

8. _____ is the process of changing data to make it more organized and easier to read.

9. A financial analyst imports a dataset to their computer from a storage device. As it’s being imported, the connection is interrupted, which compromises the data. Which of the following processes caused the compromise?

10. If a data analyst is using data that has been _____, the data will lack integrity and the analysis will be faulty.

11. Which of the following conditions are necessary to ensure data integrity?

12. A researcher asks a random sample of freshmen to describe how they feel about their first year at a university. Research assistants use predetermined criteria to assign categories to each description given: confident, Nervous, Fearful, and Insecure. What is the level of measurement used for described phenomena?

13. A measurement scale in which values are categorized to represent qualitative differences and ranked in a meaningful manner is classified as

14. Reporting the temperature of a summer day in the state of California in degrees Fahrenheit is a measurement scale of

15. What can jeopardize data integrity throughout its lifecycle?

16. Examples of variables in statistical phenomena consist

17. Measurement scale which allows the determination of differences in intervals is classified as

18. Which of the following conditions are necessary to ensure data integrity?

19. Number of students in a stats class

20. As a data analyst, you are working for a national pizza restaurant chain. You have a dataset with monthly order totals for each branch over the past year. With only this data, what questions can you answer?

In the subject of statistics, data is collected, organized, presented, and analyzed, and interpretation is made to make wise and intelligent decisions. The data is a collection of variables, whereas a variable is some kind of measure that can vary regarding time, person/object, place, etc. Let us start with the MCQ level of measurement quiz.

In statistics, data can be classified based on the level of measurement, which refers to the nature of the information captured by the data. There are four main levels of measurement: (i) nominal, (ii) ordinal, (ii) interval, and (iv) ratio.

Nominal Level

Characteristics: Categories or labels without any inherent order.
Examples: Gender (male, female), colors, types of fruits.

Ordinal Level

Characteristics: Categories with a meaningful order or rank, but the differences between the categories are not uniform.
Examples: Educational levels (high school, college, graduate), customer satisfaction ratings (poor, fair, good, excellent).

Interval Level

Characteristics: Categories with a meaningful order, and the differences between the categories are uniform, but there is no true zero point.
Examples: Temperature (measured in Celsius or Fahrenheit) and IQ scores.

Ratio Level

Characteristics: Categories with a meaningful order, uniform differences between categories, and a true zero point.
Examples: Height, weight, income, age.

Understanding the level of measurement is crucial because it determines the types of statistical analyses that can be performed on the data. Different statistical tests and methods are appropriate for each level, and using an inappropriate analysis may lead to incorrect conclusions.

In summary, nominal data involve categories without order, ordinal data have ordered categories with non-uniform differences, interval data have ordered categories with uniform differences but no true zero, and ratio data have ordered categories with uniform differences and a true zero point.

MCQ Level of Measurement

• Temperature on a centigrade scale (no absolute zero point) is a measurement scale of:
• Participants in an experiment are asked to wear headphones. Across a four-minute long interval, the experimenter presents audio clips of different instruments. Participants are asked to raise their hands every time they hear a new instrument. Their total score is the number of correct responses. The measurement scale is
• Reporting the temperature of a summer day in the state of California in degrees Fahrenheit is a measurement scale of
• Number of students in a stats class
• Classifying elementary school children as nonreaders (0), starting readers (1), or advanced readers (2) to place each child in a reading group is
• A researcher asks a random sample of freshmen to describe how they feel about their first year at a university. Research assistants use predetermined criteria to assign categories to each description given: confident, Nervous, Fearful, and Insecure. What is the level of measurement used for described phenomena?
• Which of the following conditions are necessary to ensure data integrity?
• ___________ is the process of changing data to make it more organized and easier to read.
• As a data analyst, you are working for a national pizza restaurant chain. You have a dataset with monthly order totals for each branch over the past year. With only this data, what questions can you answer?
• A data analyst is working on a project about the global supply chain. They have a dataset with lots of relevant data from Europe and Asia. However, they decided to generate new data that represents all continents. What type of insufficient data does this scenario describe?
• If a data analyst is using data that has been _________, the data will lack integrity and the analysis will be faulty.
• Which of the following conditions are necessary to ensure data integrity?
• A financial analyst imports a dataset to their computer from a storage device. As it’s being imported, the connection is interrupted, which compromises the data. Which of the following processes caused the compromise?
• What can jeopardize data integrity throughout its lifecycle?
• The data in which we study Regions is called
• A measurement scale in which values are categorized to represent qualitative differences and ranked in a meaningful manner is classified as
• The measurement scale which allows the determination of differences in intervals is classified as
• Data measurement which arises from a specific process of counting is classified as a
• The collection of observations for all variables related to some research or findings is classified as
• Examples of variables in statistical phenomena consist

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Important MCQs for Statistics with Answers

This quiz post contains Online MCQs for Statistics with answers covering variable and type of variable, Measures of central tendencies such as mean, median, mode, Weighted mean, data and type of data, sources of data, Measure of Dispersion/ Variation, Standard Deviation, Variance, Range, etc.

Online MCQs for Statistics with Answers

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PLEASE COMMENT below the MCQ with the CORRECT ANSWER and its DETAILED EXPLANATION.

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Basic statistics deals with the measure of central tendency (such as mean, median, mode, weighted mean, geometric mean, and Harmonic mean) and measures of dispersion (such as range, standard deviation, and variances).

Basic statistical methods include planning and designing the study, collecting data, arranging, and numerical and graphically summarizing the collected data.

Basic statistics are also used to perform statistical analysis to draw meaningful inferences. Basic statistics are used to extract useful information from the data. The extracted information may be useful for decision-making purposes.

A basic visual inspection of data using some graphical and numerical statistics may give some useful hidden information already available in the data. The graphical representation includes a bar chart, pie chart, dot chart, box plot, histogram, frequency polygon, scatter diagram, stem and leaf plot, cumulative frequency curve, and Pareto Chart, etc.

Companies related to finance, communication, manufacturing, charity organizations, government institutes, simple to large businesses, etc. are all examples that have a massive interest in collecting data and measuring different sorts of statistical findings. This helps them to learn from the past, notice the trends, and plan for the future.

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