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 $X_n=53$.
Summation Operator

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.

Summation Operator

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 a\\
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) &= \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}\\
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
$$c\sum\limits_{i=L}^Ux_i = \sum\limits_{i=L}Y cx_i$$

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
$$\sum\limits_{i=a}^n x_i = \sum\limits_{i=a}^{c}x_i + \sum_{i=c+1}^b x_i$$

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:
$$\sum\limits_{i_1=a_1}^{n_1} x_{i_1} \times \cdots \times \sum\limits_{i_n=a_n}^{n_n} x_{i_n} = \sum\limits_{i_1=a_1}^{n_1} \times \cdots \sum\limits_{i_1=a_1}^{n_n}x_{i_1}\times \cdots \times x_{i_n}$$
x_{i,j}$$

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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. Examples of variables in statistical phenomena consist

 
 
 
 

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

 
 
 
 

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

 
 
 
 

4. 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?

 
 
 
 

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

 
 
 
 

6. Number of students in a stats class

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

12. 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?

 
 
 
 

13. 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?

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

17. 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

 
 
 
 

18. The data in which we study Regions is called

 
 
 
 

19. What can jeopardize data integrity throughout its lifecycle?

 
 
 
 

20. 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?

 
 
 
 


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.

Level of Measurement

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

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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MCQs for Statistics with Answers

This quiz post contains Online MCQs for Statistics with answers covering variable and type of variable, Measure 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

If you found that any POSTED MCQ is/ are WRONG
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 tendencies (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.

MCQs for Statistics with Answers

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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MCQs Basic Statistics 12

This quiz contains MCQs Basic Statistics with answers covering variable and type of variable, Measure of central tendency such as mean, median, mode, Weighted mean, data and type of data, sources of data, Measure of Dispersion/ Variation, Standard Deviation, Variance, Range, etc. Let us start the MCQs Basic Statistics Quiz.

Please go to MCQs Basic Statistics 12 to view the test

If you found that any POSTED MCQ is/ are WRONG
PLEASE COMMENT below the MCQ with the CORRECT ANSWER and its DETAILED EXPLANATION.

Don’t forget to mention the MCQs Statement (or Screenshot), because MCQs and their answers are generated randomly

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

The 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.

Mcqs Basic Statistics

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, 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, noticing the trends, and planning for the future.

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