Properties of Correlation Coefficient

The coefficient of correlation is a statistic used to measure the strength and direction of the linear relationship between two Quantitative variables.

Properties of Correlation Coefficient

Understanding these properties helps us to interpret the correlation coefficient accurately and avoid misinterpretations. The following are some important Properties of Correlation Coefficient.

  • The correlation coefficient ($r$) between $X$ and $Y$ is the same as the correlation between $Y$ and $X$. that is the correlation is symmetric with respect to $X$ and $Y$, i.e., $r_{XY} = r_{YX}$.
  • The $r$ ranges from $-1$ to $+1$, i.e., $-1\le r \le +1$.
  • There is no unit of $r$. The correlation coefficient $r$ is independent of the unit of measurement.
  • It is not affected by the change of origin and scale, i.e., $r_{XY}=r_{YX}$. If a constant is added to each value of a variable, it is called a change of origin and if each value of a variable is multiplied by a constant, it is called a change of scale.
  • The $r$ is the geometric mean of two regression coefficients, i.e., $\sqrt{b_{YX}\times b_{XY}}$.
    In other words, if the two regression lines of $Y$ on $X$ and $X$ on $Y$ are written as $Y=a+bX$ and $X=c+dy$ respectively then $bd=r^2$.
  • The sign of $r_{XY}, b_{YX}$, and $b_{XY}$ is dependent on covariance which is common in the three as given below:
  • $r=\frac{Cov(X, Y)}{\sqrt{Var(X) Var(Y)}},\,\, b_{YX} = \frac{Cov(Y, X)}{Var(X)}, \,\, b_{XY}=\frac{Cov(Y, X)}{Var(Y)}$

Hence, $r_{YX}, b_{YX}$, and $b_{XY}$ have the same sign.

  • If $r=-1$ the correlation is perfectly negative, meaning as one variable increases the other increases proportionally.
  • If $r=+1$ the correlation is perfectly positive, meaning as one variable increases the other decreases proportionally.
  • If $r=0$ there is no correlation, i.e., there is no linear relationship between the variables. However, a non-linear relationship may exist but it does not necessarily mean that the variables are independent.
Properties of Correlation Coefficient

Examples of Correlation Coefficient

The following are some real-life examples of correlation coefficients (ranging from -1 to +1) to illustrate relationships between variables:

Positive Correlation ($r$ close to +1)

  • The relationship between study time and exam scores: As students spend more time studying, their exam scores tend to increase. A correlation coefficient of $r = 0.85$ indicates a strong positive relationship.
  • The relationship between advertising spending and sales revenue: Companies that invest more in advertising often see higher sales. A correlation coefficient of $r = 0.70$ suggests a strong positive link.

Negative Correlation ($r$ close to -1)

  • The relationship between hours spent on social media and academic performance: As students spend more time on social media, their grades may decline. A correlation coefficient of $r = -0.65$ indicates a moderate negative relationship.
  • The relationship between temperature and heating costs: As outdoor temperatures rise, heating costs tend to decrease. A correlation coefficient of $r = -0.90$ shows a strong negative correlation.

Weak or No Correlation ($r$ close to 0)

  • The relationship between shoe size and IQ: There is no logical connection between shoe size and intelligence. A correlation coefficient of $r = 0.05$ indicates almost no correlation.
  • The relationship between rainfall and stock market performance: Rainfall has no direct impact on stock market trends. A correlation coefficient of $r = -0.10$ suggests a very weak or negligible relationship.

Real-World Applications

  • Healthcare: The correlation between exercise frequency and heart health.
  • Economics: The correlation between unemployment rates and crime rates.
  • Education: The correlation between parental income and children’s academic achievement.
  • Environment: The correlation between carbon emissions and global temperatures.

Independence of Origin and Scale

Theorem: Correlation: Independent of Origin and Scale. Show that the correlation coefficient is independent of origin and scale, i.e., $r_{XY}=r_{uv}$.

Proof: The formula for correlation coefficient is,

$$r_{XY}=\frac{\varSigma(X-\overline{X})((Y-\overline{Y})) }{\sqrt{[\varSigma(X-\overline{X})^2][\varSigma(Y-\overline{Y})^2]}}$$

\begin{align*}
\text{Let}\quad u&=\frac{X-a}{h}\\
\Rightarrow X&=a+hu \Rightarrow \overline{X}=a+h\overline{u} \\
\text{and}\quad v&=\frac{Y-b}{K}\\
\Rightarrow Y&=b+Kv \Rightarrow \overline{Y}=b+K\overline{v}\\
\text{Therefore}\\
r_{uv}&=\frac{\varSigma(u-\overline{u})((v-\overline{v})) }{\sqrt{[\varSigma(u-\overline{u})^2][\varSigma(v-\overline{v})^2]}}\\
&=\frac{\varSigma (a+hu-a-h\overline{u}) (b+Kv-b-K\overline{v})} {\sqrt{\varSigma(a+hu-a-h\overline{u})^2\varSigma(b+Kv-b-K\overline{v})^2}}\\
&=\frac{\varSigma(hu-h\overline{u})(Kv-K\overline{v})}{\sqrt{[\varSigma(hu-h\overline{u})^2][\varSigma(Kv-K\overline{v})^2]}}\\
&=\frac{hK\varSigma(u-\overline{u})(v-\overline{v})}{\sqrt{[h^2 K^2 \varSigma(u-\overline{u})^2] [\varSigma(v-\overline{v})^2]}}\\
&=\frac{hK\varSigma(u-\overline{u})(v-\overline{v})}{hK\,\sqrt{[\varSigma(u-\overline{u})^2] [\varSigma(v-\overline{v})^2]}}\\
&=\frac{\varSigma(u-\overline{u})(v-\overline{v}) }{\sqrt{[\varSigma(u-\overline{u})^2][\varSigma(v-\overline{v})^2]}}=
r_{uv}
\end{align*}

Correlation Coefficient Range

Important Points about Correlation Analysis

  1. Non-causality: Correlation does not imply causation. If two variables are strongly correlated, it does not necessarily mean that changes in one variable cause changes in the other. This is because the correlation only measures the strength and direction of the linear relationship between two quantitative variables, not the underlying cause-and-effect relationship.
  2. Sensitive to Outliers: The correlation coefficient can be sensitive to outliers, as outliers can disproportionately influence the correlation calculation.
  3. Assumption of Linearity: The correlation coefficient measures the linear relationship between variables. It may not accurately capture non-linear relationships between variables.
  4. Scale Invariance: The correlation coefficient is independent of the scale of the data. That is, multiplying or dividing all the values of one or both variables by a constant will not affect the strength and direction of the correlation coefficient. This makes it useful for comparing relationships between variables measured in different units.
  5. Strength vs. Causation: A high correlation does not necessarily imply causation. It is because two variables are strongly correlated does not mean one causes the other. There might be a third unknown factor influencing both variables. Correlation analysis is a good starting point for exploring relationships, but further investigation is needed to establish causality.
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Layout of the Factorial Design: Two Factor $2^2$ (2024)

The layout of a factorial design is typically organized in a table format. Each row of the table represents an experimental run, while each column represents a factor or the response variable. The levels of factors are indicated by symbols such as + and – for high and low levels, respectively. The response variable values corresponding to each experimental condition are recorded in the form of a sign table.

Consider a simple example layout for a two-factor factorial design with factors $A$ and $B$.

RunFactor AFactor BResponse
1$Y_1$
2+$Y_2$
3+$Y_3$
4++$Y_4$

Layout of the Factorial Design: Two Factor in $n$ Replicates

Consider there are two factors and each factor has two levels in $n$ replicates. The layout of the factorial design will be as described below for $n$ replicates.

Layout for the factorial design Two Factor Two Level

$y_{111}$ is the response from the first factor at the low level, the second factor at the low level, and the first replicate of the trial. Similarly, $y_{112}$ represents the second replicate of the same trial, and up to $n$th observation is $n$th trial at the same level of $A$ and $B$.

Geometrical Structure of Two-Factor Factorial Design

The geometrical structure of two factors (Factor $A$ and $B$), each factor has two levels, low ($-$) and high (+). Response 1 is at the low level of $A$ and a low level of $B$, similarly, response 2 is produced at a high level of $A$ and a low level of $B$. The third response is at a low level of $A$ and a high level of $B$, similarly, the 4th response is at a high level of $A$ and a high level of $B$.

Geometrical Structure of two Factor Layout of Factorial Experiment

Real Life Example

The concentration of reactant vs the amount of the catalyst produces some response, the experiment has three replicates.

Layout of Two Factors Real Life Example

Geometrical Structure of the Example

Layout of the Factorial Design: Two Factor $2^2$ (2024)

Factor Effects

\begin{align} A &=\frac{(a+ab)-((I) +b)}{2} = \frac{100+90-80-60}{2} = 25\\
B &= \frac{(b+ab) – ((I) +a) }{2} = \frac{60+90-80-100}{2} = -15\\
AB&=\frac{((I)+ab)-(a+b)}{2} = \frac{80+90-100-60}{2}=5
\end{align}

Minus 15 ($-15$) is the effect of $B$, which shows the change in factor level from low to high bringing on the average $-15$ decrease in the response.

Reference

Montgomery, D. C. (2017). Design and Analysis of Experiments. 9th ed, John Wiley & Sons.

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Test Preparation MCQs

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

5. 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?

 
 
 
 

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

 
 
 
 

7. Examples of variables in statistical phenomena consist

 
 
 
 

8. 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?

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

12. 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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

15. 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?

 
 
 
 

16. 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?

 
 
 
 

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

 
 
 
 

18. What can jeopardize data integrity throughout its lifecycle?

 
 
 
 

19. Number of students in a stats class

 
 
 
 

20. The data in which we study Regions is called

 
 
 
 

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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