Statistics Chapter on Correlation
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Questions and Answers

What does a correlation coefficient value of r = -1 indicate?

  • Perfect negative correlation between the variables. (correct)
  • High positive correlation between the variables.
  • No relationship between the variables.
  • Moderate negative correlation between the variables.

What does perfectly positive correlation indicate when plotted on a scatter diagram?

  • Points exhibit a weak upward trend with considerable scatter.
  • All points lie on a straight line rising from the lower left to the upper right. (correct)
  • All points fall on a straight line declining from upper left to lower right.
  • All points are scattered randomly without any trend.

Which of the following explains the assumption of 'Cause & Effect Relationship' in Karl Pearson's Coefficient of Correlation?

  • No correlation can exist without a causal relationship. (correct)
  • Graphs of two unrelated variables will produce a straight line.
  • Correlation is possible even without a causal link.
  • Two independent variables always show correlation.

What is a key limitation of using Karl Pearson's Coefficient of Correlation?

<p>Only applicable to linear relationships. (A)</p> Signup and view all the answers

Which scenario demonstrates a high degree of negative correlation?

<p>Plotted points falling in a narrow band from upper left to lower right. (A)</p> Signup and view all the answers

What range of values can the coefficient of correlation (r) take?

<p>Between -1 and +1. (C)</p> Signup and view all the answers

In a scatter diagram, if the points are widely scattered with a rising tendency, what does this suggest?

<p>A low degree of positive correlation. (C)</p> Signup and view all the answers

In terms of correlation, what does an r value between +0.50 and +0.75 represent?

<p>High positive correlation. (C)</p> Signup and view all the answers

What does a correlation coefficient (r) value of 0 signify?

<p>No correlation between the variables. (B)</p> Signup and view all the answers

Which method is characterized as being simple and non-mathematical in studying the correlation between variables?

<p>Scatter diagram. (C)</p> Signup and view all the answers

Which statement is true regarding the computation of correlation coefficients?

<p>The calculation can be time consuming compared to other methods. (D)</p> Signup and view all the answers

What does a correlation coefficient value of r = 0 suggest?

<p>No relationship between the variables. (C)</p> Signup and view all the answers

If the points on a scatter diagram are clustered closely around a straight line falling from upper left to lower right, what does this indicate?

<p>High degree of negative correlation. (A)</p> Signup and view all the answers

Which of the following best describes the preparation of a scatter diagram?

<p>Each pair of values is plotted as a single dot. (B)</p> Signup and view all the answers

Which of the following is NOT a merit of Karl Pearson's Coefficient of Correlation?

<p>It guarantees a causal relationship. (C)</p> Signup and view all the answers

What is a key advantage of using scatter diagrams?

<p>They provide a visual representation of correlations. (A)</p> Signup and view all the answers

What does a positive correlation between two variables indicate?

<p>Both variables either increase or decrease together (C)</p> Signup and view all the answers

Which of the following best defines correlation analysis?

<p>It measures the degree and direction of relationships between variables (C)</p> Signup and view all the answers

What is the effect of correlation analysis on prediction reliability?

<p>It increases reliability by reducing uncertainty (C)</p> Signup and view all the answers

Which of the following is NOT an example of correlation?

<p>Relationship between shoe size and personality traits (D)</p> Signup and view all the answers

What is the term used to measure correlation?

<p>Coefficient of correlation (A)</p> Signup and view all the answers

Which factor does correlation analysis NOT help to understand?

<p>Absolute accuracy in predictions (B)</p> Signup and view all the answers

In which scenario would a negative correlation be observed?

<p>Rising unemployment associated with declining economic growth (D)</p> Signup and view all the answers

What is another term often used for correlation?

<p>Covariation (D)</p> Signup and view all the answers

What does the formula $r_c = ± ext{sqrt} rac{2c - n}{n}$ calculate?

<p>Concurrent deviation (B)</p> Signup and view all the answers

Which of the following best describes the concept of 'lag' in correlation?

<p>A time gap before a cause and effect relationship is established (C)</p> Signup and view all the answers

When adjusting pairs of items for correlation analysis, which method is used to account for delays in impact?

<p>Time lag adjustment (C)</p> Signup and view all the answers

What must be calculated to find the coefficient of correlation of trend values for long-term changes?

<p>Trend values (D)</p> Signup and view all the answers

Which of the following is a step in calculating short-term fluctuations?

<p>Deducting trend values from actual values (A)</p> Signup and view all the answers

What does the 'x' symbolize in the calculation of short-term fluctuations for the X series?

<p>Short-term fluctuations (B)</p> Signup and view all the answers

In the context of correlation in time series, why is it important to differentiate between long-term and short-term changes?

<p>They can show opposite correlation patterns (C)</p> Signup and view all the answers

What is the first step involved in calculating the correlation of long-term changes?

<p>Calculate trend values by moving average (B)</p> Signup and view all the answers

What does the Rank Correlation Coefficient measure?

<p>The correlation between two series of ranks (B)</p> Signup and view all the answers

What is indicated by an R value of -1 in Spearman's correlation?

<p>Perfect negative correlation (A)</p> Signup and view all the answers

In the formula for Rank Correlation Coefficient, what does the term $ΣD²$ represent?

<p>The sum of the differences of the ranks squared (A)</p> Signup and view all the answers

When tied ranks are present, how is the contribution to $ΣD²$ calculated?

<p>By adding the factor $ rac{m³-m}{12}$ for each tied item (B)</p> Signup and view all the answers

Which of the following is NOT a feature of Spearman's correlation coefficient?

<p>It requires normally distributed data (D)</p> Signup and view all the answers

What is one significant limitation of Spearman's Rank Correlation Coefficient?

<p>It is less powerful than parametric methods like Pearson's (C)</p> Signup and view all the answers

Which statement best describes the application of the Rank Method?

<p>It can be used for qualitative data that can only be ranked (D)</p> Signup and view all the answers

What is the significance of $ rac{1}{12}(m³-m)$ in the modified formula for rank correlation coefficient?

<p>It represents the adjustment for tied ranks (A)</p> Signup and view all the answers

What is the first step in calculating Karl Pearson's Coefficient of Correlation using actual mean deviations?

<p>Calculate the deviations from the actual mean of X series (D)</p> Signup and view all the answers

Which formula represents Karl Pearson's Coefficient of Correlation when deviations are taken from the actual mean?

<p>$r = rac{ rac{ orall xy}{ orall x} - rac{ orall y}{N}}{ orall x²}$ (B)</p> Signup and view all the answers

What must occur after squaring the deviations for both X and Y series when calculating correlation?

<p>Multiply the deviations of X series by the respective deviations of Y series (B)</p> Signup and view all the answers

Which operation is NOT involved in calculating the correlation coefficient with deviations from the assumed mean?

<p>Calculate deviations from the actual mean (C)</p> Signup and view all the answers

What does the term Σxy represent in the calculation of Karl Pearson's Coefficient of Correlation?

<p>Total of the product of deviations for X and Y series (B)</p> Signup and view all the answers

In the context of correlation coefficients, what do the variables N, $ar{X}$, and $ar{Y}$ indicate?

<p>N is the total number of data points; $ar{X}$ and $ar{Y}$ are means of X and Y respectively. (C)</p> Signup and view all the answers

How is the Coefficient of Correlation interpreted after calculations?

<p>It varies from -1 to 1, indicating the strength and direction of a linear relationship. (A)</p> Signup and view all the answers

What calculation is involved right after obtaining the total of squared deviations of Y series?

<p>Multiply the total of values in Y series with total values of X series (D)</p> Signup and view all the answers

Flashcards

Correlation

The relationship between two or more variables where a change in one variable causes a corresponding change in the other.

Correlation Analysis

A statistical technique that measures the strength and direction of the relationship between variables.

Positive Correlation

When two variables move in the same direction - if one increases, the other also increases; if one decreases, the other also decreases.

Negative Correlation

When two variables move in opposite directions - if one increases, the other decreases; if one decreases, the other increases.

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Coefficient of Correlation

A measure that quantifies the strength and direction of the relationship between two variables.

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Predictive Power of Correlation

Using correlation analysis to predict future outcomes by identifying the relationship between variables.

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Reducing Uncertainty

Correlation helps reduce uncertainty in predictions by providing a basis to understand the relationship between variables.

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Economic Applications of Correlation

Correlation analysis is used to study economic behavior by understanding the relationships between economic variables.

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Correlation coefficient (r) = +1

Indicates a perfect positive relationship between variables, meaning they increase together proportionally.

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Correlation coefficient (r) = -1

Indicates a perfect negative relationship between variables, meaning one increases as the other decreases proportionally.

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Correlation coefficient (r) = 0

Indicates no relationship between variables. Changes in one variable have no impact on the other.

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Linear Relationship Assumption

There is a clear linear relationship between the variables when plotted on a scatter diagram. It means the plotted points will form a straight line.

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Cause & Effect Relationship Assumption

The correlation is meaningless if there is no cause-and-effect relationship between the variables.

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Normality Assumption

The variables are affected by a large number of independent causes, resulting in a normal distribution.

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Merit of Pearson's Correlation

It provides both the direction and strength of the relationship between variables.

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Lag

The time gap between a cause and its effect.

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Scatter Diagram

A visual representation showing the relationship between two variables. Each point on the diagram represents a pair of values, and the overall pattern of the points reveals the degree and direction of their correlation.

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Concurrent Deviation

The simultaneous occurrence of two things that are related.

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Correlation in Time Series

A calculation that determines the correlation between two variables over time, taking into account any lag between them.

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Trend Values

A method to analyze long-term trends by representing the data with a smooth line or curve.

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Perfect Correlation

When all the points on a scatter diagram lie perfectly on a straight line, indicating a strong and direct relationship between the variables.

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High Correlation

When the points on a scatter diagram are close to a straight line, indicating a strong relationship between the variables.

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Short-Term Fluctuations

The difference between the actual value of a data point and its trend value.

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Low Correlation

When the points on a scatter diagram are scattered widely, indicating a weak relationship between variables.

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Correlation of Short-Term Changes

A method to analyze short-term changes in a time series by evaluating the deviations from the trend line.

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Adjusting Pairs of Items

The process of adjusting the timing of data points to account for any lag between cause and effect.

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No Correlation

When the points on a scatter diagram show no pattern, implying no relationship between the variables.

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Scatter Diagram Method

A method of studying the correlation between two variables by plotting them on a scatter diagram.

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Correlation Coefficient

The extent to which two variables change together, measured by a number between -1 and 1, where -1 indicates a perfect negative correlation, 0 indicates no correlation, and 1 indicates a perfect positive correlation.

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Karl Pearson's Coefficient of Correlation

A measure of how strongly two variables are related, often used to determine the strength of a linear relationship.

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Actual Mean

The average of a set of values, calculated by summing all values and dividing by the number of values.

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Taking Deviations From the Assumed Mean

The process of using a specific value (other than the actual mean) to simplify calculations, and later adjusting the result to account for the assumption.

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Deviation from Assumed Mean

The difference between each individual value and the assumed mean used in calculations.

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Deviation from Actual Mean

The difference between each value and the actual mean of the dataset.

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Correlation Coefficient Formula (Actual Mean)

The formula used to calculate Karl Pearson's Coefficient of Correlation when deviations are taken from the actual mean.

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Correlation Coefficient Formula (Assumed Mean)

The formula used to calculate Karl Pearson's Coefficient of Correlation when deviations are taken from the assumed mean.

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Rank Correlation Coefficient

The measure of the association between two ranked sets of data.

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Formula for Rank Correlation Coefficient

The formula used to calculate the Rank Correlation Coefficient, where 'D' is the difference in ranks for each pair and 'N' is the number of pairs.

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Tied Ranks Adjustment

The adjustment made to the Rank Correlation Coefficient formula when there are tied ranks (identical values) in a data set.

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Spearman's Rank Correlation Coefficient

A statistical measure that describes the strength and direction of the linear relationship between two variables.

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Distribution-Free

Spearman's Rank Correlation Coefficient isn't affected by the shape of the distribution of the data.

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Simplicity of Spearman's Rank Correlation Coefficient

Spearman's Rank Correlation Coefficient is simple to understand and apply, making it a user-friendly tool.

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Suitable for Qualitative Data

Spearman's Rank Correlation Coefficient is also suitable for analyzing qualitative data that can be ranked but not easily quantified.

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Suitable for Abnormal Data

Spearman's Rank Correlation Coefficient is also applicable when data is abnormal due to its reliance on ranks rather than raw data.

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Study Notes

Correlation

  • Correlation measures the relationship between two or more variables.
  • If a change in one variable leads to a corresponding change in another, the variables are correlated.

Examples of Relationships

  • Height and weight
  • Rainfall and wheat yield
  • Commodity price and demand
  • Husband's age and wife's age
  • Insulin dose and blood sugar
  • Advertising expenditure and sales

Correlation Analysis

  • A statistical technique to measure the degree and direction of the relationship between variables.
  • Used in economics and business to understand connections between variables like price and quantity demanded.
  • Aims to reduce prediction uncertainty.

Types of Correlation

  • Positive Correlation: Both variables move in the same direction.
    • If one increases, the other increases; if one decreases, the other decreases.
  • Negative Correlation: Variables move in opposite directions.
    • If one increases, the other decreases; if one decreases, the other increases.
  • Simple Correlation: Examines the relationship between two variables.
  • Multiple Correlation: Examines the relationship between three or more variables, considering the effect of other variables as constant.
  • Partial Multiple Correlation: Studies three or more variables, focusing on the relationship between two variables while holding others constant.
  • Total Multiple Correlation: Studies three or more variables without excluding the effect of any variable.
  • Linear Correlation: Variables change at a constant ratio. Data points form a straight line when plotted.
  • Non-linear (Curvilinear) Correlation: Variables do not change at a constant ratio. Data points form a curve when plotted.

Scatter Diagram

  • A graphical representation of bivariate data.
  • Used to visualize the degree and direction of correlation.
  • Plotting data points helps determine relationships and pattern.
    • Perfect positive correlation (r = +1): Points form a straight line going from the lower left to the upper right
    • Perfect negative correlation (r = −1): Points form a straight line from upper left to lower right
    • Positive correlation (0 < r < +1): Points generally rise from left to right
    • Negative correlation ( −1 < r < 0): Points generally fall from left to right
    • No correlation (r = 0): Points are scattered across the graph
    • Low correlation: A wide scatter or curve

Covariance

  • A measure of the relationship between two variables.
  • Independent of choice of origin (shifting the data).
  • Independent of choice of scale (multiplying or dividing the data).
  • Can take values from negative infinity to positive infinity (e.g. ∞ to +∞).
  • Doesn't give meaningful value of strength of relationship between variable.

Karl Pearson's Coefficient of Correlation (r)

  • A measure used to quantify the strength and direction of a linear relationship between two variables.
  • Values range from -1 to +1. A value of -1 indicates a perfect negative correlation and +1 indicates perfect positive correlation

Coefficient of Determination (r²)

  • Represents the proportion of variation in the dependent variable that is explained by the independent variable.
  • A value of .64 means 64% of variation in one variable is explained by the other.

Coefficient of Non-determination (K²)

  • Represents the proportion of variation in the dependent variable that is not explained by the independent variable.

Spearman's Rank Correlation (ρ)

  • Measures the correlation between ranked data.
  • Useful for non-linear relationships and non-normal distributions.

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Correlation Explained PDF

Description

This quiz explores the concept of correlation and its significance in statistical analysis. It covers positive and negative correlations, measurement techniques, and real-life examples of correlated variables. Test your understanding of how correlation informs predictions in various fields such as economics and business.

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