📌 Part A – 1 Mark Questions (Very Short Answer)
- What is correlation?
Correlation means the relationship between two or more variables. - Define positive correlation.
When one variable increases, the other also increases. Both move in the same direction. - Define negative correlation.
When one variable increases, the other decreases. They move in opposite directions. - What is simple correlation?
Simple correlation means relationship between only two variables. - What is multiple correlation?
Multiple correlation means relationship between one variable and a number of other variables. - What is partial correlation?
Partial correlation means relationship between two variables, keeping the influence of other variables constant. - What is perfect correlation?
When change in one variable results in change in another variable at the same rate, it is perfect correlation. - What is imperfect correlation?
When change in one variable results in change in another variable at different rates, it is imperfect correlation. - What is linear correlation?
When change in one variable bears a constant ratio to the change in the other variable. - What is non-linear correlation?
When change in one variable does not bear a constant ratio to the change in the other variable. - Name the most widely used method to measure correlation.
Karl Pearson's Coefficient of Correlation. - What is the symbol used for Karl Pearson's coefficient?
The symbol is 'r'. - Between what values does 'r' always lie?
'r' always lies between +1 and -1. - What does r = +1 indicate?
Perfect positive correlation. - What does r = -1 indicate?
Perfect negative correlation. - What does r = 0 indicate?
No correlation between the variables. - If r is between 0 and +1, what type of correlation is it?
Imperfect positive correlation. - If r is between 0 and -1, what type of correlation is it?
Imperfect negative correlation. - Write the formula for Karl Pearson's coefficient of correlation.
r = Σxy / √(Σx² × Σy²) , where x = (X - X̄) and y = (Y - Ȳ). - Give an example of positive correlation from daily life.
As the number of hours studied increases, exam marks also increase. - Give an example of negative correlation from daily life.
As the price of a product increases, its demand decreases.
📌 Part B – 2/4 Marks Questions (Short Answer)
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Define correlation. Explain the different types of correlation based on direction.
Correlation means the relationship between two or more variables. When one variable changes, it causes a change in the other.
Based on direction, correlation is of two types:
- Positive correlation: Both variables move in the same direction. If X increases, Y also increases. Example: Height and weight.
- Negative correlation: Variables move in opposite directions. If X increases, Y decreases. Example: Price and demand.
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Distinguish between positive and negative correlation with examples.
- Positive correlation: Variables move together in the same direction. Example: More rainfall leads to more crop production.
- Negative correlation: Variables move in opposite directions. Example: More exercise leads to less body weight.
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Explain the types of correlation based on the number of variables.
- Simple correlation: Relationship between only two variables. Example: Study hours and marks.
- Multiple correlation: Relationship between one variable and several other variables. Example: Crop yield depends on rainfall, fertilizer, and sunlight.
- Partial correlation: Relationship between two variables, keeping the effect of other variables constant. Example: Studying the relation between price and demand, keeping income constant.
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What is perfect and imperfect correlation?
- Perfect correlation: When the change in one variable results in a proportional change in the other at a constant rate. In perfect correlation, all points lie on a straight line. r = +1 or r = -1.
- Imperfect correlation: When the change in one variable does not result in a constant rate of change in the other. The points are scattered. r is between +1 and -1, but not exactly +1 or -1.
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Explain linear and non-linear correlation.
- Linear correlation: The ratio of change between the variables is constant. If plotted on a graph, the points fall approximately on a straight line.
- Non-linear correlation: The ratio of change between the variables is not constant. If plotted, the points fall on a curve, not a straight line.
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Interpret the value of 'r' in the following cases: (a) r = -0.8 (b) r = 0 (c) r = +0.3
- r = -0.8: There is a strong negative correlation between the variables. If X increases, Y will decrease significantly.
- r = 0: There is no correlation between the variables. They are independent of each other.
- r = +0.3: There is a weak positive correlation between the variables. If X increases, Y also increases, but the relationship is not very strong.
📌 Part C – 6/8 Marks Questions (Long Answer)
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Define correlation. Explain its various types in detail with examples.
Correlation is a statistical measure that shows the relationship between two or more variables. It tells us whether the variables move together and in what direction.
The main types of correlation are:
1. Based on Direction:
- Positive Correlation: Variables move in the same direction. Example: Income and expenditure. As income rises, expenditure also rises.
- Negative Correlation: Variables move in opposite directions. Example: The number of absences and exam scores. As absences increase, scores decrease.
2. Based on the Number of Variables:
- Simple Correlation: Involves only two variables. Example: Price and supply of a product.
- Multiple Correlation: Involves one variable and several others. Example: The yield of a crop depends on rain, soil quality, and fertilizer.
- Partial Correlation: Measures the relationship between two variables while keeping other variables constant. Example: Studying the link between advertising and sales, while keeping the price constant.
3. Based on the Ratio of Change:
- Perfect Correlation: The change in the variables is at a constant rate. All points lie on a straight line. r = +1 or r = -1.
- Imperfect Correlation: The change is not at a constant rate. Points are scattered. r is between -1 and +1.
4. Based on the Nature of the Relationship:
- Linear Correlation: The relationship when plotted on a graph forms a straight line.
- Non-linear (Curvilinear) Correlation: The relationship when plotted forms a curve.
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Explain Karl Pearson's Coefficient of Correlation. Calculate it for the following data:
X Y 10 20 20 40 30 60 40 80 50 100 Karl Pearson's Coefficient (r) is the most common method to measure the degree of linear correlation between two variables. Its value always lies between -1 and +1.
Formula: r = Σxy / √(Σx² × Σy²), where x = (X - X̄) and y = (Y - Ȳ).
Step 1: Calculate the means.
- ΣX = 10+20+30+40+50 = 150. N = 5. X̄ = 150/5 = 30.
- ΣY = 20+40+60+80+100 = 300. Ȳ = 300/5 = 60.
Step 2: Calculate deviations and products.
X Y x = X-X̄ y = Y-Ȳ x² y² xy 10 20 -20 -40 400 1600 800 20 40 -10 -20 100 400 200 30 60 0 0 0 0 0 40 80 10 20 100 400 200 50 100 20 40 400 1600 800 Total Σx²=1000 Σy²=4000 Σxy=2000 Step 3: Apply the formula.
r = 2000 / √(1000 × 4000)
r = 2000 / √(4,000,000)
r = 2000 / 2000
r = +1Interpretation: There is a perfect positive correlation between X and Y. As X increases, Y increases in the same proportion.
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Calculate Karl Pearson's coefficient of correlation for the following paired data:
X Y 28 22 37 32 40 33 38 34 35 30 33 26 40 29 32 31 34 34 33 38 Step 1: Calculate the means.
- ΣX = 28+37+40+38+35+33+40+32+34+33 = 350. N = 10. X̄ = 350/10 = 35.
- ΣY = 22+32+33+34+30+26+29+31+34+38 = 310. Ȳ = 310/10 = 31.
Step 2: Calculate deviations and products.
X Y x = X-X̄ y = Y-Ȳ x² y² xy 28 22 -7 -9 49 81 +63 37 32 +2 +1 4 1 +2 40 33 +5 +2 25 4 +10 38 34 +3 +3 9 9 +9 35 30 0 -1 0 1 0 33 26 -2 -5 4 25 +10 40 29 +5 -2 25 4 -10 32 31 -3 0 9 0 0 34 34 -1 +3 1 9 -3 33 38 -2 +7 4 49 -14 Total Σx²=130 Σy²=166 Σxy=67 Step 3: Apply the formula.
r = Σxy / √(Σx² × Σy²)
r = 67 / √(130 × 166)
r = 67 / √21580
r = 67 / 146.9
r = 0.456Interpretation: There is a moderate positive correlation (r = 0.456) between X and Y. This means when X increases, Y also tends to increase, but the relationship is not very strong.