CorrelationClass 11 Statistics For Economics NCERT Solutions
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Q1EXERCISES
The unit of correlation coefficient between height in feet and weight in kgs is
(i)
kg/feet
(ii)
percentage
(iii)
non-existent
Solution
The correct answer is (iii) non-existent.
As explained in the chapter, the correlation coefficient (r) is a pure number and has no unit of measurement. It measures the degree and direction of the linear relationship between two variables, regardless of the units in which the variables themselves are measured.
Q2EXERCISES
The range of simple correlation coefficient is
(i)
0 to infinity
(ii)
minus one to plus one
(iii)
minus infinity to infinity
Solution
The correct answer is (ii) minus one to plus one.
The chapter states that one of the properties of the correlation coefficient is that its value always lies between -1 and +1, inclusive. This can be represented as -1 ≤ r ≤ 1.
Q3EXERCISES
If r_xy is positive the relation between X and Y is of the type
(i)
When Y increases X increases
(ii)
When Y decreases X increases
(iii)
When Y increases X does not change
Solution
The correct answer is (i) When Y increases X increases.
A positive correlation coefficient (r > 0) indicates that the two variables move in the same direction. This means that an increase in one variable is associated with an increase in the other, and a decrease in one is associated with a decrease in the other.
Q4EXERCISES
If r_xy = 0 the variable X and Y are
(i)
linearly related
(ii)
not linearly related
(iii)
independent
Solution
The correct answer is (ii) not linearly related.
As stated in the chapter, if r = 0, it means the two variables are uncorrelated, which signifies the absence of a linear relationship between them. However, it does not rule out the possibility of a non-linear relationship, so they are not necessarily independent.
Q5EXERCISES
Of the following three measures which can measure any type of relationship
(i)
Karl Pearson's coefficient of correlation
(ii)
Spearman's rank correlation
(iii)
Scatter diagram
Solution
The correct answer is (iii) Scatter diagram.
The chapter explains that a scatter diagram is a visual tool that can present the form of any relationship, whether it is linear or non-linear. In contrast, both Karl Pearson's coefficient and Spearman's rank correlation are designed to measure the strength and direction of a linear association between variables.
Q6EXERCISES
If precisely measured data are available the simple correlation coefficient is
(i)
more accurate than rank correlation coefficient
(ii)
less accurate than rank correlation coefficient
(iii)
as accurate as the rank correlation coefficient
Solution
The correct answer is (i) more accurate than rank correlation coefficient.
The chapter suggests that Spearman's rank correlation is used when precise measurements are not possible (e.g., for attributes like honesty) or when data contains extreme values. The simple (Karl Pearson's) correlation coefficient utilizes the actual values of the data, not just their ranks. Therefore, when precise quantitative data is available without extreme outliers, the simple correlation coefficient is considered a more accurate measure because it uses all the available information.
Q7EXERCISES
Why is r preferred to covariance as a measure of association?
Solution
The correlation coefficient (r) is preferred to covariance as a measure of association for two main reasons:
-
Standardized Value: The value of 'r' is standardized and lies within a fixed range of -1 to +1. This makes it easy to interpret the strength of the relationship. A value close to 1 or -1 indicates a strong linear relationship, while a value close to 0 indicates a weak one. Covariance, on the other hand, is not standardized and its value can range from negative infinity to positive infinity, making it difficult to judge the strength of the relationship from its magnitude alone.
-
Unit-Free Measure: The correlation coefficient 'r' is a pure number and has no units. This allows for the comparison of relationships between different pairs of variables measured in different units (e.g., comparing the relationship between height and weight with the relationship between income and consumption). Covariance, however, is dependent on the units of the variables, making such comparisons difficult.
Q8EXERCISES
Can r lie outside the -1 and 1 range depending on the type of data?
Solution
No, the correlation coefficient (r) cannot lie outside the -1 and 1 range, regardless of the type of data. The chapter explicitly states as a property of the correlation coefficient: "The value of the correlation coefficient lies between minus one and plus one, -1 ≤ r ≤ 1. If, in any exercise, the value of r is outside this range it indicates error in calculation."
Q9EXERCISES
Does correlation imply causation?
Solution
No, correlation does not imply causation. The chapter strongly emphasizes this point, stating, "Correlation measures covariation, not causation. Correlation should never be interpreted as implying cause and effect relation." The presence of a correlation between two variables simply means that they tend to move together in a predictable way. This relationship could be due to a cause-and-effect link, a coincidence, or the influence of a third, unobserved variable. For example, the text mentions that a high correlation between ice-cream sales and drowning deaths does not mean one causes the other; rather, a third variable (hot weather) causes an increase in both.
Q10EXERCISES
When is rank correlation more precise than simple correlation coefficient?
Solution
The term "more precise" can be interpreted as more appropriate or reliable in certain situations. According to the chapter, Spearman's rank correlation coefficient is preferred over the simple (Karl Pearson's) correlation coefficient in the following cases:
- Qualitative Data: When dealing with attributes that cannot be measured numerically but can be ranked, such as honesty, beauty, or intelligence.
- Extreme Values (Outliers): If the dataset contains extreme values, the simple correlation coefficient can be heavily distorted. Rank correlation is not affected by outliers because it only considers the relative ordering of the data points.
- Non-linear Relationship: When the relationship between variables is clearly non-linear but monotonic (consistently increasing or decreasing), rank correlation can provide a better measure of the association than the simple correlation coefficient, which is designed specifically for linear relationships.
Q11EXERCISES
Does zero correlation mean independence?
Solution
No, a zero correlation does not necessarily mean the variables are independent. A correlation coefficient (r) of zero specifically indicates that there is no linear relationship between the two variables. However, as the chapter mentions, there might still be a strong non-linear relationship. For example, a U-shaped or inverted U-shaped relationship (like in Exercise 19, where Y = X²) can yield a correlation coefficient of zero, even though the variables are clearly dependent on each other.
Q12EXERCISES
Can simple correlation coefficient measure any type of relationship?
Solution
No, the simple (Karl Pearson's) correlation coefficient cannot measure any type of relationship. It is designed specifically to measure the strength and direction of only a linear relationship between two variables. The chapter advises that it is important to first examine a scatter diagram to see if the relationship appears linear before calculating Karl Pearson's coefficient. If the relationship is non-linear (curved), the coefficient can be misleading and may not accurately represent the strength of the association.
Q13EXERCISES
Collect the price of five vegetables from your local market every day for a week. Calculate their correlation coefficients. Interpret the result.
Solution
This is a practical activity. The steps to complete it are as follows:
-
Data Collection: Visit a local market for seven consecutive days. Each day, record the price per kg of five different vegetables (e.g., tomatoes, potatoes, onions, carrots, bell peppers).
-
Data Tabulation: Create a table with days of the week and columns for the prices of each vegetable.
-
Calculation: Choose pairs of vegetables to analyze (e.g., tomatoes and potatoes, potatoes and onions). For each pair, calculate the Karl Pearson's coefficient of correlation (r) using the formula: Where X is the price of the first vegetable, Y is the price of the second vegetable, and N is the number of days (7).
-
Interpretation of Results:
- Positive Correlation (r > 0): If the prices of two vegetables (e.g., two seasonal vegetables) tend to rise and fall together, you might find a positive correlation. This could be due to common factors like weather, transportation costs, or overall market demand.
- Negative Correlation (r < 0): If the vegetables are substitutes for each other (e.g., if the price of one goes up, people buy the other, causing its price to rise from demand while the first one's demand falls), you might observe a weak negative correlation. However, prices are often influenced by supply-side factors, so this is less common.
- No Correlation (r ≈ 0): If the prices of two vegetables move independently of each other, the correlation coefficient will be close to zero. For example, the price of potatoes (a staple) and bell peppers (less of a staple) might not be strongly related.
Q14EXERCISES
Measure the height of your classmates. Ask them the height of their benchmate. Calculate the correlation coefficient of these two variables. Interpret the result.
Solution
This is a practical activity involving data collection and analysis.
-
Data Collection: Create two lists. List X will be the height of each student in your class. List Y will be the height of their corresponding benchmate.
-
Tabulation: Create a table with two columns, one for student height (X) and one for their benchmate's height (Y).
-
Calculation: Use the Karl Pearson's correlation coefficient formula to calculate 'r' for the paired data (X, Y).
-
Interpretation:
- It is most likely that the correlation coefficient will be close to zero (r ≈ 0). This is because seating arrangements in a classroom are usually random with respect to height. There is no systematic reason why a tall person would be paired with another tall person, or a short person with another short person.
- A result close to zero would indicate that there is no linear relationship between the height of a student and the height of their benchmate. The variables are likely independent.
Q15EXERCISES
List some variables where accurate measurement is difficult.
Solution
The chapter provides several examples of variables, particularly attributes, where accurate numerical measurement is difficult or impossible. These variables are better suited for ranking. Some examples include:
- Honesty: This is a personal trait that cannot be assigned a precise numerical value.
- Beauty: Perceptions of beauty are subjective and vary between individuals and cultures, making a standard measurement impossible.
- Intelligence: While IQ tests exist, they are often debated, and intelligence is a complex concept that is difficult to capture in a single number.
- Physical Appearance: Similar to beauty, this is a subjective quality.
- Fairness: This is a moral or ethical quality that cannot be quantified.
Q16EXERCISES
Interpret the values of r as 1, -1 and 0.
Solution
The values of the correlation coefficient (r) are interpreted as follows:
-
r = 1: This indicates a perfect positive linear correlation. It means that the two variables have an exact linear relationship, and they move in the same direction. When one variable increases, the other increases by a perfectly proportional amount. On a scatter diagram, all the points would lie exactly on a straight line that slopes upwards.
-
r = -1: This indicates a perfect negative linear correlation. It means the two variables have an exact linear relationship, but they move in opposite directions. When one variable increases, the other decreases by a perfectly proportional amount. On a scatter diagram, all the points would lie exactly on a straight line that slopes downwards.
-
r = 0: This indicates no linear correlation. It means there is a complete absence of a straight-line relationship between the two variables. The variables are said to be linearly uncorrelated. However, it is important to remember that some form of non-linear relationship might still exist between them.
Q17EXERCISES
Why does rank correlation coefficient differ from Pearsonian correlation coefficient?
Solution
Spearman's rank correlation coefficient differs from Pearsonian (simple) correlation coefficient primarily because they use different inputs from the data:
-
Use of Ranks vs. Actual Values: Pearson's coefficient is calculated using the actual numerical values of the variables. In contrast, Spearman's coefficient is calculated using the ranks assigned to the values of the variables, not the values themselves. It measures the linear association between the ranks.
-
Sensitivity to Information: Because Pearson's method uses the actual values, it utilizes all the information concerning the data, including the magnitude of the differences between values. Spearman's method discards this information and only considers the ordinal position (rank) of each data point. As the chapter states, "This is due the fact that all the information concerning the data is not utilised."
-
Effect of Outliers: Pearson's coefficient is sensitive to extreme values (outliers), which can significantly affect the result. Spearman's coefficient is much less affected by outliers because an extreme value will still have a rank (e.g., 1st or last) and its large magnitude does not disproportionately influence the calculation.
Because of these differences, the two coefficients will generally give different results for the same dataset, unless the first differences of the values of items, when arranged in order, are constant.
Q18EXERCISES
Calculate the correlation coefficient between the heights of fathers in inches (X) and their sons (Y) X 65 66 57 67 68 69 70 72 Y 67 56 65 68 72 72 69 71 (Ans. r = 0.603)
Solution
To calculate the Karl Pearson's correlation coefficient (r), we will construct a table to find the necessary sums.
| X | Y | X² | Y² | XY |
|---|---|---|---|---|
| 65 | 67 | 4225 | 4489 | 4355 |
| 66 | 56 | 4356 | 3136 | 3696 |
| 57 | 65 | 3249 | 4225 | 3705 |
| 67 | 68 | 4489 | 4624 | 4556 |
| 68 | 72 | 4624 | 5184 | 4896 |
| 69 | 72 | 4761 | 5184 | 4968 |
| 70 | 69 | 4900 | 4761 | 4830 |
| 72 | 71 | 5184 | 5041 | 5112 |
| ΣX=534 | ΣY=540 | ΣX²=35788 | ΣY²=36644 | ΣXY=36118 |
Here, N (number of pairs) = 8.
We use the formula:
-
Calculate the numerator: NΣXY - (ΣX)(ΣY) = 8 * 36118 - (534 * 540) = 288944 - 288360 = 584
-
Calculate the first part of the denominator: NΣX² - (ΣX)² = 8 * 35788 - (534)² = 286304 - 285156 = 1148
-
Calculate the second part of the denominator: NΣY² - (ΣY)² = 8 * 36644 - (540)² = 293152 - 291600 = 1552
-
Calculate r:
Comment on the result:
The calculated correlation coefficient is approximately r = 0.44. This indicates a moderate positive linear relationship between the heights of fathers and their sons. It suggests that taller fathers tend to have taller sons.
(Note: The calculated value r = 0.44 differs from the answer r = 0.603 provided in the textbook. The calculation above is correct based on the given data. The discrepancy may be due to a typographical error in the question's data or the provided answer in the source.)
Q19EXERCISES
Calculate the correlation coefficient between X and Y and comment on their relationship: X -3 -2 -1 1 2 3 Y 9 4 1 1 4 9 (Ans. r = 0)
Solution
To calculate the correlation coefficient, we first find the required sums.
| X | Y | X² | Y² | XY |
|---|---|---|---|---|
| -3 | 9 | 9 | 81 | -27 |
| -2 | 4 | 4 | 16 | -8 |
| -1 | 1 | 1 | 1 | -1 |
| 1 | 1 | 1 | 1 | 1 |
| 2 | 4 | 4 | 16 | 8 |
| 3 | 9 | 9 | 81 | 27 |
| ΣX=0 | ΣY=28 | ΣX²=28 | ΣY²=196 | ΣXY=0 |
Here, N (number of pairs) = 6.
Using the formula:
- Calculate the numerator: NΣXY - (ΣX)(ΣY) = 6 * 0 - (0 * 28) = 0 - 0 = 0
Since the numerator is 0, the correlation coefficient 'r' will also be 0.
Comment on their relationship:
The correlation coefficient is r = 0. This indicates that there is no linear relationship between variables X and Y. However, observing the data (Y = X²), it is clear that there is a perfect non-linear (parabolic or U-shaped) relationship between them. This example demonstrates that a zero correlation coefficient does not mean the variables are independent; it only means they are not linearly related.
Q20EXERCISES
Calculate the correlation coefficient between X and Y and comment on their relationship X 1 3 4 5 7 8 Y 2 6 8 10 14 16 (Ans. r = 1)
Solution
To calculate the correlation coefficient, we construct a table for the required sums.
| X | Y | X² | Y² | XY |
|---|---|---|---|---|
| 1 | 2 | 1 | 4 | 2 |
| 3 | 6 | 9 | 36 | 18 |
| 4 | 8 | 16 | 64 | 32 |
| 5 | 10 | 25 | 100 | 50 |
| 7 | 14 | 49 | 196 | 98 |
| 8 | 16 | 64 | 256 | 128 |
| ΣX=28 | ΣY=56 | ΣX²=164 | ΣY²=656 | ΣXY=328 |
Here, N (number of pairs) = 6.
Using the formula:
-
Calculate the numerator: NΣXY - (ΣX)(ΣY) = 6 * 328 - (28 * 56) = 1968 - 1568 = 400
-
Calculate the first part of the denominator: NΣX² - (ΣX)² = 6 * 164 - (28)² = 984 - 784 = 200
-
Calculate the second part of the denominator: NΣY² - (ΣY)² = 6 * 656 - (56)² = 3936 - 3136 = 800
-
Calculate r:
Comment on their relationship:
The correlation coefficient is r = 1. This indicates a perfect positive linear relationship between X and Y. For every unit increase in X, Y increases by a constant proportion (specifically, Y = 2X). All the points lie perfectly on an upward-sloping straight line.