Key Concepts and Formulas for Dispersion, Correlation & Index Numbers
Measures of Dispersion, Correlation, and Index Number are crucial topics in statistics for Commerce students. Understanding these helps in analyzing data variability, relationships, and changes over time. These concepts are vital for school exams, competitive tests, and for making informed business decisions in real life.
| Concept | Definition | Example/Usage |
|---|---|---|
| Measures of Dispersion | Statistics that show how data values spread from the average. | Standard Deviation, Range, Variance |
| Correlation | Describes the relationship between two variables. | Height and Weight, Price and Demand |
| Index Number | Shows the relative change of a variable over time or location. | Consumer Price Index (CPI), Wholesale Price Index (WPI) |
Measures of Dispersion Explained
Measures of dispersion, also called measures of spread, help us understand how much data varies from the average. Common types used in Commerce and Economics are range, mean deviation, standard deviation, and variance. Each has its own calculation method and use case in analyzing numerical data.
Types of Measures of Dispersion
- Range: Difference between the highest and lowest values.
- Variance: Average of squared differences from the mean.
- Standard Deviation: Square root of variance, showing spread in original units.
- Mean Deviation: Average of absolute differences from a central value (mean or median).
- Quartile Deviation: Half the difference between the third and first quartile.
Correlation in Statistics
Correlation shows the degree and direction of the relationship between two variables. It can range from +1 (perfect positive) to -1 (perfect negative). Commerce students often use correlation to analyze trends, such as between demand and price or sales and advertisement.
Types of Correlation
- Positive Correlation: Both variables move in the same direction.
- Negative Correlation: Variables move in opposite directions.
- No Correlation: No pattern between the variables.
Index Number: Definition and Importance
Index numbers are statistical tools that measure changes in economic data over time. The most common example in daily life is the Consumer Price Index, which reflects inflation and changes in living costs. For exams, knowing types and methods of calculation is essential.
Types and Uses of Index Numbers
- Price Index Numbers: Measure average change in prices (CPI, WPI).
- Quantity Index Numbers: Show changes in quantities like production or sales.
- Value Index Numbers: Reflect changes in total value (price × quantity).
Practice MCQs on Measures of Dispersion, Correlation, and Index Number
- Which of the following is an absolute measure of dispersion?
a) Range
b) Coefficient of Variation
c) Correlation
d) Index Number
Answer: a) Range - Standard deviation is:
a) The mean of data
b) The spread of data around the mean
c) The product of means
d) Median
Answer: b) The spread of data around the mean - If the range of a data set is 45 and the minimum value is 15, what is the maximum value?
a) 30
b) 60
c) 75
d) 45
Answer: b) 60 - Which measure of dispersion is least affected by extreme values?
a) Range
b) Standard Deviation
c) Quartile Deviation
d) Mean Deviation
Answer: c) Quartile Deviation - The correlation coefficient r can vary between:
a) 0 to +1
b) –1 to +1
c) 0 to 100
d) –100 to +100
Answer: b) –1 to +1 - Consumer Price Index is an example of:
a) Quantity Index Number
b) Value Index Number
c) Price Index Number
d) Volume Index Number
Answer: c) Price Index Number - If SD = 5 and mean = 20, what is the coefficient of variation?
a) 4%
b) 10%
c) 25%
d) 15%
Answer: c) 25% - Which of the following statements is true for correlation?
a) It shows cause and effect
b) It only shows association between variables
c) It measures only dispersion
d) It measures central tendency
Answer: b) It only shows association between variables - Index numbers are mainly used to:
a) Measure central tendency
b) Measure degree of relation
c) Compare changes over time
d) Find the mean
Answer: c) Compare changes over time - Which method is used for finding the relationship between two variables graphically?
a) Lorenz Curve
b) Scatter Diagram
c) Index Number
d) Histogram
Answer: b) Scatter Diagram
Calculation Walkthroughs
Standard Deviation Example: For the numbers 10, 15, 20, 25, 30:
Step 1: Mean = (10 + 15 + 20 + 25 + 30) / 5 = 20
Step 2: Find deviations: –10, –5, 0, +5, +10
Step 3: Square: 100, 25, 0, 25, 100
Step 4: Mean of squares = (100 + 25 + 0 + 25 + 100) / 5 = 50
Step 5: SD = √50 ≈ 7.07
Correlation Coefficient Example: If the variables X (2, 4, 6) and Y (3, 6, 9) show perfect direct proportion, their correlation coefficient r = +1 (perfect positive correlation).
Application & Exam Tips
- Remember that relative measures (like coefficient of variation) compare variability across different data sets.
- In MCQs, pay attention to keywords like "least affected by extreme values."
- Practice formula application for fast calculation.
- Understand when to use absolute vs. relative dispersion measures in exam questions.
- Use the Lorenz Curve to quickly compare dispersion visually.
- Check if the question is asking for unit-based (absolute) or percentage-based (relative) answers.
Related Concepts and Further Learning
- Explore Karl Pearson Coefficient of Correlation for advanced correlation methods.
- Learn more about important index numbers and their uses in economics.
- Understand data organization for dispersion with Tabulation methods.
- Master graphical tools like the Scatter Diagram for visualizing correlation.
- Connect dispersion with central tendency using Median and related statistics.
In summary, MCQs on Measures of Dispersion, Correlation, and Index Number are essential for Commerce students preparing for academic and competitive exams. Mastery of these concepts enables confident data analysis, clarity in interpreting business and economic trends, and success in statistics-related questions. Continue practicing with Vedantu for top scores and a strong statistical foundation.
FAQs on MCQs on Measures of Dispersion, Correlation and Index Number
1. What are the common measures of dispersion?
The main measures of dispersion in statistics describe the spread or variability of a dataset. They include:
- Range: The difference between the highest and lowest values.
- Variance: The average of the squared differences from the mean.
- Standard Deviation: The square root of the variance, representing the typical distance from the mean.
- Mean Deviation: The average of the absolute differences from the mean.
- Quartile Deviation: Half the difference between the third and first quartiles.
2. Which measure of dispersion can be negative?
None of the standard measures of dispersion (range, variance, standard deviation, mean deviation, quartile deviation) can be negative. Dispersion measures quantify the spread of data; a negative value wouldn't make sense in this context. They always represent a positive distance or variation.
3. What are the characteristics of a good measure of dispersion?
A good measure of dispersion should be:
- Easy to calculate and understand: Simple formulas ensure wide applicability.
- Unambiguous: It should provide a clear and single value for the spread.
- Based on all observations: The measure should consider all data points, not just a few.
- Not greatly affected by extreme values (outliers): Robust measures are less sensitive to unusual data points.
- Suitable for further mathematical treatment: The measure should allow for further statistical analysis.
4. What is an index number in statistics?
An index number is a statistical measure that shows the relative change in a variable (or group of variables) over time or across different locations. It is usually expressed as a percentage of a base period value. Index numbers are widely used in economics to track price changes (price index), production levels (quantity index), and other economic indicators. Understanding index numbers helps in economic analysis and forecasting.
5. What is correlation and its types?
Correlation measures the strength and direction of the linear relationship between two or more variables. Types of correlation include:
- Positive correlation: As one variable increases, the other tends to increase.
- Negative correlation: As one variable increases, the other tends to decrease.
- No correlation: There is no linear relationship between the variables.
6. Which of the following measures of dispersion can attain a negative value?
None of the common measures of dispersion can be negative. Measures like range, variance, standard deviation, and mean deviation always result in non-negative values because they represent the spread or variability of data, which cannot be negative.
7. What are absolute and relative measures of dispersion?
Absolute measures of dispersion (e.g., range, standard deviation, variance) express the spread in the original units of the data. Relative measures of dispersion (e.g., coefficient of variation) express the spread relative to the average value, allowing for comparison across datasets with different units or scales. The coefficient of variation is calculated by dividing the standard deviation by the mean and multiplying by 100.
8. How are index numbers used in economics?
Index numbers are fundamental in economics for tracking economic trends. They are used to:
- Measure changes in price levels (Consumer Price Index, Wholesale Price Index).
- Monitor production levels (Industrial Production Index).
- Track changes in economic activity (leading economic indicators).
- Compare economic performance across different regions or time periods.
9. Can correlation be greater than 1?
No, the correlation coefficient (like Pearson's r) ranges from -1 to +1. A value of +1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no linear correlation. A value greater than 1 or less than -1 is not possible.
10. Which measure of dispersion is least affected by extreme values?
The quartile deviation (or interquartile range) is least affected by extreme values (outliers). It focuses on the middle 50% of the data, making it more robust than measures like the range or standard deviation, which are sensitive to outliers.
11. Why is standard deviation considered more reliable than range?
Standard deviation is considered more reliable than range because it uses all data points in its calculation, unlike range, which only considers the minimum and maximum values. This makes the standard deviation less sensitive to outliers and provides a more comprehensive representation of data variability.
12. In what scenarios would mode-based dispersion be useful?
Mode-based dispersion, while less common than mean-based measures, can be useful for categorical data or datasets with a clearly defined mode (most frequent value). It helps analyze how data is clustered around the most frequent value, providing insight into the concentration of observations.
13. How does coefficient of variation help in risk analysis?
The coefficient of variation (CV), a relative measure of dispersion, helps compare the risk associated with investments or projects with different scales or units. A higher CV indicates higher volatility or risk. It normalizes the variability relative to the mean, facilitating meaningful comparison.
14. When should weighted index numbers be preferred over simple index numbers?
Weighted index numbers are preferred over simple index numbers when the items within a dataset have varying importance or weights. For example, in a price index, if some goods account for a larger portion of consumer spending, they should receive a higher weight. Using weights gives a more realistic and representative index.
15. How do changes in scale or units affect each measure of dispersion?
Changes in scale affect absolute measures (range, standard deviation, variance) proportionally. Relative measures (coefficient of variation) are unaffected by changes in scale because the effect on the mean and standard deviation cancels out. Changes in units require appropriate conversions in the measures.
