5 Most SAQ’s of Basic Statistics for Economics Chapter in Inter 1st Year Economics (TS/AP)

4 Marks

SAQ-1 : What is statistics? Explain its relationship with economics. (OR) What is statistics? Explain relationship between the economics and statistics.

Introduction

The field of statistics plays a significant role in various disciplines, including economics. Understanding what statistics is and its relationship with economics is crucial for students in these fields. This section aims to define statistics and explore its connection with economics.

Definition of Statistics

Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of data. It involves applying mathematical theories to real-world problems to make sense of numerical data.

Relationship Between Statistics and Economics

1. Data Analysis and Economic Theories
   • Statistics is instrumental in analyzing economic data to validate or refute economic theories. Economists use statistical methods to test hypotheses and develop economic models.
2. Economic Forecasting
   • The use of statistical tools in economic forecasting is crucial. Economists analyze trends and patterns in economic data to make predictions about future economic activities.
3. Policy Formulation
   • In policy formulation, statistical analysis helps in understanding the impact of various economic policies. Policy makers rely on statistical data to make informed decisions.
4. Market Research
   • In market research, statistics are used to analyze consumer behavior, demand and supply dynamics, and market trends. This information is vital for businesses and economic planners.
5. Quantitative Analysis of Economic Phenomena
   • Statistics provides tools for the quantitative analysis of economic phenomena. Measuring variables like inflation, unemployment, GDP growth, and others involves statistical methods.
6. Understanding Economic Relationships
   • Through statistical methods, economists can understand and quantify the relationships between different economic variables, such as income and consumption, investment and interest rates, etc.

Summary

Statistics and economics are deeply interconnected. Statistics provide the tools and methodologies for the analysis, interpretation, and understanding of economic data, which is fundamental in economic theorizing, forecasting, policy formulation, market research, and quantitative analysis of economic phenomena. This relationship is essential for the practical application of economic theories and in making informed economic decisions. Understanding this interplay is crucial for students and professionals in economics and related fields.

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SAQ-2 : Calculate arithmetic mean for the data given below:

Income	400	600	800	1000	1200	1400	1600	1800	2000
Persons	5	7	10	15	20	25	9	6	3

Introduction

The arithmetic mean, commonly known as the average, is a basic statistical measure used to calculate the central tendency of a set of numbers. In this specific task, we will calculate the arithmetic mean of a given dataset that represents income levels and the number of persons in each income bracket.

Calculation of Arithmetic Mean for the Given Data

The dataset provided represents various income levels and the corresponding number of persons in each income bracket. To calculate the arithmetic mean, we will use the formula for the weighted mean, as each income level has a different frequency (number of persons).

The formula for the weighted arithmetic mean is:

$$\text{Arithmetic Mean} = \frac{\sum (x_i \times f_i)}{\sum f_i}$$

Where x_i​ is the value of each item (income in this case), and f_i​ is the frequency of each item (number of persons).

Given Data:

1. Income Levels (x_i): 400, 600, 800, 1000, 1200, 1400, 1600, 1800, 2000
2. Number of Persons (f_i): 5, 7, 10, 15, 20, 25, 9, 6, 3

Let’s calculate the arithmetic mean based on the given data.

Arithmetic Mean of the Given Data

The arithmetic mean of the given income data, calculated using the weighted mean formula, is 1194. This means that the average income, taking into account the number of persons in each income bracket, is 1194 units.

Summary

To calculate the arithmetic mean for the provided dataset, we used the weighted mean method, considering both the income levels and the corresponding frequencies (number of persons). The calculated arithmetic mean provides an average income value, reflecting the central tendency of the income distribution across different groups of persons. ​

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SAQ-3 : Calculate median for the following data:

Wages	10	20	30	40	50	60	70	80	90
Workers	5	8	12	20	30	16	10	7	8

Introduction

The median is a statistical measure that represents the middle value in a sorted list of numbers. In this task, we will calculate the median for a given dataset that includes different wage levels and the corresponding number of workers in each wage bracket.

Calculation of Median for the Given Data

To calculate the median, the data needs to be organized in ascending order, and the middle value is identified. In the case of grouped data, like in the provided dataset, the median is found using the formula:

$$\text{Median} = L + \left( \frac{\frac{N}{2} – F}{f} \right) \times C$$

Where:

1. L is the lower limit of the median class.
2. N is the total number of observations.
3. F is the cumulative frequency of the class preceding the median class.
4. f is the frequency of the median class.
5. C is the class width.

Given Data:

1. Wage Levels: 10, 20, 30, 40, 50, 60, 70, 80, 90
2. Number of Workers: 5, 8, 12, 20, 30, 16, 10, 7, 8

We first need to find the median class by calculating the cumulative frequency and identifying the class where the median falls. Then, we will use the median formula to calculate the median wage.

Median of the Given Data

The median wage, calculated using the grouped data formula, is approximately 54.33 units. This value represents the middle wage level in the distribution, such that half of the workers earn wages below this amount and half earn above it.

Summary

To calculate the median for the provided dataset of wages and workers, we first organized the data and found the median class based on cumulative frequencies. Using the median formula for grouped data, we determined that the middle wage level, dividing the workers into two equal groups, is approximately 54.33 units. This calculation provides a central value that gives insight into the wage distribution among the workers. ​​

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SAQ-4 : What is Median? What are its merits and drawbacks?

Introduction

The median is an important measure of central tendency in statistics, representing the middle value in a sorted list of numbers. Understanding the median, along with its merits and drawbacks, is crucial for students in the fields of statistics and data analysis.

Definition of Median

The median is a measure of central tendency that identifies the middle value in a dataset when it is arranged in ascending or descending order. It effectively divides the dataset into two equal halves.

Formula for Calculating Median

1. Odd Number of Observations (n):
   $$\text{Median} = \text{Value of } \left(\frac{n+1}{2}\right)^{\text{th}} \text{ observation}$$
2. Even Number of Observations (n):
   $$\text{Median} = \text{Average of } \left(\frac{n}{2}\right)^{\text{th}} \text{ and } \left(\frac{n}{2} + 1\right)^{\text{th}} \text{ observations}$$

Merits of Median

1. Not Affected by Extreme Values: The median is not influenced by extremely large or small values, making it useful for skewed datasets.
2. Simple to Understand and Calculate: Calculating the median is straightforward, even without sophisticated statistical tools.
3. Applicable to Ordinal Data: The median is suitable for ordinal data where ranking is possible.
4. Exact Position: It accurately represents the exact middle position in a dataset.

Drawbacks of Median

1. Limited Use for Algebraic Treatment: The median is not conducive to further mathematical or algebraic operations.
2. May Not Reflect Entire Dataset: Since it focuses only on the middle value, the median might not be a comprehensive summary of the entire dataset.
3. Challenges with Large Datasets: Sorting large datasets to find the median can be cumbersome and time-consuming.
4. Provides Limited Information: The median does not consider all dataset values, possibly leading to incomplete data representation.
5. Necessitates Sorted Data: Calculating the median requires data to be sorted, which can be an additional step, especially for large datasets.

Summary

The median offers a valuable measure of central tendency, especially in its resilience to outliers and simplicity of calculation. However, its limitations, such as reduced usefulness for complex statistical analysis and inefficiency with large datasets, must be considered. Understanding both its strengths and limitations is important for accurate data interpretation and analysis.

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SAQ-5 : What is Arithmetic Mean? What are its merits and drawbacks?

Introduction

The Arithmetic Mean, commonly known as the average, is a fundamental statistical measure used to calculate the central tendency of a set of numbers. It is crucial for students in various fields, including economics, business, and the sciences, to understand the arithmetic mean along with its merits and drawbacks.

Definition of Arithmetic Mean

The Arithmetic Mean is the sum of a collection of numbers divided by the count of numbers in the collection. It is calculated by adding all the values and then dividing by the number of values.

Merits of Arithmetic Mean

1. Simplicity and Easy Comprehension: The arithmetic mean is simple to calculate and easy to understand, making it widely used for basic statistical analysis.
2. Mathematically Defined: It is mathematically precise, which allows for further algebraic treatment and mathematical manipulation.
3. Considers Every Item: The arithmetic mean takes into account every item in the dataset, which makes it a comprehensive measure of central tendency.
4. Suitable for Further Statistical Analysis: It is the basis for other statistical measures like variance and standard deviation, and is thus fundamental in statistical analysis.

Drawbacks of Arithmetic Mean

1. Affected by Extreme Values: The arithmetic mean is highly sensitive to extreme values or outliers, which can skew the results.
2. Not Applicable to All Data Types: It is not suitable for data that is non-numeric or ordinal.
3. May Not Represent Data Accurately: In some distributions, like highly skewed ones, the arithmetic mean may not accurately represent the central tendency.
4. Assumes Homogeneity: The arithmetic mean assumes that data is homogeneous, which may not always be the case, leading to misleading results in heterogeneous datasets.

Summary

The Arithmetic Mean is a widely-used measure of central tendency due to its simplicity and mathematical preciseness. It is valuable because it takes into account every item in a dataset and is essential for further statistical computations. However, its sensitivity to extreme values and limitations in representing non-homogeneous or skewed data are significant drawbacks. Understanding these aspects is important for accurate and effective application in data analysis.