AIOU 1350 Solved Assignment Autumn 2025

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ALLAMA IQBAL OPEN UNIVERSITY

(Department of Statistics)

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WARNING

1. Plagiarism or hiring of ghost writer(s) for solving the assignment(s) will debar the student from award of degree/certificate if found at any stage.

2. Submitting assignment(s) borrowed or stolen from other(s) as one's own will be penalized as defined in the "Aiou Plagiarism Policy".

Assignment Submission Schedule
6 Credit Hours	Due Date	3 Credit Hours	Due Date
Assignment 1	15-12-2025	Assignment 1	08-01-2026
Assignment 2	08-01-2026
Assignment 3	30-01-2026	Assignment 2	20-02-2026
Assignment 4	20-02-2026

Course: Introduction to Business Statistics (1350)	Semester: Autumn-2025
Level: I.Com

Please read the following instructions for writing your assignments. (SSC, HSSC & BA Programmes)
1. All questions are compulsory and carry equal marks but within a question the marks are distributed according to its requirements.
2. Read the question carefully and then answer it according to the requirements of the questions.
3. Late submission of assignments will not be accepted.
4. Your own analysis and synthesis will be appreciated.
5. Avoid irrelevant discussion/information and reproducing from books, study guide of allied material.

Total Marks: 100	Pass Marks: 40

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ASSIGNMENT No. 1

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Q1(a). Distinguish between primary and secondary data and give different sources from which these are obtained. ▶

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Difference Between Primary and Secondary Data

When it comes to research, data is the backbone of any study. But not all data is the same. Researchers usually rely on two main types: primary data and secondary data. Both are useful, but they differ in how they are collected and the purpose they serve. Let’s break it down in simple terms.

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What is Primary Data?

Primary data is the information that is collected directly by the researcher for the first time. It is original, fresh, and gathered to solve a specific problem. Since it is firsthand data, it is more reliable and accurate, but it can also take more time and money to collect.

Sources of Primary Data:

• Surveys and questionnaires filled out by people
• Interviews where researchers directly ask questions
• Observation of behaviors, events, or conditions
• Experiments conducted under controlled settings
• Focus groups that provide collective opinions

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What is Secondary Data?

Secondary data is the information that has already been collected and published by someone else. It is usually gathered for a purpose other than your own study. This type of data is cheaper and faster to access, but it may not be as specific or accurate as primary data.

Sources of Secondary Data:

• Books and academic journals
• Government reports and census data
• Company reports and financial statements
• Websites, online databases, and digital libraries
• Newspapers and magazines

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Conclusion

In simple words, primary data is collected directly by you, while secondary data is collected by someone else and then used for your research. Primary data is accurate and specific but costly, while secondary data is quick and affordable but may not fully match your needs. Smart researchers often use a mix of both to get the best results.

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Q1(b). Explain the applications of statistics in various areas of business and commerce, such as market research, financial analysis, production management, and quality control. ▶

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Applications of Statistics in Business and Commerce

Statistics plays a key role in modern business and commerce. It helps companies make informed decisions by turning raw data into meaningful insights. Whether it is about understanding customers, planning production, or improving product quality, statistics provides tools that support smart business strategies. Below are some major areas where statistics is widely applied.

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1. Market Research

Businesses need to know what customers want, how much they are willing to pay, and what products they prefer. Through surveys, sampling, and data analysis, statistics helps companies understand consumer behavior and market trends. For example, a company can use statistical techniques to estimate the demand for a new product or to segment its customers into different groups for targeted marketing.

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2. Financial Analysis

Finance is one of the most data-driven areas of business. Companies use statistical tools to analyze profits, expenses, risks, and investments. For instance, probability and regression analysis help in forecasting sales, predicting stock price movements, and managing risks. Statistics also supports budgeting by identifying patterns in revenues and costs over time.

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3. Production Management

Efficient production is essential for reducing costs and meeting customer demand. Statistics helps managers in areas like demand forecasting, inventory control, and resource allocation. By analyzing past production data, managers can plan how much to produce, when to produce, and how to optimize the use of machinery and labor.

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4. Quality Control

Delivering quality products is crucial for business success. Statistical techniques such as control charts and sampling inspection are used to monitor and maintain product quality. For example, factories often test a small sample from each batch of production to ensure that the entire batch meets quality standards. This reduces waste, saves costs, and builds customer trust.

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Conclusion

In short, statistics is not just about numbers, it is about making better business decisions. From understanding customers in market research to managing finances, from improving production efficiency to ensuring product quality, statistics guides businesses toward growth and competitiveness.

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Q2(a). Discuss the role of tabulation in data analysis. Explain the different parts of a statistical table and how they contribute to effective data presentation and interpretation. ▶

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Role of Tabulation in Data Analysis

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Tabulation is an essential step in data analysis because it organizes raw data into a clear, systematic format. When data is arranged in rows and columns, it becomes easier to understand, compare, and interpret. Without tabulation, large amounts of data can be confusing and overwhelming, making it difficult to identify patterns, trends, or relationships between variables. By summarizing data effectively, tabulation helps researchers and decision-makers draw meaningful conclusions and communicate findings clearly.

Parts of a Statistical Table

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Title: The title of a statistical table gives a concise description of the data being presented. It helps readers understand the scope, content, and time period of the data immediately, providing context before they examine the details.

Headings: Headings label the columns (and sometimes rows) of the table. Column headings describe the type of data in each column, while row headings indicate the categories being compared. Accurate headings prevent confusion and make the table easier to read.

Stub or Row Labels: The stub is usually the first column of the table and lists the categories or variables under study. It serves as a reference for the data in the table, allowing readers to compare different groups easily.

Body of the Table: The body contains the actual data, presented in a structured manner. Numbers, percentages, or other measurements are placed in their corresponding rows and columns. Consistent formatting in the body ensures clarity and helps highlight important patterns.

Footnotes or Remarks: Footnotes provide extra information about the data, such as explanations of symbols, abbreviations, or special conditions. They improve understanding and reduce the risk of misinterpretation.

Source of Data: Including the data source adds credibility and allows readers to verify the information. It also shows whether the data is from primary research or secondary sources.

By combining these parts effectively, a statistical table transforms raw data into meaningful information. It simplifies complex datasets, highlights trends, and supports better analysis and decision-making in business, research, and everyday applications.

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Q2(b). What is a frequency distribution? How is it constructed? ▶

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Frequency Distribution

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A frequency distribution is a way of organizing data to show how often each value or group of values occurs in a dataset. It provides a clear picture of the distribution of data, making it easier to understand patterns, identify trends, and analyze the dataset effectively. Instead of looking at a long list of raw numbers, a frequency distribution summarizes the data in a concise format, which can be used for further analysis like drawing graphs or calculating statistical measures.

Construction of a Frequency Distribution

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Constructing a frequency distribution involves several steps:

1. Collect and arrange the data: Start with a complete set of data and organize it in ascending or descending order to make analysis easier.

2. Decide the number of classes: For continuous data, divide the data range into a suitable number of classes or intervals. The number of classes is usually determined based on the size of the dataset; too many or too few classes can make the table less informative.

3. Determine class width: The class width is the difference between the upper and lower boundaries of each class. It can be calculated by dividing the range of the data (maximum value minus minimum value) by the number of classes.

4. Set class limits: Define the lower and upper limits of each class to ensure that all data points fall into a specific class. For example, if a class is 10–19, all values from 10 up to 19 are included.

5. Tally the frequencies: Count how many data points fall into each class. This is called the frequency, which indicates how often a value or range of values appears in the dataset.

6. Create the frequency table: Construct a table with columns for the class intervals and the corresponding frequencies. Additional columns for relative frequency or cumulative frequency can also be included for more detailed analysis.

Once the frequency distribution is prepared, it becomes easier to visualize the data using graphs like histograms, bar charts, or pie charts, and to calculate key statistical measures like mean, median, and mode. It is a fundamental tool for summarizing and interpreting data effectively.

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Q3(a). Define a frequency polygon and describe the process of creating a frequency polygon from a frequency distribution. Explain the relationship between a histogram and a frequency polygon. ▶

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Frequency Polygon

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A frequency polygon is a graphical representation of a frequency distribution. It is created by plotting the midpoints of each class interval on the x-axis and their corresponding frequencies on the y-axis, and then connecting the points with straight lines. A frequency polygon helps visualize the shape of the data distribution and makes it easier to compare multiple distributions on the same graph.

Creating a Frequency Polygon from a Frequency Distribution

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The process of creating a frequency polygon involves the following steps:

1. Prepare a frequency distribution: Start with a frequency table that includes class intervals and their corresponding frequencies.

2. Determine class midpoints: For each class interval, calculate the midpoint by adding the lower and upper limits of the class and dividing by two. The midpoints will be plotted on the x-axis.

3. Plot the points: On a graph, mark the midpoints on the x-axis and plot their corresponding frequencies on the y-axis.

4. Connect the points: Join the plotted points with straight lines. To close the polygon, a point with zero frequency is added before the first class and after the last class.

Relationship Between a Histogram and a Frequency Polygon

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A histogram is a bar graph where each bar represents a class interval and its height shows the frequency of that class. The frequency polygon, on the other hand, is a line graph that connects the midpoints of the tops of the bars of the histogram. In essence, the frequency polygon provides a smooth outline of the histogram, highlighting the overall pattern of the distribution. Both tools are used to visualize data, but the frequency polygon is particularly useful for comparing two or more distributions on the same graph.

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Q3(b). What are the key differences between a histogram and a bar chart? When is it more appropriate to use a histogram versus a bar chart for data presentation? ▶

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Key Differences Between a Histogram and a Bar Chart

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While histograms and bar charts both use rectangular bars to represent data, they are used in different contexts and have distinct features:

1. Nature of Data: A histogram is used for continuous data, where the data values are in intervals or ranges, such as ages, heights, or test scores. A bar chart is used for categorical or discrete data, such as types of fruits, brands, or countries.

2. Bar Arrangement: In a histogram, the bars are placed adjacent to each other with no gaps, reflecting the continuous nature of the data. In a bar chart, bars are separated by gaps to emphasize that the categories are distinct and not continuous.

3. Bar Width: In histograms, all bars have equal width that corresponds to the class intervals. In bar charts, the width of the bars is arbitrary, and the focus is on comparing the height of the bars rather than the interval size.

4. Horizontal Axis: For histograms, the x-axis represents the continuous intervals of the variable. For bar charts, the x-axis represents distinct categories, which may not have a numerical relationship.

5. Purpose: Histograms are primarily used to show the shape, distribution, and spread of continuous data, whereas bar charts are used to compare frequencies or quantities across different categories.

When to Use a Histogram vs. a Bar Chart

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Use a histogram when you want to analyze the distribution of numerical data and see patterns such as skewness, symmetry, or clustering. It is ideal for large datasets with continuous variables. Use a bar chart when you want to compare quantities across distinct categories or groups, making it easy to visualize differences between items. For example, a histogram is suitable for showing the distribution of students’ test scores, while a bar chart is appropriate for showing the number of students in different departments.

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Q4(a). Explain how to find the median for a dataset with an even number of observations. Provide an example dataset with an even number of values and demonstrate the calculation of the median. ▶

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Finding the Median for a Dataset with an Even Number of Observations

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The median is the middle value of a dataset when the numbers are arranged in ascending or descending order. When the dataset has an even number of observations, there is no single middle value. In this case, the median is calculated by taking the average of the two middle values.

Steps to Find the Median:

1. Arrange the dataset in ascending order.

2. Identify the two middle values. For a dataset with n values, the middle positions are n/2 and (n/2) + 1.

3. Calculate the median by finding the average of these two middle values.

Example:

Consider the dataset: 12, 7, 9, 15, 10, 8

Step 1: Arrange in ascending order:

7, 8, 9, 10, 12, 15

Step 2: Identify the two middle values:

There are 6 observations, so the middle positions are 6/2 = 3 and (6/2) + 1 = 4. The 3rd and 4th values are 9 and 10.

Step 3: Calculate the median:

Median = (9 + 10) / 2 = 19 / 2 = 9.5

So, the median of the dataset is 9.5.

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Q4(b). Calculate the mean number of siblings based on the frequency distribution.

Number of Siblings	Frequency
0	15
1	25
2	18
3	12
4	8

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Calculating the Mean Number of Siblings from a Frequency Distribution

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To calculate the mean from a frequency distribution, we use the formula:

\( \text{Mean} = \frac{\sum (f \cdot x)}{\sum f} \)

Where \(x\) is the value of the variable (number of siblings) and \(f\) is the frequency.

Given Frequency Distribution:

Number of Siblings (x)	Frequency (f)
0	15
1	25
2	18
3	12
4	8

Step 1: Multiply each value by its frequency

\(0 \cdot 15 = 0\)

\(1 \cdot 25 = 25\)

\(2 \cdot 18 = 36\)

\(3 \cdot 12 = 36\)

\(4 \cdot 8 = 32\)

Step 2: Sum the products

\(\sum (f \cdot x) = 0 + 25 + 36 + 36 + 32 = 129\)

Step 3: Sum the frequencies

\(\sum f = 15 + 25 + 18 + 12 + 8 = 78\)

Step 4: Calculate the mean

\( \text{Mean} = \frac{129}{78} \approx 1.65 \)

Therefore, the mean number of siblings is approximately 1.65.

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Q5(a). Discuss the empirical relationship between mean, median and mode. ▶

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Empirical Relationship Between Mean, Median, and Mode

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In statistics, the mean, median, and mode are measures of central tendency used to describe the center of a dataset. While each measure is calculated differently, there is an empirical relationship that often applies to moderately skewed, unimodal distributions.

Empirical Formula:

\( \text{Mode} \approx 3 \times \text{Median} - 2 \times \text{Mean} \)

This formula helps estimate the mode when the mean and median are known and is particularly useful for skewed distributions.

Interpretation:

1. Symmetrical Distribution: The dataset is balanced, so:

\( \text{Mean} = \text{Median} = \text{Mode} \)

2. Positively Skewed Distribution (tail on the right):

\( \text{Mode} \lt \text{Median} \lt \text{Mean} \)

3. Negatively Skewed Distribution (tail on the left):

\( \text{Mean} \lt \text{Median} \lt \text{Mode} \)

This relationship shows how the mean is affected by extreme values, the median represents the midpoint of the data, and the mode indicates the most frequent value. It provides a quick way to understand the central tendency in skewed datasets and helps in estimating one measure if the other two are known.

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Q5(b). Consider the numbers 4 and 9. Calculate the arithmetic mean, geometric mean, and harmonic mean of these two numbers. What inequality relationship do you observe between AM, GM and HM? ▶

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Calculating AM, GM, and HM for Two Numbers

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Consider the numbers: 4 and 9. We will calculate the Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) using MathJax.

1. Arithmetic Mean (AM)

The formula for the arithmetic mean of two numbers \(a\) and \(b\) is:

\( AM = \frac{(a + b)}{2} \)

Substitute \(a = 4\) and \(b = 9\):

\( AM = \frac{(4 + 9)}{2} = \frac{13}{2} = 6.5 \)

2. Geometric Mean (GM)

The formula for the geometric mean of two numbers \(a\) and \(b\) is:

\( GM = \sqrt{a \cdot b} \)

Substitute \(a = 4\) and \(b = 9\):

\( GM = \sqrt{4 \cdot 9} = \sqrt{36} = 6 \)

3. Harmonic Mean (HM)

The formula for the harmonic mean of two numbers \(a\) and \(b\) is:

\( HM = \frac{2ab}{a + b} \)

Substitute \(a = 4\) and \(b = 9\):

\( HM = \frac{2 \cdot 4 \cdot 9}{4 + 9} = \frac{72}{13} \approx 5.538 \)

Observed Inequality Relationship

For any two positive numbers, the following relationship holds:

\( HM \leq GM \leq AM \)

Here, we have:

\( HM \approx 5.538 \leq GM = 6 \leq AM = 6.5 \)

This confirms that the harmonic mean is the smallest, the geometric mean is in the middle, and the arithmetic mean is the largest.

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Q5(c). Explain in detail how to select a suitable measure of central tendency. ▶

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Selecting a Suitable Measure of Central Tendency

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Measures of central tendency, such as mean, median, and mode, summarize a dataset by identifying a central value. Choosing the most appropriate measure depends on the nature of the data and the objective of the analysis. Here’s a detailed guide to selecting a suitable measure:

1. Consider the Type of Data

The type of data nominal, ordinal, interval, or ratio plays a major role in deciding the measure of central tendency:

Nominal Data: These are categorical data without any inherent order, such as eye color or brand names. The mode is the only meaningful measure because it represents the most frequently occurring category.

Ordinal Data: These data have a clear order but unequal intervals, such as rankings or satisfaction levels. The median is preferred because it identifies the middle value without assuming equal spacing between ranks. The mode can also be reported if the most common category is of interest.

Interval and Ratio Data: These are numerical data with equal intervals. The mean is generally the most informative measure because it uses all data points and is suitable for further statistical analysis. The median is preferred when the data are skewed, and the mode is useful for identifying the most common value.

2. Examine the Distribution of Data

The shape of the data distribution affects the choice of measure:

Symmetrical Distribution: If the data are roughly symmetrical, the mean, median, and mode are close. The mean is commonly used because it considers all values.

Skewed Distribution: If the data are skewed (long tail on one side), the median is more reliable because it is not affected by extreme values. The mean may be misleading in such cases.

3. Presence of Outliers

Outliers can heavily influence the mean, making it unrepresentative of the dataset. In such cases, the median is a better choice because it is resistant to extreme values. The mode remains unaffected by outliers, but it may not reflect the dataset's center accurately.

4. Purpose of Analysis

The objective of the analysis can guide the choice of measure:

If you need a precise average for calculations or further statistical analysis, the mean is preferred.

If you want to identify the central value without being influenced by extremes, the median is suitable.

If the goal is to highlight the most common occurrence or category, the mode is appropriate.

5. Data Characteristics and Practical Considerations

Sometimes, data may have multiple modes (bimodal or multimodal), making the mode less informative. In such cases, the median or mean may be more representative. If the dataset is small, the mean can be skewed by even one extreme value, so median or mode may be preferred.

Summary Table:

Data Type / Situation	Recommended Measure
Nominal data	Mode
Ordinal data	Median or Mode
Symmetrical interval/ratio data	Mean
Skewed interval/ratio data or presence of outliers	Median
Most frequent occurrence is of interest	Mode

By considering the type of data, distribution, outliers, and the purpose of analysis, you can select the most suitable measure of central tendency for accurate and meaningful interpretation of data.

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ASSIGNMENT No. 2
(Units 6 to 9)

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Q1(a). Define various measures of dispersion and give their formulae. ▶

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Measures of Dispersion in Statistics

Measures of dispersion are used to describe the spread or variability in a dataset. They indicate how much the data values deviate from the central value such as the mean or median. Here are the main measures:

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1. Range

The range is the simplest measure of dispersion. It is the difference between the largest and smallest values in the dataset.

Formula:

\( \text{Range} = \text{Maximum value} - \text{Minimum value} \)

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2. Mean Deviation (MD)

Mean deviation measures the average of the absolute differences between each observation and the mean (or median).

Formula for mean deviation about mean:

\( MD = \frac{\sum_{i=1}^{n} |x_i - \bar{x}|}{n} \)

Where \(x_i\) is each observation, \(\bar{x}\) is the mean, and \(n\) is the number of observations.

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3. Variance (\(\sigma^2\) or \(s^2\))

Variance measures the average of the squared differences from the mean.

Population variance:

\( \sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N} \)

Sample variance:

\( s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1} \)

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4. Standard Deviation (SD)

Standard deviation is the square root of the variance. It is widely used because it is in the same units as the data.

Formulas:

Population SD: \( \sigma = \sqrt{\sigma^2} \)

Sample SD: \( s = \sqrt{s^2} \)

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5. Coefficient of Variation (CV)

CV is a relative measure of dispersion, expressed as a percentage. It shows the extent of variability relative to the mean.

Formula:

\( CV = \frac{\text{SD}}{\text{Mean}} \times 100 \)

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6. Quartile Deviation (QD) / Semi-Interquartile Range

This measures the spread of the middle 50% of the data using quartiles.

Formula:

\( QD = \frac{Q_3 - Q_1}{2} \)

Where \(Q_1\) is the first quartile and \(Q_3\) is the third quartile.

These measures help not only to understand the central tendency of data but also how widely the values are spread. They are essential for statistical analysis in business, research, and everyday decision-making.

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Q1(b). The following table gives the marks of students:

Classes	f
30 - 39	8
40 - 49	87
50 - 59	190
60 - 69	86
70 - 79	20

Calculate quartile deviation and mean deviation from the mean. ▶

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Frequency Distribution and Calculations

Class Interval	Frequency (f)	Midpoint (x)	\(f * x\)	\(|x - \bar{x}| \)	\(f * |x - \bar{x}| \)
30 - 39	8	34.5	276	20.54	164.32
40 - 49	87	44.5	3861.5	10.54	916.98
50 - 59	190	54.5	10355	0.54	102.6
60 - 69	86	64.5	5547	9.46	813.56
70 - 79	20	74.5	1490	19.46	389.2
Total			21529.5		2386.66

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Mean: \( \bar{x} = \frac{\sum f x}{\sum f} = \frac{21529.5}{391} \approx 55.04 \)

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Quartile Deviation (QD):

\( Q_1 = 50 + \frac{97.75 - 95}{190} \cdot 10 \approx 50.15 \)

\( Q_3 = 60 + \frac{293.25 - 285}{86} \cdot 10 \approx 60.96 \)

\( QD = \frac{Q_3 - Q_1}{2} = \frac{60.96 - 50.15}{2} \approx 5.41 \)

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Mean Deviation from Mean (MD):

\( MD = \frac{\sum f |x - \bar{x}|}{\sum f} = \frac{2386.66}{391} \approx 6.11 \)

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Q2(a). Write down the uses and limitations of index numbers. ▶

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Uses of Index Numbers

1. They help measure changes in economic variables like prices, production, wages, or sales over time.

2. They are useful for comparing economic data across different time periods or regions.

3. They assist in inflation analysis by tracking price level changes.

4. Policymakers and businesses use them for planning, budgeting, and decision-making.

5. They make it easier to summarize large amounts of statistical data into a single figure for analysis.

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Limitations of Index Numbers

1. They may not fully reflect quality changes in products, leading to inaccurate conclusions.

2. Choice of base year can affect results; an inappropriate base year may distort trends.

3. They rely on accurate and complete data, which is not always available.

4. Aggregating different items into one index can hide variations within individual components.

5. They cannot explain the causes of changes; they only indicate the magnitude of change.

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Q2(b). Construct chain indices for the following years, taking 2000 as a base.

Item	Year
	2000	2001	2002	2003	2004
Wheat	2.80	3.40	3.60	4.00	4.20
Rice	2.95	3.60	2.90	2.75	2.75
Maize	3.10	3.50	3.40	4.50	3.70

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Chain Indices for Wheat, Rice, and Maize (Base Year 2000)

The chain index formula is:

\[ \text{Chain Index} = \frac{\text{Price in current year}}{\text{Price in previous year}} \times 100 \]

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Item	Year
	2000	2001	2002	2003	2004
Wheat	100	121.43	105.88	111.11	105.00
Rice	100	122.03	80.56	94.83	100.00
Maize	100	112.90	97.14	132.35	82.22

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Chain Indices for Wheat, Rice, and Maize (Base Year 2000)

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Wheat:

2001: \( \frac{3.40}{2.80} \times 100 = 121.43 \)

2002: \( \frac{3.60}{3.40} \times 100 = 105.88 \)

2003: \( \frac{4.00}{3.60} \times 100 = 111.11 \)

2004: \( \frac{4.20}{4.00} \times 100 = 105.00 \)

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Rice:

2001: \( \frac{3.60}{2.95} \times 100 = 122.03 \)

2002: \( \frac{2.90}{3.60} \times 100 = 80.56 \)

2003: \( \frac{2.75}{2.90} \times 100 = 94.83 \)

2004: \( \frac{2.75}{2.75} \times 100 = 100.00 \)

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Maize:

2001: \( \frac{3.50}{3.10} \times 100 = 112.90 \)

2002: \( \frac{3.40}{3.50} \times 100 = 97.14 \)

2003: \( \frac{4.50}{3.40} \times 100 = 132.35 \)

2004: \( \frac{3.70}{4.50} \times 100 = 82.22 \)

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These chain indices show the relative change in prices from the previous year for each item.

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Q3(a). What is a scatter diagram? Describe its role in the theory of regression. ▶

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Scatter Diagram and Its Role in Regression

A scatter diagram is a graphical representation used to show the relationship between two variables. Each point on the diagram represents a pair of values, with one variable plotted along the horizontal axis (x-axis) and the other along the vertical axis (y-axis).

The scatter diagram helps to visually inspect the type and strength of the relationship between the two variables. For example, the points may suggest a positive relationship (both variables increase together), a negative relationship (one increases while the other decreases), or no clear relationship at all.

Role in the Theory of Regression:

1. Preliminary Analysis: Before performing regression analysis, a scatter diagram helps determine whether a linear or non-linear relationship exists between the variables.

2. Assumption Checking: Regression assumes a certain pattern in the relationship between variables. The scatter diagram can reveal whether this assumption is reasonable.

3. Direction and Strength: Observing the scatter diagram gives an idea about the direction (positive or negative) and the approximate strength (tight clustering vs. wide spread) of the relationship, which guides regression model fitting.

4. Outlier Detection: Scatter diagrams make it easier to spot unusual observations or outliers that may affect regression results.

In summary, the scatter diagram is a visual tool that complements regression analysis, helping analysts understand relationships between variables, guide model selection, and interpret results effectively.

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Simple Example of a Scatter Diagram

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Q3(b). Find the regression line of expenditure on sales and interpret it.

Sales	35	33	40	46	58	41	32	43	45
Expenditure	11	26	25	20	17	19	25	24	21

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We have the following data:

Sales (X)	35	33	40	46	58	41	32	43	45
Expenditure (Y)	11	26	25	20	17	19	25	24	21

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Step 1: Compute Means

Mean of Sales (\(\bar{X}\)):

\(\bar{X} = \frac{35+33+40+46+58+41+32+43+45}{9} = \frac{373}{9} \approx 41.44\)

Mean of Expenditure (\(\bar{Y}\)):

\(\bar{Y} = \frac{11+26+25+20+17+19+25+24+21}{9} = \frac{188}{9} \approx 20.89\)

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Step 2: Compute Deviations, Products and Squares

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X	Y	X - X̄	Y - Ȳ	(X-X̄)(Y-Ȳ)	(X-X̄)²
35	11	-6.44	-9.89	63.64	41.47
33	26	-8.44	5.11	-43.12	71.22
40	25	-1.44	4.11	-5.92	2.07
46	20	4.56	-0.89	-4.06	20.80
58	17	16.56	-3.89	-64.43	274.20
41	19	-0.44	-1.89	0.83	0.19
32	25	-9.44	4.11	-38.81	89.12
43	24	1.56	3.11	4.85	2.44
45	21	3.56	0.11	0.39	12.68
Total				-87.63	513.19

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Step 3: Compute Slope (b) and Intercept (a)

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Slope:

\(b = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2} = \frac{-87.63}{513.19} \approx -0.171\)

Intercept:

\(a = \bar{Y} - b\bar{X} = 20.89 - (-0.171 \times 41.44) \approx 27.98\)

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Step 4: Regression Line

The regression line of expenditure on sales is:

\(Y = 27.98 - 0.171X\)

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Step 5: Interpretation

• The slope (\(-0.171\)) indicates that for every 1-unit increase in sales, expenditure decreases by approximately 0.171 units on average.
• The intercept (27.98) represents the estimated expenditure when sales are zero.
• This regression line can be used to predict expenditure based on sales and shows a slight negative relationship in this dataset.

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Q4(a). Define the term correlation and the product-moment moment coefficient of correlation. ▶

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Correlation and the Product-Moment Coefficient of Correlation

1. Correlation:

Correlation is a statistical measure that describes the degree and direction of a relationship between two variables. It tells us how one variable changes when the other variable changes. Correlation can be:

• Positive: Both variables increase or decrease together.
• Negative: One variable increases while the other decreases.
• Zero: No relationship exists between the variables.

2. Product-Moment Coefficient of Correlation (Pearson's r):

The product-moment coefficient of correlation, denoted by r, measures the strength and direction of a linear relationship between two variables X and Y. It is calculated using the formula:

\[ r = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum (X_i - \bar{X})^2 \sum (Y_i - \bar{Y})^2}} \]

Where:

• X_i, Y_i are individual observations
• X̄, Ȳ are the means of X and Y
• Σ denotes summation over all observations

The value of r ranges between -1 and +1:

• r = +1: Perfect positive linear relationship
• r = -1: Perfect negative linear relationship
• r = 0: No linear relationship

The product-moment coefficient of correlation helps us understand both the direction and strength of the linear association between two variables, making it a key tool in statistical analysis.

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Q4(b). Find the coefficient of correlation between X and Y and interpret it.

X	3.0	6.5	5.3	8.6	1.2	4.2	2.9
Y	1.9	3.2	2.7	4.5	1.0	2.0	1.7

▶

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Step 1: Compute Means

\[\bar{X} = \frac{3.0 + 6.5 + 5.3 + 8.6 + 1.2 + 4.2 + 2.9}{7} \approx 4.53\]

\[\bar{Y} = \frac{1.9 + 3.2 + 2.7 + 4.5 + 1.0 + 2.0 + 1.7}{7} \approx 2.41\]

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Step 2: Compute Deviations, Products, and Squares

X	Y	X - X̄	Y - Ȳ	(X-X̄)(Y-Ȳ)	(X-X̄)²	(Y-Ȳ)²
3.0	1.9	-1.53	-0.51	0.78	2.34	0.26
6.5	3.2	1.97	0.79	1.56	3.88	0.62
5.3	2.7	0.77	0.29	0.22	0.59	0.08
8.6	4.5	4.07	2.09	8.50	16.57	4.37
1.2	1.0	-3.33	-1.41	4.69	11.09	1.99
4.2	2.0	-0.33	-0.41	0.14	0.11	0.17
2.9	1.7	-1.63	-0.71	1.16	2.66	0.50
Total				16.05	37.24	7.99

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Step 3: Correlation Coefficient

\[ r = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum (X_i - \bar{X})^2 \sum (Y_i - \bar{Y})^2}} = \frac{16.05}{\sqrt{37.24 \times 7.99}} \approx 0.988 \]

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Step 4: Interpretation

• The correlation coefficient r ≈ 0.988 indicates a very strong positive linear relationship between X and Y.
• As X increases, Y increases almost proportionally.

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Q5(a). Explain the classical, a posteriori and mathematical definition of probability. What are the axioms of probability? ▶

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Probability: Definitions and Axioms

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1. Classical Definition of Probability:

The classical definition of probability, also called the theoretical probability, is based on the assumption that all outcomes of an experiment are equally likely. It is defined as:

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\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]

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For example, the probability of getting a head when tossing a fair coin is 1/2 because there is one favorable outcome (head) out of two equally likely outcomes (head or tail).

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2. A Posteriori (Empirical) Definition of Probability:

The a posteriori or empirical probability is based on actual experiments or observations. It is defined as:

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\[P(E) = \frac{\text{Number of times event } E \text{ occurs}}{\text{Total number of trials}} \]

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For example, if a die is rolled 100 times and the number 4 appears 18 times, the empirical probability of getting a 4 is 18/100 = 0.18.

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3. Mathematical (Axiomatic) Definition of Probability:

The mathematical definition, also called the axiomatic approach, defines probability based on a set of axioms introduced by Kolmogorov. It provides a formal framework for all types of probability, including theoretical and empirical.

Let S be the sample space and P be a function assigning probabilities to events. Then:

• P(E) ≥ 0 for every event E (non-negativity)
• P(S) = 1 (the probability of the sample space is 1)
• If E1, E2, E3, ... are mutually exclusive events, then P(E1 ∪ E2 ∪ E3 ∪ ...) = P(E1) + P(E2) + P(E3) + ... (additivity)

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Axioms of Probability:

• Non-negativity: For any event E, P(E) ≥ 0.
• Normalization: The probability of the entire sample space is 1, i.e., P(S) = 1.
• Additivity: For mutually exclusive events E1, E2, ..., the probability of their union is the sum of their probabilities: P(E1 ∪ E2 ∪ ...) = P(E1) + P(E2) + ...

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These definitions and axioms provide the foundation for all probability calculations in statistics and probability theory.

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Q5(b). Ten vegetable cans, all of the same size, have lost their labels. It is known that 5 contain tomatoes and 5 contain corn. If 5 are selected at random, what is the probability that all contain tomatoes? What is the probability that 3 or more contain tomatoes? ▶

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Probability Problem: Selecting Vegetable Cans

We have 10 vegetable cans: 5 contain tomatoes and 5 contain corn. We select 5 cans at random. We want to find:

(a) Probability that all 5 selected cans contain tomatoes

The total number of ways to choose 5 cans from 10 is:

\( \binom{10}{5} = 252 \)

The number of ways to choose 5 tomato cans from 5 is:

\( \binom{5}{5} = 1 \)

So, the probability that all 5 selected cans contain tomatoes is:

\( P(\text{all tomatoes}) = \frac{\binom{5}{5}}{\binom{10}{5}} = \frac{1}{252} \approx 0.00397 \)

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(b) Probability that 3 or more contain tomatoes

“3 or more” means 3, 4, or 5 tomato cans. We calculate each case:

3 tomatoes: \( \binom{5}{3} \times \binom{5}{2} = 10 \times 10 = 100 \)

4 tomatoes: \( \binom{5}{4} \times \binom{5}{1} = 5 \times 5 = 25 \)

5 tomatoes: \( \binom{5}{5} \times \binom{5}{0} = 1 \times 1 = 1 \)

Total favorable outcomes = 100 + 25 + 1 = 126

Total possible outcomes = \(\binom{10}{5} = 252\)

So, the probability that 3 or more contain tomatoes is:

\( P(\text{3 or more tomatoes}) = \frac{126}{252} = 0.5 \)

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• All tomatoes: \(0.00397\)
• 3 or more tomatoes: \(0.5\)

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