Statistical Description Of Data - 2 - Free MCQ Practice Test with solutions,

• Pie-diagram: is a circular statistical graphic that is divided into slices to illustrate numerical proportion.


• Usage: Pie-diagrams are used to represent quantitative data in a circle, where each slice represents a proportion of the whole.


• Components: Each component is represented as a slice of the pie, with the size of each slice proportional to the value it represents.


• Comparison: Pie-diagrams are useful for comparing different components and their relation to the total, making it easy to see the distribution of data at a glance.


• Qualitative vs Quantitative: While pie-diagrams can be used for qualitative data, they are more commonly used for quantitative data where each component has a numerical value.

• Arranges observations in an increasing order: This is one way to organize data in a frequency distribution. It helps in determining the range of values present in the data set.


• Arranges observation in terms of a number of groups: Another way to present a frequency distribution is by grouping the data into intervals or classes. This makes it easier to analyze the data and identify patterns.


• Relates to a measurable characteristic: A frequency distribution is based on a measurable characteristic or variable. It helps in understanding the distribution of values and the frequency of each value.


• All these: A frequency distribution can involve arranging observations in increasing order, grouping observations, and relating to a measurable characteristic. All these aspects are essential in creating a comprehensive frequency distribution.

• Grouped Frequency Distribution: This type of frequency distribution is used when dealing with continuous variables that have a wide range of values. The data is grouped into intervals or classes, and the frequency of values falling within each interval is recorded.


• Simple Frequency Distribution: This type of frequency distribution is used when dealing with discrete variables that have a limited number of distinct values. Each value is listed along with its frequency of occurrence.


• Grouped vs. Simple: While both types of frequency distributions serve the purpose of organizing and summarizing data, grouped frequency distributions are more commonly used for continuous variables due to their nature of having a wide range of values.

The correct answer is option A: Grouped frequency distribution.

• Frequency distribution: Frequency distribution is a way to organize data into different classes or intervals showing the number of occurrences in each class.


• Shares distribution: When shares are distributed among different individuals or entities, it represents a count or number of shares allocated to each party.


• Discrete variable: In this case, the number of shares given to each entity is a whole number or an integer value. It cannot be divided into smaller parts. Therefore, it is an example of a discrete variable.


• Continuous variable: A continuous variable would be something that can take any value within a certain range, such as height or weight. Shares distribution does not fall into this category as it involves distinct, separate values.


• Attribute: An attribute is a characteristic or property of an object or entity. While shares distribution can be considered an attribute, it is more appropriately categorized as a discrete variable in the context of frequency distribution.


• Conclusion: Therefore, the distribution of shares is an example of the frequency distribution of a discrete variable.

• Continuous Variable: The distribution of profits of a blue-chip company is a continuous variable because profits can take on any value within a certain range. Profits can be in decimals and can vary continuously.


• Discrete Variable: A discrete variable would be something like the number of employees in a company, which can only take on whole number values.


• Attributes: Attributes are characteristics or properties of data. In this case, the profits of a blue-chip company can be considered as an attribute of the company's financial performance.

• Definition: Time series analysis involves studying price fluctuations, production of commodities, and deposits in banks over a period of time to identify patterns and trends.


• Purpose: The main goal of time series analysis is to make forecasts or predictions based on past data.


• Methods: Various statistical techniques such as trend analysis, moving averages, exponential smoothing, and autoregressive integrated moving average (ARIMA) models are used in time series analysis.


• Applications: Time series analysis is widely used in economics, finance, weather forecasting, and many other fields to make informed decisions and predictions based on historical data.


• Importance: By analyzing time series data, businesses and researchers can gain insights into the underlying patterns and factors influencing the variables being studied.

• Mutually Inclusive Classification: This type of classification is usually meant for a discrete variable. Discrete variables are variables that can only take specific values.


• Discrete Variable: A discrete variable is a variable that can only take on a finite number of values or a countable number of values. Examples include the number of students in a class or the number of cars in a parking lot.


• Continuous Variable: In contrast, a continuous variable can take on an infinite number of values within a certain range. Examples include height, weight, or temperature.


• Attribute: An attribute is a characteristic or property of an object that can be used for classification or identification. It can be either discrete or continuous.

• Definition: Mutually exclusive classification refers to the categorization of data into distinct groups where each data point belongs to only one category.



• Applicability: Mutually exclusive classification is usually meant for a continuous variable.



• Continuous Variable: A continuous variable is a type of variable that can take any value within a certain range. When applying mutually exclusive classification to a continuous variable, each data point is placed into a single category that is distinct from the others.



• Discrete Variable: While mutually exclusive classification can also be applied to discrete variables, it is more commonly used with continuous variables due to the nature of continuous data.



• Attribute: Mutually exclusive classification can also be used for attributes, especially when dealing with categorical data that needs to be separated into distinct groups.



• Conclusion: In summary, mutually exclusive classification is typically meant for a continuous variable, where each data point is assigned to a single category that is separate from the others.

• What is LCB?




• The LCB stands for Lower Confidence Bound.


• It is a statistical measure used in quality control to estimate the lower limit of a process parameter.




• Relationship between LCB and LCL:




• The LCB is a lower limit to the Lower Confidence Limit (LCL).


• It helps in determining the lower boundary within which the true value of the process parameter is expected to lie with a certain level of confidence.




• Options provided:




• A: An upper limit to LCL - This is incorrect as the LCB is a lower limit, not an upper limit.


• B: A lower limit to LCL - This is the correct option as explained above.


• C: (a) and (b) - This is incorrect as the LCB is only a lower limit to LCL.


• D: (a) or (b) - This is incorrect as the LCB is specifically a lower limit to LCL.

• Definition: Mathematical statistics is a branch of statistics that deals with the development of statistical methods.


• Focus: It focuses on the theoretical aspects of statistics, such as probability theory and sampling theory.


• Applications: Mathematical statistics is used in various fields such as finance, biology, engineering, and social sciences.


• Statistical Methods: It involves the development and evaluation of statistical methods for data analysis and interpretation.


• Research: Researchers in mathematical statistics work on improving existing statistical techniques and developing new ones.


• Advanced Topics: Advanced topics in mathematical statistics include regression analysis, hypothesis testing, and Bayesian statistics.


• Education: Many universities offer courses and degrees in mathematical statistics to train professionals in this field.


• Importance: Mathematical statistics plays a crucial role in making informed decisions based on data and improving the accuracy of predictions.

• Length of a class: The length of a class in statistics refers to the range of values within that class.


• UCB and LCB: UCB stands for Upper Class Boundary and LCB stands for Lower Class Boundary. The difference between these two boundaries gives the range of values in a class.


• Therefore, the length of a class is: The difference between the Upper Class Boundary (UCB) and Lower Class Boundary (LCB) of that class.

• Less than Cumulative Frequency: This refers to the total frequency of all the values that are less than a particular class boundary.


• More than Cumulative Frequency: This refers to the total frequency of all the values that are greater than a particular class boundary.


• Relationship: The sum of the less than cumulative frequency and the more than cumulative frequency for a particular class boundary will always add up to the total frequency.


• Explanation: This relationship holds true because the less than cumulative frequency accounts for all the values below the class boundary, while the more than cumulative frequency accounts for all the values above the class boundary. Therefore, when you add these two frequencies together, you cover all the values in the dataset, resulting in the total frequency.


• Conclusion: The correct answer is option A: Total frequency.

• Definition: Frequency density is the ratio of the frequency of a class interval to the width of the class interval.

• Formula: Frequency Density = Class Frequency / Class Length

• Frequency density helps in comparing different class intervals with varying widths.


• It is useful in creating frequency histograms and frequency polygons.


• By calculating frequency density, we can standardize the data and make comparisons easier.

• Frequency density is commonly used in statistics and data analysis to represent the distribution of data.


• It provides a way to visualize and interpret the frequency distribution of a dataset.

• Relative frequency: Relative frequency is a ratio that compares the frequency of a particular class to the total number of elements in a dataset.


• Lies between 0 and 1: The relative frequency for a particular class cannot be less than 0 or greater than 1. This is because it is a ratio and ratios are always expressed as fractions or decimals between 0 and 1.


• Lies between 0 and 1, both inclusive: Inclusive means that the endpoints of the interval are included. So, when we say the relative frequency lies between 0 and 1, both 0 and 1 are possible values for the relative frequency.


• Lies between –1 and 0: This statement is incorrect because relative frequency cannot be negative. It is always a non-negative value between 0 and 1.


• Lies between –1 to 1: This statement is incorrect because relative frequency cannot be greater than 1. It is always a non-negative value between 0 and 1.

• Histogram: The mode of a distribution can be determined by looking at the peak or highest point on a histogram. The value that occurs most frequently in the data set is the mode.


• Less than type ogives: Mode can also be found by constructing less than type ogives and identifying the class interval with the highest frequency. The mode is the midpoint of this class interval.


• More than type ogives: Similarly, mode can be obtained from more than type ogives by finding the class interval with the highest frequency and determining the mode as the midpoint of this interval.


• Frequency polygon: Frequency polygons can also help in identifying the mode by locating the highest point on the polygon, which corresponds to the mode of the distribution.

Therefore, the mode of a distribution can be obtained by analyzing histograms, less than type ogives, more than type ogives, or frequency polygons to identify the value or class interval that occurs most frequently in the data set.

• Step 1: Determine the cumulative frequency



First, calculate the cumulative frequency of the distribution. This can be done by adding up the frequencies as you move down the table.




• Step 2: Identify the median class



Find the class interval that contains the median value. This is usually the class interval where the cumulative frequency is closest to half of the total frequency.




• Step 3: Calculate the median



Once you have identified the median class, you can calculate the median using the formula:




Median = L + ((n/2 - F) / f) * w




Where:




• L = Lower boundary of the median class


• n = Total frequency


• F = Cumulative frequency of the class before the median class


• f = Frequency of the median class


• w = Width of the class interval

By following these steps, you can obtain the median of a distribution using less than type ogives. This method is particularly useful when dealing with grouped data and can provide a more accurate representation of the central tendency of the data.

• Definition: A histogram is a graphical representation of the distribution of numerical data. It consists of a series of adjacent rectangles, or bars, each representing a class interval.



• Class Frequencies: In a histogram, the height of each bar represents the frequency of data points within that class interval. By comparing the heights of the bars, we can easily compare the frequencies of different classes.



• Visual Representation: The visual representation of a histogram makes it easy to see patterns and trends in the data. This allows for quick comparison of class frequencies.



• Ease of Interpretation: Histograms provide a clear and concise way to compare class frequencies, making it a useful tool for data analysis and interpretation.



• Statistical Analysis: Histograms are often used in statistical analysis to understand the distribution of data and compare the frequencies of different classes.

Therefore, a histogram is the most suitable method for comparing class frequencies as it provides a visually appealing and easy-to-interpret representation of the data.

• Frequency Curve: A frequency curve is a smooth curve that represents the distribution of a set of data. It shows the frequency of values within intervals.



• Frequency Polygon: A frequency polygon is a line graph that displays the frequency of each value or group of values in a data set. It is created by connecting the midpoints of the tops of the bars in a histogram.



• Histogram: A histogram is a graphical representation of the distribution of numerical data. It consists of a series of rectangles whose areas are proportional to the frequencies of a variable.



• Frequency Curve as a limiting form: A frequency curve is considered a limiting form because it is a smoothed version of a frequency polygon or histogram. As the number of data points increases, the frequency curve becomes smoother and approaches a continuous curve.



• Relationship between Frequency Curve, Frequency Polygon, and Histogram: Both frequency polygon and histogram are used to represent the distribution of data in a visual way, while the frequency curve is a more refined and smooth version of these representations.

• Mixed: This type of frequency curve shows a mix of different patterns and shapes, making it difficult to identify a specific trend or pattern.


• Inverted J-shaped: This type of frequency curve starts low, rises sharply, and then drops off quickly, resembling the shape of an inverted letter J.


• U-shaped: This type of frequency curve forms a U shape, with low values at the beginning and end, and a peak in the middle.


• Bell-shaped: This type of frequency curve is symmetric and forms a bell shape, with a single peak in the middle and tapering off towards the ends.

• Type of frequency curve: Bell-shaped frequency curve

• Bell-shaped frequency curve: The distribution of profits of a company follows a bell-shaped frequency curve. This means that most of the profits are concentrated around the average value, with fewer profits at the extreme ends.


• Normal distribution: A bell-shaped frequency curve is also known as a normal distribution. It is a common pattern found in various natural phenomena and statistical data.


• Characteristics: The bell-shaped curve is symmetrical around the mean value of profits, with a gradual decrease in frequency as we move away from the mean in either direction.


• Use in business: Understanding the distribution of profits in a company can help in making strategic decisions related to pricing, marketing, and financial planning.


• Conclusion: Therefore, the distribution of profits of a company following a bell-shaped frequency curve indicates a stable and predictable pattern in the allocation of profits.

Calculation:

• Out of 1000 persons, 25% were industrial workers. So, the number of industrial workers = 25% of 1000 = 250.


• The rest were agricultural workers, so the number of agricultural workers = 1000 - 250 = 750.


• 300 persons enjoyed world cup matches on TV.


• 30% of the people who had not watched world cup matches were industrial workers. So, the number of people who had not watched world cup matches = 1000 - 300 = 700.


• Number of industrial workers who had not watched world cup matches = 30% of 700 = 210.


• Number of agricultural workers who had enjoyed world cup matches on TV = Total number of people who enjoyed world cup matches - Number of industrial workers who had enjoyed world cup matches = 300 - 210 = 90.

Therefore, the number of agricultural workers who had enjoyed world cup matches on TV is 90.

• Total women in the area: 40%


• Total coffee drinkers: 45%


• Male coffee drinkers: 20%

• Since the total women in the area are 40%, the total men in the area would be 60%.


• Out of the total population, 45% are coffee drinkers.


• Out of the total coffee drinkers, 20% are men. Therefore, the percentage of women who are coffee drinkers would be 25% (45% - 20%).


• Now, to find the percentage of female non-coffee drinkers, we subtract the percentage of female coffee drinkers from the total percentage of women in the area. So, 40% - 25% = 15%.

• The percentage of female non-coffee drinkers in the area is 15%.

• Raw Materials: 12 units out of total 90 units (12+20+35+23)


• Labour: 20 units out of total 90 units


• Direct Production: 35 units out of total 90 units


• Others: 23 units out of total 90 units

• Central angle = (Cost of component / Total cost) * 360 degrees

• Find the central angles for the largest and smallest components


• Subtract the smaller angle from the larger angle to get the difference


• Central angle for the largest component is for Direct Production = (35/90) * 360 = 140 degrees


• Central angle for the smallest component is for Raw Materials = (12/90) * 360 = 48 degrees


• Difference = 140 - 48 = 92 degrees

see the data of less than or equal to 3
ie , 0 , 1 ,2  ,3
15+19+22+31=87

greater than1500  
36 + 7 = 43 
so percentage 43/86 = 50 %

no of students who got marks more than 30 = Below 50 - Below 30
just do 100 - 65 =   35

The correct option is Option A.

1995

70% male students = 2100 (70% of 3000)

30% female students = 900

65% read commerce = 1950

35% read science = 1050

20% of female read science = 180

80% of female read commerce = 720

Male students read commerce = 1230 (1950 - 720)

Male students read science = 870

2000

75% male students = 2700 (75% of 3600)

25% female students = 900

40% read science = 1440

60% read science = 2160

50% of male students read commerce = 1350

50% of male students read science = 1350

Female students of science = 90

Female students of commerce = 810

x = female commerce students / female science students 

   = 720 + 810 / 180 + 90 

   = 1530 / 270 

   = 5.666

y = male commerce students / female commerce students 

   = 1230 + 1350 / 870 + 1350

   = 2580 / 2220

   = 1.162

So, x > y 

Hence proved

• Total female workers = 20% of total employees


• Married female workers = 40% of total female workers


• Female non-Trade Union members = 30


• Total male workers = 80% of total employees


• Male Trade Union members = 500


• Married male workers = 50% of total male workers


• Unmarried non-member male employees = 60


• Unmarried non-member employees = 80

• Total married male workers = 50% of total male workers = 0.5 * 600 = 300


• Married male non-members = Total married male workers - Male Trade Union members = 300 - 500 = -200

• Total married female workers = 40% of total female workers = 0.4 * 120 = 48


• Married female non-members = Total married female workers - Female non-Trade Union members = 48 - 30 = 18

• Ratio = Married Male Non-Members / Married Female Non-Members = 200 / 18 = 100 / 9 = 11.11