SUMMARY: The chapter on Statistics in Class 9 Mathematics introduces students to the collection, presentation, and analysis of data. KEY TOPICS: data collection, frequency distribution, graphical representation of data, bar graphs, histograms, frequency polygons, measures of central tendency, mean, median, mode
What is the mode of the following data set: 3, 7, 3, 2, 9, 3, 5?
A3
B2
C5
D7
Check answerHide answer
Correct answer: Option 1 — 3
Q21 Mark
In a frequency distribution, what does the term 'class interval' refer to?
AThe range of values in a data set
BThe number of times a value occurs
CThe midpoint of a group of data
DThe grouping of data into ranges
Check answerHide answer
Correct answer: Option 4 — The grouping of data into ranges
Q31 Mark
Which graphical representation is best suited for displaying the frequency distribution of continuous data?
ABar graph
BHistogram
CPie chart
DLine graph
Check answerHide answer
Correct answer: Option 2 — Histogram
Q41 Mark
If the mean of a data set is 20 and the sum of all observations is 100, how many observations are there in the data set?
A4
B5
C10
D15
Check answerHide answer
Correct answer: Option 2 — 5
Q51 Mark
In a given data set, if the median is 15 and the data is arranged in ascending order, which of the following statements is true?
AHalf of the data points are less than 15
BAll data points are greater than 15
CThe mode must be 15
DThe mean must be less than 15
Check answerHide answer
Correct answer: Option 1 — Half of the data points are less than 15
Short Answer Questions5 questions
Q63 Marks
What is the difference between primary data and secondary data in the context of data collection?
View sample solutionHide solution
Primary data is collected firsthand by the researcher for a specific purpose, while secondary data is information that has already been collected and published by others. Examples include surveys for primary data and government reports for secondary data.
Q73 Marks
Explain how to construct a frequency distribution table using a given set of data. Provide an example.
View sample solutionHide solution
To construct a frequency distribution table, first, organize the data into intervals (classes) and then count how many data points fall into each interval. For example, if the data set is {1, 2, 2, 3, 4, 4, 4, 5}, the intervals could be 1-2, 3-4, and 5, with frequencies of 3, 4, and 1 respectively.
Q83 Marks
What are bar graphs and how do they differ from histograms?
View sample solutionHide solution
Bar graphs are used to represent categorical data with rectangular bars, where the length of each bar is proportional to the value it represents. Histograms, on the other hand, are used for continuous data and display the frequency of data within certain ranges (bins) without gaps between the bars.
Q93 Marks
Define mean, median, and mode. How are they different from each other?
View sample solutionHide solution
Mean is the average of a data set, calculated by dividing the sum of all values by the number of values. Median is the middle value when the data is arranged in ascending order, while mode is the value that appears most frequently in the data set. They differ in their calculation and the type of data they best represent.
Q103 Marks
How can a frequency polygon be constructed from a frequency distribution table?
View sample solutionHide solution
To construct a frequency polygon, first plot the midpoints of each interval from the frequency distribution table on a graph, with the frequency on the vertical axis. Then, connect the points with straight lines. This visual representation helps in understanding the distribution of data more clearly.
Long Answer Questions5 questions
Q116 Marks
Explain the process of collecting data for a survey. What are the different methods of data collection, and how do they impact the reliability of the data?
View sample solutionHide solution
Data collection for a survey involves several steps, including defining the objective, selecting the target population, and choosing a method for data collection. Common methods include surveys, interviews, observations, and experiments. Each method has its strengths and weaknesses; for instance, surveys can reach a larger audience quickly but may suffer from response bias, while interviews provide in-depth information but are time-consuming. The reliability of the data can be affected by the method chosen, as well as the sample size and selection process. Ensuring a representative sample is crucial for the validity of the results.
Q126 Marks
Define frequency distribution and explain how to construct a frequency distribution table using a given set of data. Provide an example.
View sample solutionHide solution
A frequency distribution is a summary of how often each value occurs in a dataset. To construct a frequency distribution table, first, organize the data into classes or intervals. Then, count the number of observations that fall into each class and record these counts in the table. For example, if we have the data set: 2, 3, 3, 5, 5, 5, 7, we can create intervals like 2-3, 4-5, and 6-7. The frequency distribution table would show that the interval 2-3 has a frequency of 3, the interval 4-5 has a frequency of 3, and the interval 6-7 has a frequency of 1.
Q136 Marks
Discuss the different graphical representations of data, including bar graphs and histograms. How do these representations help in understanding data?
View sample solutionHide solution
Graphical representations of data, such as bar graphs and histograms, are essential for visualizing information. A bar graph displays categorical data with rectangular bars, where the length of each bar represents the frequency of each category. In contrast, a histogram is used for continuous data and shows the frequency distribution of numerical data by dividing it into intervals. These visual tools help in quickly identifying trends, patterns, and outliers in the data, making it easier for analysts to interpret and communicate findings effectively.
Q146 Marks
What are measures of central tendency? Explain how to calculate the mean, median, and mode of a given dataset with an example.
View sample solutionHide solution
Measures of central tendency are statistical measures that describe the center of a dataset. The mean is calculated by summing all values and dividing by the number of observations. The median is the middle value when the data is arranged in ascending order, and if there is an even number of observations, it is the average of the two middle values. The mode is the value that appears most frequently in the dataset. For example, for the dataset: 1, 2, 2, 3, 4, the mean is (1+2+2+3+4)/5 = 2. The median is 2, and the mode is also 2, as it appears most frequently.
Q156 Marks
Explain the concept of a frequency polygon and how it is constructed. What advantages does it have over other graphical representations?
View sample solutionHide solution
A frequency polygon is a graphical representation that uses line segments to connect points representing the frequencies of each class interval in a frequency distribution. To construct a frequency polygon, first, create a frequency distribution table, then plot the midpoints of each class interval against their corresponding frequencies. Finally, connect the points with straight lines. One advantage of a frequency polygon over other graphical representations, like histograms, is that it allows for easier comparison between different datasets, as multiple frequency polygons can be overlaid on the same graph, providing a clearer visual comparison.
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): The mean is always greater than or equal to the mode in a frequency distribution.
Reason (R): The mean is influenced by all values in the data set, while the mode is only the most frequently occurring value.
Show explanationHide explanation
Correct answer: Option 3 —
A is true, but R is false.
Q171 Mark
Assertion (A): A histogram can be used to represent categorical data.
Reason (R): Histograms display the frequency of numerical data in intervals.
Show explanationHide explanation
Correct answer: Option 4 —
A is false, but R is true.
Q181 Mark
Assertion (A): The median is the middle value of a data set when arranged in ascending order.
Reason (R): The median divides the data set into two equal halves.
Show explanationHide explanation
Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q191 Mark
Assertion (A): Frequency polygons can be constructed using the midpoints of the intervals of a histogram.
Reason (R): Frequency polygons provide a visual representation of the frequency distribution using line segments.
Show explanationHide explanation
Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q201 Mark
Assertion (A): Data collection is not necessary for statistical analysis.
Reason (R): Statistical analysis relies on data to draw conclusions and make decisions.
Show explanationHide explanation
Correct answer: Option 4 —
A is false, but R is true.
Statement-Based Questions5 questions
Q211 Mark
Statement 1: The mean is the sum of all observations divided by the number of observations.
Statement 2: The mode is the value that appears most frequently in a data set.
Show answerHide answer
Correct answer: Option 1 —
Both statements are true.
Q221 Mark
Statement 1: A histogram is used to represent categorical data.
Statement 2: A bar graph can represent both categorical and continuous data.
Show answerHide answer
Correct answer: Option 3 —
Only Statement 2 is true.
Q231 Mark
Statement 1: In a frequency distribution, the total frequency is always equal to the number of observations.
Statement 2: The median is the average of the highest and lowest values in a data set.
Show answerHide answer
Correct answer: Option 2 —
Only Statement 1 is true.
Q241 Mark
Statement 1: A frequency polygon is created by connecting the midpoints of the top of the bars in a histogram.
Statement 2: The range is calculated by subtracting the smallest value from the largest value in a data set.
Show answerHide answer
Correct answer: Option 1 —
Both statements are true.
Q251 Mark
Statement 1: Data can be collected through surveys and experiments.
Statement 2: The mode is always the largest value in a data set.
Show answerHide answer
Correct answer: Option 2 —
Only Statement 1 is true.
