Samacheer Kalvi Class 8 Maths Solutions Chapter 6 Statistics Exercise 6.2

Tamilnadu Samacheer Kalvi 8th Maths Solutions Chapter 6 Statistics Ex 6.2

 

Question 1. Which of the following data can be represented in a histogram?
(i) The number of mountain climbers in the age group 20 to 60 in TamilNadu.
(ii) Production of cycles in different years.
(iii) The number of students in each class of a school.
(iv) The number votes polled from 7 am to 6 pm in an election.
(v) The wickets fallen from 1 over to 50th over in a one day cricket match.
Answer:
(i) Yes
(ii) No
(iii) No
(iv) Yes
(v) Yes. A histogram is best for data that can be grouped into continuous ranges, like age or time. When data has distinct, separate categories, a bar graph is usually a better choice.
In simple words: Histograms work well when numbers change smoothly, like ages or election times. They don't work for things like specific years or different classes, which are not continuous.

🎯 Exam Tip: Remember that histograms are used for continuous data (like age groups, height ranges), while bar graphs are for discrete data (like number of students in a class, production in specific years).

 

Question 2. Fill in the blanks:
(i) The total area of the histogram is _______ to the total frequency of the given data.
(ii) A graph that displays data that changes continuously over the periods of time is _______.
(iii) Histogram is a graphical representation of _______ data.
Answer:
(i) proportional
(ii) Histogram
(iii) grouped. Histograms visually show how often different values appear in grouped data, making it easy to see patterns.
In simple words: For histograms, the total space inside the bars matches the total count of all data. A histogram shows numbers that keep changing smoothly. It is a picture of data that has been put into groups.

🎯 Exam Tip: Understand that the area of a histogram bar is proportional to the frequency of that class interval, and the total area represents the total frequency.

 

Question 3. In a village, there are 570 people who have cell phones. An NGO survey their cell phone usage. Based on this survey a histogram is drawn. Answer the following questions.
(i) How many people use the cell phone for less than 3 hours?
(ii) How many of them use the cell phone for more than 5 hours?
(iii) Are people using cell phone for less than 1 hour?
Answer:
(i) People using less than 3 hours means those in the 0-1 hour and 1-2 hour intervals. From the histogram, this is \( 110 + 220 = 330 \) people.
(ii) People using more than 5 hours means those in the 5-6 hour and 6-7 hour intervals. From the histogram, this is \( 100 + 50 = 150 \) people.
(iii) No. The histogram starts from 1 hour, meaning there is no data shown for usage less than 1 hour. This indicates no one in the survey used a phone for less than 1 hour.
In simple words: Look at the graph. For less than 3 hours, add the bars before the 3-hour mark. For more than 5 hours, add the bars after the 5-hour mark. If a time range has no bar, it means no one used it for that amount of time.

🎯 Exam Tip: When reading histograms, carefully check the x-axis for class intervals and the y-axis for frequencies to ensure you add up the correct bars for each question.

 

Question 4. Draw a histogram for the following data.

Answer: To draw the histogram, we use the class intervals on the X-axis and the number of students on the Y-axis. The data is a continuous frequency distribution, which is suitable for a histogram. Each bar will touch the next, showing the continuous nature of the data.
In simple words: We put the class intervals (like 0-10) on the bottom line (X-axis) and the number of students on the side line (Y-axis). Then, draw tall bars for each interval, making sure they touch each other.

🎯 Exam Tip: When drawing a histogram, ensure that there are no gaps between the bars for continuous data, and labels for both axes are clear.

 

Question 5. Construct a histogram from the following distribution of total marks of 40 students in a class.

Answer: The given data is a continuous distribution of marks. We will use the marks on the X-axis and the number of students on the Y-axis to construct the histogram. Each class interval will have a bar whose height shows the number of students in that mark range. The bars will touch, as marks are continuous.
In simple words: We draw a histogram by putting marks on the bottom line and the number of students on the side line. For each range of marks, we draw a bar as tall as the number of students in that range.

🎯 Exam Tip: Always make sure your class intervals are continuous when drawing a histogram. If they are not, you might need to adjust them first.

 

Question 6. The distribution of heights (in cm) of 100 people is given below. Construct a histogram and the frequency polygon imposed on it.

Answer: The given distribution is discontinuous because there is a gap between the upper limit of one class and the lower limit of the next (e.g., 135 to 136). To convert it into a continuous distribution, we adjust the class intervals. We find the gap between adjacent intervals (136 - 135 = 1). We then subtract half of this gap (0.5) from the lower limit of each class and add half of the gap (0.5) to the upper limit of each class.
Lower boundary \( = \) lower limit \( - \frac{1}{2} \) (gap between adjacent class interval)
\( = 125 - \frac{1}{2} (1) = 124.5 \)
Upper boundary \( = \) Upper limit \( + \frac{1}{2} \) (gap between the adjacent class interval)
\( = 135 + \frac{1}{2} (1) = 135.5 \)
The new frequency table with continuous intervals is:We then draw the histogram using these new continuous height intervals on the X-axis and frequency on the Y-axis. To superimpose the frequency polygon, we find the midpoint of the top of each histogram bar and connect these midpoints with line segments. We also extend the polygon to the midpoints of imaginary classes at each end to bring it to the X-axis.
In simple words: First, we fix the height ranges so there are no gaps between them. We do this by taking half the gap between ranges and adding/subtracting it. Then, we draw the histogram with these new ranges. For the frequency polygon, we put dots in the middle of the top of each bar and connect them with lines.

🎯 Exam Tip: Always adjust discontinuous class intervals to continuous ones before drawing a histogram or frequency polygon. This ensures accuracy and proper graphical representation.

 

Question 7. In a study of dental problem, the following data were obtained. Represent the above data by a frequency polygon.

Answer: To draw a frequency polygon, we first need to find the midpoint of each class interval. The midpoint is calculated as (Lower limit + Upper limit) / 2. Then, we plot these midpoints against their corresponding frequencies.
Finding the midpoints of the class interval we get:The points to be plotted are A(5,5), B(15, 13), C(25, 25), D(35, 14), E(45, 30), G(65, 43), H(75, 50). To complete the frequency polygon, we add two imaginary classes, one before the first and one after the last, both with a frequency of zero. This allows the polygon to start and end on the X-axis, forming a closed shape. In this case, an imagined class of 80-90 is used, extending the polygon to its midpoint on the X-axis.
In simple words: First, for each age group, find the middle number. Then, plot these middle numbers with the number of patients on a graph. Connect these dots with straight lines. To make it a full shape, add two extra points at the very beginning and end, on the bottom line, where the count is zero.

🎯 Exam Tip: Always find the class midpoints correctly for each interval to plot the frequency polygon accurately. Remember to connect the polygon to the X-axis by including midpoints of zero-frequency classes at both ends.

 

Question 8. The marks obtained by 50 students in Mathematics are given below (j) Make a frequency distribution table taking a class size of 10 marks (ii) Draw a histogram and a frequency polygon.

Answer: First, we need to find the highest and lowest marks from the given data to determine the range.
Minimum marks obtained = 08
Maximum marks obtained = 89
Range = Maximum marks – Minimum marks
\( = 89 - 08 = 81 \)
Given a class size of 10 marks, we can calculate the number of possible intervals:
Number of possible intervals \( = \frac{\text{Range}}{\text{Class size}} \)
\( = \frac{81}{10} = 8.1 \approx 9 \) intervals.
Next, we create a frequency distribution table with class intervals of size 10 and use tally marks to count the number of students in each interval:Now we have the continuous frequency table for drawing the histogram:We will draw the histogram taking class interval in x-axis and frequency in y-axis. The frequency polygon is then drawn by connecting the midpoints of the top of each histogram bar, extending to the x-axis at both ends with imagined classes having zero frequency.
In simple words: First, find the smallest and largest marks. Then, divide the marks into groups of 10. Count how many students fall into each group using tally marks to make a table. Use this table to draw the histogram (bars touching each other) and a frequency polygon (connecting the middle tops of the bars with lines).

🎯 Exam Tip: When given raw data, always calculate the range and choose appropriate class intervals to create a meaningful frequency distribution table before drawing any graphs.

 

Question 9. Data is a collection of _____
(a) numbers
(b) words
(c) measurements
(d) all of the options
Answer: (d) all of the options
In simple words: Data can be any kind of information, like numbers, words, or measurements. It's just facts or details.

🎯 Exam Tip: Remember that "data" is a broad term that includes all forms of information gathered for study, not just numerical values.

 

Question 10. The number of times an observation occurs in the given data is called _____
(a) tally marks
(b) data
(c) frequency
(d) none of the options
Answer: (c) frequency
In simple words: When we count how often something appears in our data, that count is called its frequency.

🎯 Exam Tip: Frequency is a fundamental concept in statistics, referring to how often each value or group of values appears in a dataset.

 

Question 11. The difference between the largest value and the smallest value of the given data is _____
(a) range
(b) frequency
(c) variable
(d) none of the options
Answer: (a) range
In simple words: The range tells us how spread out our data is, from the smallest number to the largest number.

🎯 Exam Tip: The range is a simple measure of data dispersion, found by subtracting the minimum value from the maximum value.

 

Question 12. The data that can take values between a certain range is called _____
(a) ungrouped
(b) grouped
(c) frequency
(d) none of the options
Answer: (b) grouped
In simple words: When we put data into categories or sections based on their values, it is called grouped data.

🎯 Exam Tip: Grouped data is used to summarize large datasets by dividing observations into class intervals.

 

Question 13. Inclusive series is a _____ series.
(a) continuous
(b) discontinuous
(c) both
(d) none of the options
Answer: (b) discontinuous
In simple words: In an inclusive series, there are small gaps between the end of one group and the start of the next group. This means the numbers don't flow smoothly.

🎯 Exam Tip: In an inclusive series, the upper limit of one class interval does not overlap with the lower limit of the next, leading to gaps in the data representation.

 

Question 14. In a class interval the upper limit of one class is the lower limit of the other class. This is _____ series.
(a) Inclusive
(b) exclusive
(c) ungrouped
(d) none of the options
Answer: (b) exclusive
In simple words: An exclusive series is where the end of one group is the exact same as the start of the next group, so there are no gaps.

🎯 Exam Tip: Exclusive series are continuous, with class intervals like 0-10, 10-20, where the upper limit is excluded from the first class and included in the next.

 

Question 15. The graphical representation of ungrouped data is _____
(a) histogram
(b) frequency polygon
(c) pie chart
(d) all of the options
Answer: (c) pie chart
In simple words: A pie chart is good for showing parts of a whole, like how different categories share a total amount, especially for data that is not grouped.

🎯 Exam Tip: Pie charts are best for displaying proportions of a whole for categorical or ungrouped data, where each slice represents a category's share.

 

Question 16. Histogram is a graph of a _____ frequency distribution.
(a) continuous
(b) discontinuous
(c) discrete
(d) none of the options
Answer: (a) continuous
In simple words: Histograms are used for numbers that can be any value within a range, like heights or temperatures, showing how often they appear in a smooth way.

🎯 Exam Tip: Always remember that histograms are specifically designed for continuous data and continuous frequency distributions, unlike bar graphs which handle discrete data.

 

Question 17. A _____ is a line graph for the graphical representation of the continuous frequency distribution.
(a) frequency polygon
(b) histogram
(c) pie chart
(d) bar graph
Answer: (a) frequency polygon
In simple words: A frequency polygon uses lines to connect dots that show how often different numbers appear in a continuous set of data.

🎯 Exam Tip: A frequency polygon provides a clear visual representation of the shape of a distribution and can be drawn by connecting the midpoints of the tops of histogram bars.

 

Question 18. The graphical representation of grouped data is _____
(a) bar graph
(b) pictograph
(c) pie chart
(d) histogram
Answer: (d) histogram
In simple words: When we put data into groups, a histogram is the best picture to show how much is in each group, with bars touching each other.

🎯 Exam Tip: Histograms are uniquely suited for visualizing the distribution of grouped data, especially when the data is continuous.