Let’s start working on the questions in exercise 3.2 of class 7, Data Handling of NCERT book. In this chapter, we will solve questions on how to find mean, median, and mode. Let’s look at some of the basics before going into further details.
The mean, median, and mode measures are a data set’s key aspects. The mean is the average, calculated by adding all the numbers in a data set and then dividing by the count of numbers. The median is the middle value when the data is ordered from smallest to largest. If there is an even number of values, it is calculated by taking the average of the two middle numbers. The mode is the most frequently occurring number in the data set.
Class 8 Maths for Chapter 3 Exercise 3.2 Data Handling – NCERT Book Solutions
1. The scores in mathematics test (out of 25) of 15 students is as follows: 19, 25, 23, 20, 9, 20, 15, 10, 5, 16, 25, 20, 24, 12, 20
Find the mode and median of this data. Are they same?
Solution:
To find the mode and median of the given data, we first arrange the scores in ascending order and then calculate the mode (the most frequently occurring score) and the median (the middle score in the ordered list).
Given Data:
Scores (out of 25) of 15 students: 19, 25, 23, 20, 9, 20, 15, 10, 5, 16, 25, 20, 24, 12, 20
Arranged Data in Ascending Order:
5, 9, 10, 12, 15, 16, 19, 20, 20, 20, 20, 23, 24, 25, 25
Finding the Mode:
The score that appears most frequently is 20, occurring four times.
Therefore, the mode of the data is 20.
Finding the Median:
There are 15 scores, so the median is the middle score. The formula to find the median position in an ordered list is (Number of data points + 1) / 2.
For 15 data points, the median position = (15 + 1) / 2 = 16 / 2 = 8th score.
Looking at the ordered data, the 8th score is 20.
Therefore, the median of the data is 20.
Comparing the Mode and Median:
Both the mode and median of this data are the same, 20.
2. The runs scored in a cricket match by 11 players is as follows: 6, 15, 120, 50, 100, 80, 10, 15, 8, 10, 15
Find the mean, mode and median of this data. Are the three same?
To find the mean, mode, and median of the given cricket match data, we first calculate each of these measures of central tendency and then compare them to see if they are the same.
Solution:
Given Data:
Runs scored by 11 players: 6, 15, 120, 50, 100, 80, 10, 15, 8, 10, 15
Finding the Mean (Average):
The mean is the total runs divided by the number of players.
Total runs = 6 + 15 + 120 + 50 + 100 + 80 + 10 + 15 + 8 + 10 + 15 = 429
Number of players = 11
Mean = Total runs / Number of players = 429 / 11 ˜ 39
Finding the Mode:
The mode is the score that appears most frequently. In this data, 15 appears three times, which is more frequent than any other score.
Therefore, the mode of the data is 15.
Finding the Median:
First, arrange the data in ascending order: 6, 8, 10, 10, 15, 15, 15, 50, 80, 100, 120
There are 11 scores, so the median is the middle score. The formula to find the median position is (Number of data points + 1) / 2.
Median position = (11 + 1) / 2 = 12 / 2 = 6th score
Looking at the ordered data, the 6th score is 15.
Therefore, the median of the data is 15.
Comparing the Mean, Mode, and Median:
The mean is approximately 39, the mode is 15, and the median is also 15. Therefore, the mode and median are the same, but the mean is different.
3. The weights (in kg.) of 15 students of a class are: 38, 42, 35, 37, 45, 50, 32, 43, 43, 40, 36, 38, 43, 38, 47
(i) Find the mode and median of this data.
(ii) Is there more than one mode?
Solution:
To analyze the weights of the 15 students, we’ll first find the mode and median of the data and then check if there’s more than one mode.
Given Data:
Weights of 15 students (in kg): 38, 42, 35, 37, 45, 50, 32, 43, 43, 40, 36, 38, 43, 38, 47
Finding the Mode:
The mode is the weight that appears most frequently. In this data, 38 and 43 both appear three times.
Therefore, the modes of the data are 38 and 43, indicating there are two modes.
Finding the Median:
First, arrange the data in ascending order: 32, 35, 36, 37, 38, 38, 38, 40, 42, 43, 43, 43, 45, 47, 50
There are 15 data points, so the median is the middle value. The formula to find the median position is (Number of data points + 1) / 2.
Median position = (15 + 1) / 2 = 16 / 2 = 8th value
Looking at the ordered data, the 8th value is 40.
Therefore, the median of the data is 40 kg.
(i) The mode is 38 and 43, and the median is 40.
(ii) Yes, there is more than one mode in this data set.
4. Find the mode and median of the data: 13, 16, 12, 14, 19, 12, 14, 13, 14
To find the mode and median of the given data, we first arrange the numbers in order and then identify the most frequently occurring number (mode) and the middle number (median).
Solution:
Given Data:
13, 16, 12, 14, 19, 12, 14, 13, 14
Finding the Mode:
The mode is the number that appears most frequently. In this data, the number 14 appears three times, more frequently than any other number.
Therefore, the mode of the data is 14.
Finding the Median:
First, arrange the data in ascending order: 12, 12, 13, 13, 14, 14, 14, 16, 19
There are 9 data points, so the median is the middle value. The median position for an odd number of data points is (Number of data points + 1) / 2.
Median position = (9 + 1) / 2 = 10 / 2 = 5th value
Looking at the ordered data, the 5th value is 14.
Therefore, the median of the data is 14.
The mode of the data is 14, and the median is also 14.
5. Tell whether the statement is true or false:
(i) The mode is always one of the numbers in a data.
(ii) The mean is one of the numbers in a data.
(iii) The median is always one of the numbers in a data.
(iv) The data 6, 4, 3, 8, 9, 12, 13, 9 has mean 9.
Solution:
We’ll evaluate each statement to determine if it’s true or false based on the properties of mode, mean, and median.
(i) The mode is always one of the numbers in a data.
The mode is defined as the number that appears most frequently in a data set. Therefore, it must be a number from the data set.
Statement (i) is True.
(ii) The mean is one of the numbers in a data.
The mean is the average of all numbers in a data set. It may or may not be a number that is actually in the data set.
Statement (ii) is False.
(iii) The median is always one of the numbers in a data.
The median is the middle value when a data set is ordered. For an odd number of data points, it is one of the numbers in the data set. For an even number of data points, it is the average of the two middle numbers, which may not be a number in the data set.
Statement (iii) is False.
(iv) The data 6, 4, 3, 8, 9, 12, 13, 9 has mean 9.
To verify this, calculate the mean: (6 + 4 + 3 + 8 + 9 + 12 + 13 + 9) / 8 = 64 / 8 = 8.
Statement (iv) is False. The mean of the data is 8, not 9.
Worksheet for Exercise 3.2 Data Handling
- The scores in mathematics test (out of 25) of 15 students is as follows: 19, 25, 23, 20, 9, 20, 15, 10, 5, 16, 25, 20, 24, 12, 20. Find the mode and median of this data. Are they the same?
- The runs scored in a cricket match by 11 players is as follows: 6, 15, 120, 50, 100, 80, 10, 15, 8, 10, 15. Find the mean, mode, and median of this data. Are the three the same?
- The weights (in kg) of 15 students of a class are: 38, 42, 35, 37, 45, 50, 32, 43, 43, 40, 36, 38, 43, 38, 47. (i) Find the mode and median of this data. (ii) Is there more than one mode?
- Find the mode and median of the data: 13, 16, 12, 14, 19, 12, 14, 13, 14.
- Tell whether the statement is true or false: (i) The mode is always one of the numbers in a data. (ii) The mean is one of the numbers in a data. (iii) The median is always one of the numbers in a data. (iv) The data 6, 4, 3, 8, 9, 12, 13, 9 has mean 9.
Answers
- Mode: 20, Median: 19, They are not the same.
- Mean: 39.27, Mode: 15, Median: 15, They are not the same.
- (i) Mode: 38 and 43, Median: 38. (ii) Yes, there are two modes.
- Mode: 14, Median: 14.
- (i) True. (ii) False. (iii) False. (iv) False.
