Directions: for the following three (03) items: Read the following information and answer the three items that follow: Marks Number of students Physics Mathematics 10 – 20 8 10 20 – 30 11 21 30 – 40 30 38 40 – 50 26 15 50 – 60 15 10 60 - 70 10 6
What is the mean of marks in Physics?
40.9
Calculating the Mean of Physics Marks from Grouped Data
To find the mean of marks from a frequency distribution table like the one provided for Physics marks, we follow these steps:
- Identify the class intervals and their corresponding frequencies (number of students).
- Calculate the midpoint for each class interval. The midpoint is the average of the lower and upper limits of the class.
- Multiply the frequency of each class by its midpoint.
- Sum all the products calculated in step 3 (\(\Sigma f_i x_i\)).
- Sum all the frequencies (\(\Sigma f_i\)).
- Divide the sum of products by the sum of frequencies to get the mean using the formula: \( \text{Mean} = \frac{\Sigma f_i x_i}{\Sigma f_i} \)
Let's apply these steps to the Physics marks data provided:
| Marks (Class Interval) | Number of Students (\(f_i\)) | Midpoint (\(x_i\)) | \(f_i \times x_i\) |
|---|---|---|---|
| 10 – 20 | 8 | \( \frac{10+20}{2} = 15 \) | \( 8 \times 15 = 120 \) |
| 20 – 30 | 11 | \( \frac{20+30}{2} = 25 \) | \( 11 \times 25 = 275 \) |
| 30 – 40 | 30 | \( \frac{30+40}{2} = 35 \) | \( 30 \times 35 = 1050 \) |
| 40 – 50 | 26 | \( \frac{40+50}{2} = 45 \) | \( 26 \times 45 = 1170 \) |
| 50 – 60 | 15 | \( \frac{50+60}{2} = 55 \) | \( 15 \times 55 = 825 \) |
| 60 – 70 | 10 | \( \frac{60+70}{2} = 65 \) | \( 10 \times 65 = 650 \) |
| Total | \( \Sigma f_i = 8+11+30+26+15+10 = 100 \) | \( \Sigma f_i x_i = 120+275+1050+1170+825+650 = 4090 \) |
Now, we apply the mean formula:
\( \text{Mean} = \frac{\Sigma f_i x_i}{\Sigma f_i} = \frac{4090}{100} = 40.9 \)
The mean of the marks in Physics is 40.9.
Revision Table: Key Statistics Terms
| Term | Definition | Relevance in Question |
|---|---|---|
| Mean | The average of a dataset, calculated by summing all values and dividing by the number of values. For grouped data, it's the sum of (frequency × midpoint) divided by the sum of frequencies. | The question asks for the mean of Physics marks. |
| Frequency Distribution | A table that shows how often each value or range of values occurs in a dataset. | The provided data for Physics and Mathematics is in frequency distribution format. |
| Class Interval | A range of values within a frequency distribution, like 10-20 or 20-30 in the marks data. | The marks are given in class intervals. |
| Frequency | The number of times a value or a value within a class interval occurs in the dataset. | The number of students in each marks range represents the frequency. |
| Midpoint | The central value of a class interval, used to represent the interval in calculations for grouped data mean. | Calculated for each marks interval to find the mean. |
Additional Information on Grouped Data Statistics
Besides the mean, other statistical measures like Median and Mode can also be calculated for grouped data, although the methods are slightly different from those used for ungrouped data.
- Median for Grouped Data: The median is the middle value. For grouped data, it is estimated using a formula that involves the median class (the class containing the \(\frac{N}{2}\)-th observation), the lower boundary of the median class, the cumulative frequency of the class before the median class, the frequency of the median class, and the class width.
- Mode for Grouped Data: The mode is the value that appears most frequently. For grouped data, the mode is estimated using a formula that involves the modal class (the class with the highest frequency), the lower boundary of the modal class, the frequencies of the modal class and the classes immediately before and after it, and the class width.
Understanding these measures helps in getting a complete picture of the distribution of data, such as the Physics marks in this case.
Similar Questions
A random sample of 20 people is classified in the following table according to their ages:
Age
Frequency
15 – 25
2
25 – 35
4
35 – 45
6
45 – 55
5
55 - 65
3
What is the mean age of this group of people?
If the mode of the scores 10, 12, 13, 15, 15, 13, 12, 10, x is 15, then what is the value of x?
The numbers 4 and 9 have frequencies x and (x - 1) respectively. If their arithmetic mean is 6, then what is the value of x?
If M is the mean of n observations x 1- k, x 2- k, x 3- k, _ _ _, x n- k, where k is any real number, then what is the mean of x 1, x 2, x 3, _ _ _, x n?
The following tables gives the frequency distribution of number of peas per pea pod of 198 pods:
Number of peas
1
2
3
4
5
6
7
Frequency
4
33
76
50
26
8
1
What is the median of this distribution?Consider the following discrete frequency distribution:
What is the value of median of the distribution?x
1
2
3
4
5
6
7
8
f
3
15
45
57
50
36
25
9
What is the mean of the marks ?
What is the median of the marks?
What is the sum of the deviations measured from the median?
The observations 4, 1, 4, 3, 6, 2, 1, 3, 4, 5, 1, 6 are outputs of 12 dices thrown simultaneously. If m and M are means of lowest 8 observations and highest 4 observations respectively, then what is (2m + M) equal to ?
Important Questions from Measures of Central Tendency
A random sample of 20 people is classified in the following table according to their ages:
Age
Frequency
15 – 25
2
25 – 35
4
35 – 45
6
45 – 55
5
55 - 65
3
What is the mean age of this group of people?
If the difference of mode and median is 36, then the difference of median and mean is:
In a Mathematics test 15 students scored 80 marks, 20 students scored 75 marks, 28 students scored 65 marks and 25 students scored 60 marks, mode of the score is:
If the mode of the scores 10, 12, 13, 15, 15, 13, 12, 10, x is 15, then what is the value of x?
The average of 25 numbers is 36. If the average of first 13 numbers is 32 and that of the last 13 numbers is 39, then the 13th number is
