The mean height of 25 boys in a class is 150 cm, and the mean height of 35 girls in the same class is 145 cm. The combined mean height of 60 students in the class is ______ cm (approximately).
147
Calculating Combined Mean Height in Statistics
This problem asks us to find the average height of all the students in a class when we know the average height and the number of boys and girls separately. This is a common type of problem involving the concept of a combined mean.
We are given the following information about the students in the class:
- Number of boys ($n_b$): 25
- Mean height of boys ($\bar{x}_b$): 150 cm
- Number of girls ($n_g$): 35
- Mean height of girls ($\bar{x}_g$): 145 cm
- Total number of students ($N$): $n_b + n_g = 25 + 35 = 60$
To find the combined mean height of all 60 students, we need to use the formula for the combined mean of two groups.
The formula for the combined mean ($\bar{X}$) is:
\( \bar{X} = \frac{(n_b \times \bar{x}_b) + (n_g \times \bar{x}_g)}{n_b + n_g} \)
This formula essentially calculates the total sum of heights for all students (boys and girls combined) and then divides by the total number of students.
Step-by-Step Combined Mean Calculation
Let's plug the given values into the formula:
- Calculate the sum of heights for the boys: \( n_b \times \bar{x}_b = 25 \times 150 \)
- Calculate the sum of heights for the girls: \( n_g \times \bar{x}_g = 35 \times 145 \)
- Calculate the total sum of heights: \( (25 \times 150) + (35 \times 145) \)
- Calculate the total number of students: \( 25 + 35 \)
- Divide the total sum of heights by the total number of students to get the combined mean.
Let's perform the calculations:
\( \text{Sum of heights for boys} = 25 \times 150 = 3750 \text{ cm} \)
\( \text{Sum of heights for girls} = 35 \times 145 \)
| Calculation | Result |
|---|---|
| \( 35 \times 145 \) | 5075 |
\( \text{Sum of heights for girls} = 5075 \text{ cm} \)
Now, calculate the total sum of heights:
\( \text{Total sum of heights} = 3750 + 5075 = 8825 \text{ cm} \)
The total number of students is \( 25 + 35 = 60 \).
Finally, calculate the combined mean height:
\( \bar{X} = \frac{8825}{60} \)
| Division | Result |
|---|---|
| \( \frac{8825}{60} \) | \( 147.0833... \) |
The calculated combined mean height is approximately \( 147.08 \) cm.
The question asks for the approximate combined mean height in cm. Looking at the options, 147 is the closest value to 147.0833...
Revision Table: Key Statistics Concepts
| Term | Definition | Formula/Calculation Example |
|---|---|---|
| Mean (Average) | The sum of all values divided by the number of values. | For values \(x_1, x_2, ..., x_n\): \( \bar{x} = \frac{\sum x_i}{n} \) |
| Combined Mean | The mean of a dataset formed by combining two or more groups, where the mean and size of each group are known. | For two groups: \( \bar{X} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2} \) |
| Sum of Values | For a group with mean \( \bar{x} \) and size \( n \), the sum of values is \( n \times \bar{x} \). | If \( n=10 \) and \( \bar{x}=5 \), sum is \( 10 \times 5 = 50 \). |
Additional Information on Combined Mean
The combined mean is a weighted average. The mean of each group is weighted by its size. Larger groups have a greater influence on the combined mean than smaller groups.
In this problem, the girls group is larger (35 girls) than the boys group (25 boys). Therefore, the combined mean height is expected to be closer to the mean height of the girls (145 cm) than the mean height of the boys (150 cm).
- Mean height of boys: 150 cm
- Mean height of girls: 145 cm
- Combined mean height: Approximately 147 cm
The combined mean (147 cm) is indeed closer to 145 cm than 150 cm, which makes sense because there are more girls than boys. This confirms our calculation is likely correct.
Understanding combined mean is important in statistics for calculating overall averages from subgroup data, which is common in surveys, research, and educational assessments.
Similar Questions
The mean of the 5 smallest numbers from a group is 15 while the mean of all the numbers of the group taken together is 17. If the mean of the numbers leaving the smallest five out is 18.25, how many numbers were there in the group in all?
The total weight of school bags of 35 students of a class was 449.75 kg. What is the average weight of each school bag if each student carried only one school bag?
- The mean of three numbers is 15. The range of this data set is 8 while the difference between the two smallest numbers is 1. The greatest of the three numbers is:
Four numbers, when arranged in ascending order, are w, x, y and z. The average of the smallest three numbers is 18, while the average of the largest three was 22. What is the range of the data?
Samit was given some money to take care of his travel during a 6-day sales drive he had to undertake. However, he had to increase his stay by another 4 days and as a result his average daily travel allowance went down by Rs. 56. What was the amount that was sanctioned to him in the beginning?
The average of three numbers is 6. The average of the first two is 5 while the average of the last two is 8. The three numbers are:
The average marks obtained by Raghav in 15 tests are 25. Zubeida has maintained an average of 23 so far but has taken only 10 of the tests. If each test is out of 30, how much must Zubeida score on average in the remaining 5 tests to still have a chance to match Raghav’s performance?
What is the mean of first 60 natural numbers?
What is the mean of first hundred natural numbers?
The average marks obtained by Raghav in 12 tests is 24. Zubeida has maintained an average of 23 so far but has taken only 9 of the tests. If each test is out of 30, at least how much must Zubeida score in any one of the remaining three tests to still have a chance to match Raghav’s performance?
Important Questions from Average
Average of 40 numbers is 71, if the number 100 replaced by 140, then average is increased by
The captain of a football team of 11 members is 28 years old and the goalkeeper is 4 years older than him. If the ages of these two are removed, then the average age of the remaining players is two years less than the average age of the whole team. What is the average age of the team?
There are two Classes A and B having 25 and 30 students respectively. In Class-A the highest score is 21 and lowest score is 17. In Class-B the highest score is 30 and lowest score is 22. Four students are shifted from Class-A to Class-B.
Consider the following statements:
1. The average score of Class-B will definitely decrease.
2. The average score of Class-A will definitely increase.
Which of the above statements is/are correct?
The average weight of A, B, Cis 40 kg, the average weight of B, D, Eis 42 kg and the weight of Fis equal to that of B. What is the average weight of A, B, C, D, Eand F?
A cow costs more than 4 goats but less than 5 goats. If a goat costs between Rs. 600 and Rs. 800, which of the following is a most valid conclusion?
