Question

The mean height of 25 boys in a class is \(150 \mathrm{~cm}\), and the mean height of 35 girls in the same class is \(145 \mathrm{~cm}\). The combined mean height of 60 students in the class is (approximately). (1) 143 (2) 146 (3) 147 (4) 145

Short answer

Answer: The approximate combined mean height of 60 students is 147 cm.

Step-by-step solution

Understanding the mean formula

The mean height of a group of people can be calculated by dividing the total height of the group (sum of individual heights) by the total number of people in the group.

Calculate the total height of boys

We are given the mean height of 25 boys is 150 cm. So, we can calculate the total height of all boys using the formula: total height = mean height * number of boys. Here, total height of boys = 150 * 25 = 3750 cm.

Calculate the total height of girls

Similarly, we know the mean height of 35 girls is 145 cm. Using the formula mentioned in Step 2: total height of girls = 145 * 35 = 5075 cm.

Calculate the combined total height and number of students

Now, we can add the total height of boys and girls to find the combined total height of 60 students. Combined total height = total height of boys + total height of girls = 3750 + 5075 = 8825 cm. The combined number of students in the class is 25 boys + 35 girls = 60 students.

Calculate the combined mean height

Finally, we can calculate the combined mean height by dividing the combined total height by the combined number of students: combined mean height = combined total height / combined number of students = 8825 / 60 ≈ 147 cm. So, the approximate combined mean height of 60 students in the class is 147 cm, which corresponds to option (3).

Key concepts

Mean Height

The mean height is a simple yet powerful concept used to describe the average height of a group of people. To determine the mean height, you sum up the individual heights and divide by the number of people in that group. This gives you a single number that represents a typical value for the dataset. For example, if you know that the mean height of a group of 25 boys is 150 cm, this suggests that 150 cm is representative of their height. In the context of the given exercise, understanding how to calculate mean is crucial as it sets the groundwork for calculating combined means.

Combined Mean

The combined mean is used when you have more than one group and you want to find the overall average for everyone involved. It helps in finding the average when you mix different groups together.
To find it, first calculate the total height of each group separately. Then, sum these totals, and finally, divide by the overall number of individuals across all groups.

• In the exercise, the boys and girls form two separate groups, and their heights are averaged to find their individual means.
• Next, these means are used to calculate the total heights for each group, which are then summed together.

This process helps derive a mean which accurately represents the height of all 60 students together.

Total Height of a Group

The total height of a group refers to the sum of the individual heights of everyone in that group. Knowing the total height is essential because it's a step needed to find the mean height. In our exercise, we start by calculating the total height from the mean height and number of individuals.

• Given that the mean height of the 25 boys is 150 cm, their total height is calculated as 150 cm (mean) * 25 (boys), resulting in 3750 cm.
• Similarly, the mean height of the 35 girls is 145 cm which leads to a total height of 145 cm * 35, equalling 5075 cm.

Adding these two totals will give the combined total height, critical for finding the combined mean.

Calculation with Different Sample Sizes

Calculating means when dealing with different sample sizes can be a bit tricky. It requires giving proper weight to each group based on its size which ensures a fair representation.
In this exercise, we handle different sample sizes fairly by calculating the total height and using these totals for finding means, regardless of how many individuals are in each group.

• The boys' group of 25 and the girls' group of 35 have different numbers, yet each of their contributions to the combined total height is crucial.
• After calculating the combined total height, dividing by the total number of students (60) will give a combined mean that fairly represents all individuals.

This ensures that larger groups do not disproportionately influence the overall mean.