Question

If $2,7,10,$ and 1 are the number of students who weigh (in $\mathrm{kg}$ ) $35,30,42,$ and $40,$ respectively, what is the mean weight of these 20 students?

Short answer

To find the mean weight, divide the total cumulative weight (2*35 + 7*30 + 10*42 + 1*40) by the total number of students (20).

Step-by-step solution

Determine the total weight of the students in each group

First, calculate the total weight for each group of students by multiplying the number of students by their respective weights. For the group weighing 35 kg: there are 2 students, so total weight is 2 * 35 kg. Repeat for each weight group.

Sum the total weights of all groups

Add the total weights from all groups to find the cumulative weight of all students. This is the sum of the products from Step 1.

Divide the cumulative weight by the total number of students

To find the mean weight, divide the cumulative weight of all students by the total number of students, which is 20.

Key concepts

Mean Calculation in Statistics

Understanding the concept of the mean, or average, is essential in statistics. It is the value obtained by dividing the sum of a set of numbers by the count of numbers in the set. When we talk about mean weight calculation specifically, it involves the same fundamental process. Imagine you want to know the average weight of a particular group. To do this, you must sum the total weight of all the individuals in the group and then divide that sum by the total number of individuals.

For example, if you had five friends who weighed 50, 60, 70, 80, and 90 kg, the mean weight would be the sum of these weights divided by 5, which would give us 70 kg. This number represents the central tendency of your friends' weights. The mean gives us an idea of the middle ground of our data set, which can be particularly useful for comparability and understanding the data distribution as a whole.

Cumulative Weight Computation

When dealing with sets of data where multiple entities share the same value, such as several students with the same weight, it may be more efficient to compute the cumulative weight rather than adding each individual's weight separately. This is done by multiplying the weight by the number of entities with that weight and then adding all the products together.

In practical terms, suppose you have three students, each weighing 40 kg. Instead of adding 40 kg three times, you would calculate the cumulative weight by multiplying 40 kg by the number of students: 3*40 kg, giving us 120 kg. This method saves time and simplifies the process when you're dealing with large groups or datasets. It's a basic principle of multiplication and addition that forms the foundation of cumulative computations in statistics.

Statistical Data Analysis

The process of exploring data to find underlying patterns and to assess how well these patterns hold up is known as statistical data analysis. It involves collecting, summarizing, and interpreting data to make informed decisions or predictions. In the context of our mean weight example, once we calculate the mean weight of the student group, we can use this information to analyze various aspects.

For instance, by knowing the average weight, schools could tailor their nutritional programs or PE activities to better suit the student body’s needs. In broader terms, this kind of analysis can inform health policies, market research, and much more. The power of statistical data analysis comes from its ability to turn raw numbers into actionable insights by using various methods, including mean calculation, to describe and infer patterns and trends.

Basic Statistical Concepts

Even the most complex statistical analyses are built on a foundation of basic statistical concepts. These include measures of central tendency like the mean, median, and mode, each of which provides different insights into the distribution of a dataset. Variability is another cornerstone, with the range, interquartile range, variance, and standard deviation telling us how spread out our data is.

Understanding these concepts enables students to handle more sophisticated analyses, such as hypothesis testing and regression modeling. In classroom settings or everyday life, a firm grasp of these basics can help make sense of numerical information and lead to more informed judgments and decisions. Whenever we apply these statistical tools, we're engaging in a practice that's crucial across numerous professional fields and in research efforts worldwide.