Problem 28 - Algebra 2

Question

a. Data Collection Make a table showing the number of siblings of each student in the class. b. Find the mean and standard deviation of the data.

Step-by-Step Solution

Answer

To answer these questions, tabulate the data first. Then calculate the mean by summing all the data points and dividing by the number of data points. Finally, calculate the standard deviation by calculating the average of the squared differences from the mean and square rooting the result.

Step 1: Data Tabulation

First, collect the data required for the exercise, i.e., the number of siblings of each student. Every student in the class will represent a row in this table, and the cells in the table should represent how many siblings each student has. Tabulate the data in a systematic manner.

Step 2: Calculation of Mean

Now that you have the data, you can calculate the mean (average). To do this, sum up the total number of siblings for all the students and then divide it by the total number of students. We can denote this calculation as: \[ Mean = \frac{{\sum (Number Of Siblings)}}{{Total Number Of Students}}. \]

Step 3: Calculation of Standard Deviation

The standard deviation is a measure of how spread out numbers are--it shows variation or 'dispersion' from the mean. To calculate it, first, find the variance (the average of the squared differences from the mean), and then take the square root of it. We can denote this calculation as: \[ Standard Deviation = \sqrt{\frac{{\sum (Number Of Siblings - Mean)^2}}{{Total Number Of Students}}}. \] Hence, first, calculate the difference between each data point and the mean, then square the result, sum all up, divide by the total number of students, and finally take the square root of the outcome.

Key Concepts

Data Collection

Data collection is the foundation of any statistical analysis. It involves gathering and measuring information on variables of interest in a systematic way. In the given exercise, the variable of interest is the number of siblings each student has. The goal is to make a table with this information for all students in the class.
First, each student's number of siblings is recorded, ensuring that no data is missing, as that could skew the results. Once every student has reported their siblings, you write down each entry correspondingly, which may look like a simple list or a more organized table.
Data tabulation helps to visualize the data in a structured format, making it easier to apply further statistical analysis. This meticulous approach renders your data both reliable and easy to interpret.

Mean Calculation

The mean, often referred to as the average, is a central tendency metric that provides a quick snapshot of the data distribution. To calculate the mean of siblings in the class, you first need the complete data set collected.
Sum up all the siblings recorded in your table, which gives the total number of siblings for the entire class. Say there are 30 students, and their sibling count sums to 60. Thus, the mean is calculated by dividing the total siblings by the number of students.

• The formula for mean: \[ Mean = \frac{{60}}{{30}} = 2. \]

This suggests that, on average, students in the class have two siblings. The simplicity of this computation underscores how accessible and practical the mean is for representing a dataset succinctly.

Standard Deviation Calculation

Standard deviation is a key statistical measure that gives insight into data variability or dispersion. It tells you how much the individual entries in your dataset deviate from the mean.
Once the mean is calculated, we assess the variance by determining the average of the squared differences between each data point and the mean.

• For example, if a student has four siblings, the difference from the mean is 2 (4 minus the mean of 2), and the squared difference is 4.
• This process is repeated for each student's data.
• Add all squared differences and divide by the number of students.

Finally, take the square root of this variance to find the standard deviation.

• For instance, if the variance is calculated to be 1, the standard deviation is \( \sqrt{1} = 1 \).

This result shows that the sibling counts deviate by about one sibling either side of the mean. Understanding standard deviation helps gauge the spread of your data around the mean, pinpointing whether the values are tightly clustered or widely scattered.