Problem 5 - Algebra Form and Function

Question

Find the mean and standard deviation of the data set. $$ 12,-8,13,-15,60,-72,23,-13 $$

Step-by-Step Solution

Answer

Answer: The mean of the data set is 0 and the standard deviation is approximately 16.9.

Step 1: Calculate the mean of the data set

To find the mean, we need to add all the values in the data set and divide the sum by the total number of values. In this case, there are 8 values in our data set: $$ Mean = \frac{12 + (-8) + 13 + (-15) + 60 + (-72) + 23 + (-13)}{8} $$

Step 2: Calculate the mean value

Now, let's calculate the mean: $$ Mean = \frac{12 - 8 + 13 - 15 + 60 - 72 + 23 - 13}{8} = \frac{0}{8} = 0 $$ The mean of the data set is 0.

Step 3: Calculate the variance of the data set

To find the variance, we first need to find the square of the difference between each value and the mean, and then find the average of these squared differences. We'll use the mean we calculated in Step 2: $$ Variance = \frac{(12 - 0)^2 + (-8 - 0)^2 + (13 - 0)^2 + (-15 - 0)^2 + (60 - 0)^2 + (-72 - 0)^2 + (23 - 0)^2 + (-13 - 0)^2}{8} $$

Step 4: Calculate the variance value

Now, let's calculate the variance: $$ Variance = \frac{12^2 + 8^2 + 13^2 + 15^2 + 60^2 + 72^2 + 23^2 + 13^2}{8} = \frac{2288}{8} = 286 $$ The variance of the data set is 286.

Step 5: Calculate the standard deviation

Finally, to find the standard deviation, we need to take the square root of the variance we calculated in Step 4: $$ Standard \ Deviation = \sqrt{286} \approx 16.9 $$ The standard deviation of the data set is approximately 16.9.

Key Concepts

Understanding Data Sets

A data set is simply a collection of numbers or values that we want to analyze. In this exercise, the given data set is

• 12, -8, 13, -15, 60, -72, 23, -13

Each number in a data set is referred to as a data point or value. Understanding the composition of a data set is crucial for performing any mathematical calculations such as finding the mean or variance.

When given a data set, the first step is generally to understand how these numbers relate to one another by calculating their average, which provides a central value that summarizes the entire set.

Exploring Variance

Variance measures how much the values in a data set deviate from the mean. It's a fascinating concept that tells us how spread out the numbers are. A high variance means values are scattered widely from the mean, while a low variance indicates they are close together.

To compute variance, we follow a systematic approach:

• First, find the mean of the data set, as done in the original solution.
• Next, calculate the difference between each data point and the mean.
• Square each of these differences to eliminate negative values.
• Finally, average these squared differences to find the variance.

Using this method, one can frame the extent of variations in the data, making variance a valuable tool in descriptive statistics.

Executing Mathematical Calculations

Mathematical calculations are the core of statistical analysis. In this exercise, we encountered basic calculations to determine the mean, variance, and standard deviation.

These steps are essential:

• Adding all data points to calculate the total sum.
• Dividing the total sum by the number of data points to find the mean.
• Computing the variance using squared differences from the mean, as explained earlier.
• Finally, calculating the standard deviation by taking the square root of the variance.

These calculations help uncover insights such as central tendencies and variability in data, enabling better decision-making based on numerical findings.