Problem 6 - Algebra 2

Question

Find the variance and standard deviation of each set of data to the nearest tenth. {400, 300, 325, 275, 425, 375, 350}

Step-by-Step Solution

Answer

Variance: 2500.0, Standard Deviation: 50.0

Step 1: Calculate the Mean

First, we need to find the mean (average) of the data set. Add all the numbers together and divide by the number of values.\[\text{Mean} = \frac{400 + 300 + 325 + 275 + 425 + 375 + 350}{7} = \frac{2450}{7} = 350\]

Step 2: Find the Deviations

Subtract the mean from each number in the data set to find the deviation of each number.\[400 - 350 = 50, \quad 300 - 350 = -50, \quad 325 - 350 = -25\]\[275 - 350 = -75, \quad 425 - 350 = 75, \quad 375 - 350 = 25, \quad 350 - 350 = 0\]

Step 3: Square the Deviations

Square each deviation to eliminate negative numbers and emphasize larger deviations.\[50^2 = 2500, \quad (-50)^2 = 2500, \quad (-25)^2 = 625\]\[(-75)^2 = 5625, \quad 75^2 = 5625, \quad 25^2 = 625, \quad 0^2 = 0\]

Step 4: Calculate the Variance

The variance is the average of the squared deviations. Add all the squared deviations and divide by the number of data points.\[\text{Variance} = \frac{2500 + 2500 + 625 + 5625 + 5625 + 625 + 0}{7} = \frac{17500}{7} \approx 2500.0\]

Step 5: Calculate the Standard Deviation

The standard deviation is the square root of the variance. Take the square root of 2500.\[\text{Standard Deviation} = \sqrt{2500} = 50.0\]

Key Concepts

Mean Calculation

To find the variance and standard deviation of a data set, we first need to calculate the mean, which is simply the average of the data points. The mean gives us a central value of the data set around which all other numbers are spread.
To calculate the mean, follow these steps:

• Add all the numbers in the data set.
• Divide the sum by the count of the numbers in the data set.

For example, in the data set \{400, 300, 325, 275, 425, 375, 350\}, the calculation would be: \[\text{Mean} = \frac{400 + 300 + 325 + 275 + 425 + 375 + 350}{7} = \frac{2450}{7} = 350\]The mean is 350, which represents the middle point of our data set.

Data Deviation

Once we know the mean, we need to find out how much each data point deviates from it. This helps us understand the spread of the data around the mean. A deviation tells us whether a number is above or below the mean and by how much.
Here's how to calculate each data deviation:

• Subtract the mean from each data point.

For the example mean of 350:

• 400 - 350 = 50
• 300 - 350 = -50
• 325 - 350 = -25
• 275 - 350 = -75
• 425 - 350 = 75
• 375 - 350 = 25
• 350 - 350 = 0

These deviations tell us how much each number differs from the mean, both in positive and negative directions.

Squared Deviations

After determining the deviations, we need to square each one. Squaring the deviations serves a special purpose: it removes any negative signs (since a negative number squared is positive) and gives more weight to larger deviations. Bigger deviations from the mean have a more substantial impact on our final measurements—variance and standard deviation.
Let's square the deviations from our example:

• 50² = 2500
• (-50)² = 2500
• (-25)² = 625
• (-75)² = 5625
• 75² = 5625
• 25² = 625
• 0² = 0

These squared deviations are now ready to be averaged in order to calculate the variance.