Problem 68 - Algebra 2

Question

Find the mean and the standard deviation for each set of values. $$ 6 \quad 1 \quad 9 \quad 12 \quad 4 \quad 15 \quad 21 \quad 7 \quad 8 \quad 8 $$

Step-by-Step Solution

Answer

The mean of the given set is 9.1 and the standard deviation is 4.09.

Step 1: Calculate the Mean

To find the mean, sum up all the numbers and then divide by the amount of numbers. So with the given set of values: \( \frac{6 + 1 + 9 + 12 + 4 + 15 + 21 + 7 + 8 + 8}{10} = 9.1 \)

Step 2: Calculate the Variance

Variance is the average of the squared differences from the Mean. To work out the variance, first, find the difference between each number and the mean; second, square each result; third, work out the average of those squared differences. For this set of numbers: Variance \( = \frac{(6-9.1)^2 + (1-9.1)^2 + (9-9.1)^2 + (12-9.1)^2 + (4-9.1)^2 + (15-9.1)^2 + (21-9.1)^2 + (7-9.1)^2 + (8-9.1)^2 + (8-9.1)^2}{10} = 16.69 \)

Step 3: Find the Standard Deviation

Standard Deviation is the square root of the Variance. So, for our data: Standard Deviation = \( \sqrt{16.69} = 4.09 \)

Key Concepts

Understanding the Mean

The mean, also known as the average, is a central concept in statistics. It describes the central tendency or the typical value within a data set. To calculate the mean, you sum up all the numbers in the set and then divide this total by the number of values. In the case of the provided values (6, 1, 9, 12, 4, 15, 21, 7, 8, 8), the sum is 91. This total is then divided by 10, as there are 10 numbers in the set, resulting in a mean of 9.1.

• Sum of values: 91
• Number of values: 10
• Mean: 91 ÷ 10 = 9.1

Understanding the mean helps us get a sense of the overall level of the data. It is useful in many real-world contexts when comparing different data sets or understanding trends.

Grasping the Concept of Variance

Variance is a statistical measure that tells us how much the numbers in our data set differ from the mean. First, you calculate how far each number in the set is from the mean and then square these differences to avoid negative values. Finally, you find the average of these squared differences. In our example:

• Calculate the difference of each number from 9.1 and square these results
• Find the average of these squared differences

So, with the data provided and the mean being 9.1, the calculated variance is 16.69. This tells us that, on average, the numbers in our data set differ from the mean by this squared amount. High variance means numbers are more spread out, while low variance indicates they're mostly close to the mean.

The Role of Standard Deviation

Standard deviation is derived from variance and offers a concrete measure of how spread out numbers in a data set are around the mean. It is simply the square root of the variance, which transforms the units back to the original data scale. For our data set, the variance is 16.69, so the standard deviation is calculated as follows:

• Standard Deviation = \( \sqrt{16.69} \)
• This results in a standard deviation of approximately 4.09

Standard deviation is very useful because it gives us a sense of variability within the data on the same scale as the data values themselves. In practice, a small standard deviation means that most numbers are close to the mean, while a larger value indicates a wider spread. Thus, standard deviation can help in predicting future deviations or variations from the expected data.