Problem 70 - Algebra 2

Question

Find the mean and the standard deviation for each set of values. $$ 21\quad 29 \quad 35 \quad 26 \quad 25 \quad 28 \quad 27 \quad 51 \quad 24 \quad 34 $$

Step-by-Step Solution

Answer

The mean is 31 and the standard deviation is approximately 7.68.

Step 1: Calculate the Mean

First, add all the numbers in the data set: 21 + 29 + 35 + 26 + 25 + 28 + 27 + 51 + 24 + 34. Then, divide the result by the count of numbers, which is 10 in this case.

Step 2: Subtract Mean and Square the Result

Subtract the mean from each number in the data set and square the result. This gives the squared differences.

Step 3: Average of Squared Differences

Find the mean of these squared differences. Add up all of the squared differences and divide by the count of numbers, which is 10.

Step 4: Calculate Standard Deviation

Finally, take the square root of the mean of the squared differences. This is the standard deviation of the data set.

Key Concepts

Calculating Mean

Calculating the mean, also known as the average, is a fundamental concept in data set analysis. To find the mean of a data set, you begin by summing all the values together. For instance, if you have a set of numbers like \(21, 29, 35, 26, 25, 28, 27, 51, 24, 34\), you'll add them up to get \(300\). Once you have the total sum, the next step is to divide this sum by the total count of numbers in the set. In our example, there are 10 numbers, so you divide \(300\) by \(10\), resulting in a mean of \(30\).

Understanding the mean helps in summarizing the central tendency of the data and provides a simple point of reference for comparing individual values to the overall set.

Data Set Analysis

Data set analysis involves a deeper examination and interpretation of a set of numbers to uncover patterns or insights. A robust analysis often includes calculating key metrics such as mean, median, mode, range, and standard deviation. Each of these metrics offers a unique insight into the characteristics of the data set.

By understanding the mean, you grasp the general trend of the data. However, look beyond the mean to understand variation through measures like standard deviation, or to identify common values through the mode.

• Mean gives an overall average.
• Median provides the middle value.
• Mode highlights the most frequent occurrence.
• Range shows the difference between maximum and minimum.
• Standard deviation measures variability.

Effective data set analysis does not rely on just one metric, but rather combines these statistics to provide a comprehensive view of the data.

Standard Deviation Calculation

Standard deviation is a key indicator of the spread or dispersion within a data set. To calculate standard deviation, follow these steps:

First, determine the mean of your data set. Then subtract this mean from each individual number to find the variance from the mean. Each of these differences is squared, minimizing the effect of negative differences. Sum up all these squared differences to get the total variance.

In the next step, calculate the mean of these squared differences. This is the variance of the data. Finally, take the square root of this variance to obtain the standard deviation. For our example data set of \(21, 29, 35, 26, 25, 28, 27, 51, 24, 34\), the standard deviation quantifies the amount by which each number deviates from the mean. A higher standard deviation indicates more variability within the data set, while a lower one signifies that the numbers are closer to the mean.