Question

Find the mean and standard deviation of the data set. $$ 16,66,30,99,74,50,35,7 $$

Short answer

Answer: The mean of the data set is 47.125, and the standard deviation is approximately 29.77.

Step-by-step solution

Calculate the mean

To find the mean, add all the data points together and then divide by the number of data points. In this case, there are 8 data points. $$ \text{Mean} = \frac{16+66+30+99+74+50+35+7}{8} $$

Evaluate the mean

Now, we will calculate the mean by adding the numbers and dividing by 8. $$ \text{Mean} = \frac{377}{8} = 47.125 $$

Calculate the squared differences from the mean

Next, we need to find the squared difference of each data point from the mean. To do this, subtract the mean from each data point, square the result, and write down the values. $$ (16-47.125)^2, (66-47.125)^2, (30-47.125)^2, (99-47.125)^2, (74-47.125)^2, (50-47.125)^2, (35-47.125)^2, (7-47.125)^2 $$

Evaluate the squared differences

Now, we will calculate the values obtained in the previous step. $$ 972.5156, 355.8906, 292.8906, 2687.8906, 718.8906, 8.3906, 146.8906, 1607.8906 $$

Calculate the variance

To find the variance, add up all the squared differences and then divide by the number of data points. $$ \text{Variance} = \frac{972.5156+355.8906+292.8906+2687.8906+718.8906+8.3906+146.8906+1607.8906}{8} $$

Evaluate the variance

Now, we will calculate the variance by adding the numbers and dividing by 8. $$ \text{Variance} = \frac{7090.2494}{8} = 886.2812 $$

Calculate the standard deviation

To find the standard deviation, take the square root of the variance. $$ \text{Standard Deviation} = \sqrt{886.2812} $$

Evaluate the standard deviation

Finally, calculate the value of the standard deviation. $$ \text{Standard Deviation} = \approx 29.77 $$ The mean of the data set is 47.125, and the standard deviation is approximately 29.77.

Key concepts

Mean Calculation

The mean is a fundamental statistical concept that represents the average of a set of numbers. Calculating the mean is straightforward. You simply add up all the numbers in a data set and then divide by the number of data points.

In our example data set, which is \( \{16, 66, 30, 99, 74, 50, 35, 7\} \), there are 8 numbers. The sum of these numbers is 377. Therefore, to find the mean, you calculate \( \frac{377}{8} \). This results in a mean of 47.125.

The mean provides a central value that can be helpful during data analysis to understand the overall distribution of the data points.

Variance

Variance is a measure of how spread out the numbers in a data set are. It tells us how much the individual data points deviate from the mean, on average.

To calculate the variance, follow these steps:

• Subtract the mean from each data point to find differences.
• Square each of these differences to make them positive.
• Add up the squared differences.
• Finally, divide by the number of data points.

For our data set, the calculated squared differences are \(972.5156, 355.8906, 292.8906, 2687.8906, 718.8906, 8.3906, 146.8906, 1607.8906\). The sum of these is 7090.2494, and dividing by 8 gives a variance of 886.2812.

A higher variance indicates that the data points are more spread out from the mean, while a lower variance suggests they are closer to the mean.

Data Analysis

Data analysis involves examining, cleaning, transforming, and modeling data with the goal of discovering useful information. It helps us make informed decisions based on data patterns and insights.

Key steps in data analysis include:

• Collecting and inputting data correctly.
• Conducting statistical computations like mean and variance.
• Interpreting results to find patterns and anomalies.
• Visualizing data for better understanding.

For instance, calculating the mean and variance are parts of statistical analysis, enabling us to assess the central tendency and dispersion of our data set. Understanding these statistics allows analysts to grasp the underlying trends and to make predictions or decisions based on the findings.

Effective data analysis is essential in research, business, and academic settings, where making sense of data can lead to strategic advantages and knowledge advancement.