Question

For the data sets in Exercises \(1-3,\) calculate the sample variance, \(s^{2}\), using (1) the definition formula and (2) the computing formula. Then calculate the sample standard deviation, s. \(n=5\) measurements: 2,1,1,3,5

Short answer

Answer: The sample variance (s²) of the given data set is 2.8, and the sample standard deviation (s) is approximately 1.67.

Step-by-step solution

Calculating the mean (x̅)

First, we need to calculate the mean of these measurements. x̅ = \(\frac{2+1+1+3+5}{5}\) = \(\frac{12}{5}\) = 2.4

Calculate the sample variance using the definition formula (s²_1)

Now, let's apply the definition formula for the sample variance: s²_1 = \(\frac{\sum_{i=1}^n (x_i - x̅)^2}{n-1}\) To calculate by definition formula, find the (x_i - x̅)² for each measurement. (x1 - x̅)² = (2 -2.4)² = 0.16 (x2 - x̅)² = (1 -2.4)² = 1.96 (x3 - x̅)² = (1 -2.4)² = 1.96 (x4 - x̅)² = (3 -2.4)² = 0.36 (x5 - x̅)² = (5 -2.4)² = 6.76 Now, we substitute these values into the formula: s²_1 = \(\frac{0.16 + 1.96 + 1.96 + 0.36 + 6.76}{5-1}\) = \(\frac{11.2}{4}\) = 2.8

Calculate the sample variance using the computing formula (s²_2)

Now, let's apply the computing formula for the sample variance: s²_2 = \(\frac{(\sum_{i=1}^n x_i^2) - (\frac{(\sum_{i=1}^n x_i)^2}{n})}{n-1}\) Calculate the sum of square of each measurement and sum of the measurements. Σx² = 2²+1²+1²+3²+5² = 4+1+1+9+25 = 40 Σx = 2+1+1+3+5 =12 Now, we substitute these values into the formula: s²_2 = \(\frac{40 - (\frac{12^2}{5})}{5-1}\) = \(\frac{40 - (\frac{144}{5})}{4}\) = \(\frac{11.2}{4}\) = 2.8

Calculate the sample standard deviation (s)

Now we can calculate the sample standard deviation by taking the square root of the sample variance. s = \(\sqrt{s^2}\)= \(\sqrt{2.8}\) ≈ 1.67 Finally, we have found the sample variance using both formulas, and it is s² = 2.8. The sample standard deviation is s ≈ 1.67.

Key concepts

Sample Standard Deviation

Understanding the sample standard deviation involves recognizing it as a measure of data dispersion within a set. It shows how much the individual data points deviate from the mean.

• The **standard deviation** is the square root of the **variance**. Therefore, once you have computed the variance, finding the standard deviation is straightforward.
• For the given exercise, we found the sample variance ( s²) to be 2.8.

By calculating the square root of the sample variance, we determine that the standard deviation, denoted as s, is approximately 1.67. This means that, on average, each data point deviates from the mean by about 1.67 units. It's a crucial statistic as it gives insight into the consistency of the dataset. A smaller standard deviation indicates that the data points are close to the mean, while a larger one suggests more spread-out values.

Variance Calculation Methods

Variance is a measure of how spread out a set of numbers is. In statistical analysis, there are different methods to calculate sample variance.**Definition Formula**: This involves a step-by-step approach:

• First, find the mean ( \( \bar{x} \)) of the data set.
• Then, subtract this mean from each data point, and square the result.
• Add all these squared differences together, and divide by the number of data points minus one (n-1).

This step-by-step method is clear and systematic.**Computing Formula**: Alternatively, there's a shortcut method:

• Calculate the sum of the squares of each data point.
• Subtract the squared sum of the data points divided by the number of data points ( \( n \)).
• Divide the result by ( \( n-1 \)).

Both formulas result in the same variance value, 2.8, making either method a valid choice depending on context and ease of use. However, using the computing formula can often save time.

Statistical Analysis

Statistical analysis is essential in understanding patterns, relationships, and trends in data. "Sample variance" and "standard deviation" are both central to this process as they help in interpreting how data points relate to each other and the dataset overall. In this exercise, the sample variance indicates how individual measurements deviate from the average measurement:

• A variance of 2.8 shows there is some variability among the data points.
• Standard deviation further clarifies this variability by converting it into a standard metric that can be easily interpreted.

Such metrics enable researchers or students to make informed judgments about their data. Conducting statistical analysis allows for better decision-making and prediction. It is widely used across various fields, including economics, engineering, and social sciences.