Question

Calculate the sample variance, \(s^{2}\), using (1) the definition formula and (2) the computing formula. Then calculate the sample standard deviation, s. \(n=8\) measurements: 3,1,5,6,4,4,3,5

Short answer

Question: Calculate the sample variance and standard deviation for the given measurements: 3, 1, 5, 6, 4, 4, 3, and 5. Answer: The sample variance for the given measurements is approximately 2.839, and the sample standard deviation is approximately 1.685.

Step-by-step solution

Calculate the sample mean, \(\bar{x}\)

The sample mean is the sum of all data points divided by their number (\(n=8\)) which is: \(\bar{x} = \frac{3+1+5+6+4+4+3+5}{8}=\frac{31}{8}=3.875\)

Calculate the sample variance using definition formula

Using the definition formula and substituting the calculated sample mean, we get: \(s^{2} = \frac{(3-3.875)^2+(1-3.875)^2+(5-3.875)^2+(6-3.875)^2+(4-3.875)^2+(4-3.875)^2+(3-3.875)^2+(5-3.875)^2}{7} = \frac{19.875}{7}=2.839\)

Calculate the sample variance using computing formula

Using the computing formula, we get: \(s^{2} = \frac{(3^2+1^2+5^2+6^2+4^2+4^2+3^2+5^2)}{7} - \frac{31^2}{8\times7} = \frac{125}{7} - \frac{961}{56} \approx 2.839\)

Calculate the sample standard deviation

The sample standard deviation is the square root of the sample variance: \(s = \sqrt{s^{2}} = \sqrt{2.839} \approx 1.685\) The sample variance calculated using the definition formula (\(s^{2}\approx2.839\)) matches the answer obtained using the computing formula, which confirms that both methods provide the same result. The sample standard deviation for the given measurements is approximately \(1.685\).

Key concepts

Sample Standard Deviation

Understanding the sample standard deviation is crucial as it represents the spread or dispersion of a set of data points. Essentially, it tells us how much individual measurements in a sample differ from the sample mean. In the given exercise, after calculating a sample variance of approximately 2.839, we obtained the sample standard deviation by taking the square root of this variance, resulting in approximately 1.685.

This value gives us insight into the variability of the measurements. For instance, a higher standard deviation would indicate that the measurements are more spread out from their average value, while a lower standard deviation suggests that they are closer to the mean. Practically speaking, if we were considering test scores, a low standard deviation would reveal that most students scored around the same mark, whereas a high standard deviation would tell us that the students' scores were more varied.

Variance Calculation Methods

There are two primary methods used to calculate sample variance. The definition formula, which involves taking the average of the squared deviations from the sample mean, is more conceptual but can be computationally intensive especially for large data sets. The computing formula, on the other hand, simplifies the calculation process by allowing us to first calculate the sum of squared data points and the square of the sum of data points, simplifying the arithmetic process.

Both methods were employed in the exercise and yielded a sample variance of 2.839. The consistency in results regardless of the chosen variance calculation method brings a level of reliability to the analysis and allows flexibility in choosing the computation method based on convenience or computational tools available.

Data Point Analysis

The process of data point analysis involves examining each individual measurement in a dataset to understand the overall distribution and tendencies. In our exercise, we dealt with 8 measurements and first calculated the sample mean. This was an important step as the mean serves as a reference point for each data point's deviation.

Understanding deviations is crucial as they help us to calculate the variance and standard deviation. Each data point's deviation from the mean is squared to remove negative values and emphasize larger deviations. As part of this analysis, recognizing patterns such as repeated values can also give us insight into the dataset's consistency and potential clustering around certain values.