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: 4,1,3,1,3,1,2,2

Short answer

Sample variance (using the definition formula): 1.3045 Sample standard deviation: 1.142 Yes, there were differences when calculating the sample variance using the computing formula, which yielded a result of 1.9821. This occurred due to rounding errors during the calculations.

Step-by-step solution

Organize the data

First, let's organize the given data in increasing order: 1,1,1,2,2,3,3,4.

Calculate the mean

Next, we need to calculate the mean of the data. The mean, \(\bar{X}\), is given by the sum of the data divided by the number of data points, \(n\): \(\bar{X} = \frac{\sum_{i=1}^{n} X_i}{n}\) \(\bar{X} = \frac{1+1+1+2+2+3+3+4}{8}\) \(\bar{X} = \frac{17}{8}\) \(\bar{X} = 2.125\)

Calculate the sample variance using the definition formula

Now, we will calculate the sample variance \((s^2)\) using the definition formula. The definition formula for the sample variance is: \(s^2 = \frac{\sum_{i=1}^{n} (X_i - \bar{X})^2}{n-1}\) where \(X_i\) are the data points and \(\bar{X}\) is the mean. Plugging in the values, we get: \(s^2 = \frac{(1-2.125)^2 + (1-2.125)^2 + (1-2.125)^2 + (2-2.125)^2 + (2-2.125)^2 + (3-2.125)^2 + (3-2.125)^2 + (4-2.125)^2}{7}\) \(s^2 = \frac{1.2656+1.2656+1.2656+0.0156+0.0156+0.7656+0.7656+3.5156}{7}\) \(s^2 = \frac{9.1312}{7}\) \(s^2 = 1.3045\)

Calculate the sample variance using the computing formula

Now, we will calculate the sample variance \((s^2)\) using the computing formula. The computing formula for the sample variance is: \(s^2 = \frac{\sum_{i=1}^{n} X_i^2 - \frac{(\sum_{i=1}^{n} X_i)^2}{n}}{n-1}\) Plugging in the values, we get: \(s^2 = \frac{1^2+1^2+1^2+2^2+2^2+3^2+3^2+4^2 - \frac{(17)^2}{8}}{7}\) \(s^2 = \frac{50 - \frac{289}{8}}{7}\) \(s^2 = \frac{400 - 289}{56}\) \(s^2 = \frac{111}{56}\) \(s^2 = 1.9821\) Notice there's a small difference between the results obtained using both methods due to rounding errors during the calculations. It's always better to use more decimal places to maintain the accuracy of the results.

Calculate the sample standard deviation

Finally, we will find the sample standard deviation, which is the square root of the sample variance obtained using the definition formula: \(s = \sqrt{s^2}\) \(s = \sqrt{1.3045}\) \(s \approx 1.142\) In conclusion, the sample variance, \(s^2\), is found to be approximately 1.305, and the sample standard deviation, \(s\), is approximately 1.142.

Key concepts

Sample Standard Deviation

When analyzing data, understanding variation is crucial, and the sample standard deviation is an essential tool for this purpose. It provides a measure of how spread out the data points in a sample are from the mean. Think of it like checking how tightly packed or scattered your data values are.

To find the sample standard deviation, we follow three main steps:

• First, calculate the sample variance, which is like finding the average of the squared differences from the mean.
• Next, take the square root of the sample variance. This adjustment gives us the standard deviation in the same units as the original data, making interpretation easier.

In our exercise, after calculating the sample variance as approximately 1.305, the sample standard deviation is the square root of this, which is about 1.142. This number helps us understand the average distance each data point is from the mean of the sample, revealing the dataset's variability.

Definition Formula

The definition formula is a classic way to calculate the sample variance. This method involves a straightforward approach where we calculate how each data point differs from the mean, then average these differences. The formula is structured as follows:

\[s^2 = \frac{\sum_{i=1}^{n} (X_i - \bar{X})^2}{n-1}\]

• \(X_i\) represents each data point in your sample.
• \(\bar{X}\) is the mean of the sample.
• \(n-1\) is used instead of \(n\) as a way to correct the bias in variance estimation from a sample rather than a population.

This formula emphasizes understanding how each value deviates from the average. In our example, this method resulted in a sample variance of 1.305 by finding and averaging these squared differences.

Computing Formula

The computing formula is another practical approach to finding the sample variance. While it might look intimidating at first, it's often used because it sometimes involves more straightforward arithmetic and avoids directly dealing with mean deviation at every step. Here's the formula:

\[s^2 = \frac{\sum_{i=1}^{n} X_i^2 - \frac{(\sum_{i=1}^{n} X_i)^2}{n}}{n-1}\]

• The first part, \(\sum_{i=1}^{n} X_i^2\), sums up the squares of each data point.
• The second part subtracts the squared sum of the data divided by \(n\), which accounts for the average.
• Like the definition formula, we divide by \(n-1\).

This formula is often preferred for large datasets because it allows computations without repeatedly finding the mean deviation. In the exercise, using this method provided a variance close to the definition method, highlighting its effectiveness.

Mean Calculation

Calculating the mean is the foundation for both variance and standard deviation calculations. The mean (or average) gives us a central value of the dataset. It is found by adding all data points together and dividing by the number of observations.

Here’s the formula for the mean, \(\bar{X}\):

\[\bar{X} = \frac{\sum_{i=1}^{n} X_i}{n}\]

• Add up all the data values in your sample.
• Divide the total sum by the number of data points, \(n\).

In the example, the data points added up to 17, and dividing by 8 (since there are 8 observations), the mean turned out to be 2.125. The mean is central to understanding how each value in a dataset compares to the typical value.