Question

You are given \(n=5\) measurements: 2,1,1,3,5 a. Calculate the sample mean, \(\bar{x}\). b. Calculate the sample variance, \(s^{2},\) using the formula given by the definition. c. Find the sample standard deviation, \(s\). d. Find \(s^{2}\) and \(s\) using the computing formula. Compare the results with those found in parts \(\mathrm{b}\) and \(\mathrm{c} .\)

Short answer

Answer: The steps involved in calculating the sample standard deviation for a set of measurements are: 1. Calculate the sample mean by summing up the measurements and dividing by the total number of measurements. 2. Calculate the sample variance using the definition formula, which involves subtracting the mean from each measurement, squaring the differences, and dividing the sum by the total number of measurements minus 1. 3. Take the square root of the sample variance to find the sample standard deviation. 4. Alternatively, calculate the variance and standard deviation using the computing formula, and compare the results with the previous method to ensure consistency.

Step-by-step solution

Calculate the sample mean

To calculate the sample mean, add up the given measurements and divide the result by the total number of measurements, \(n=5\). In this case, we have \(\bar{x} = \frac{2+1+1+3+5}{5}\).

Calculate the sample variance

Using the definition formula for the sample variance, we have: \(s^{2} = \frac{\sum_{i=1}^{n}(x_{i} - \bar{x})^2}{n-1}\). We plug in the numbers and get \(s^{2} = \frac{(2-\bar{x})^2 + (1-\bar{x})^2 + (1-\bar{x})^2 + (3-\bar{x})^2 + (5-\bar{x})^2}{5-1}\).

Find the sample standard deviation

To find the sample standard deviation, we simply take the square root of our sample variance: \(s = \sqrt{s^2}\).

Calculate the variance and standard deviation using the computing formula

The computing formula for variance is given by: \(s^{2} = \frac{\sum_{i=1}^{n} x_i^2}{n-1} - \frac{n\bar{x}^2}{(n-1)n}\). We plug in our numbers, we get \(s^{2}=\frac{2^2+1^2+1^2+3^2+5^2}{5-1} - \frac{5\bar{x}^2}{(5-1) \cdot 5}\). Finally, to calculate the sample standard deviation using the computing formula, we take the square root of our new variance, \(s = \sqrt{s^2}\). Now we compare this result with those found in parts \(\mathrm{b}\) and \(\mathrm{c}\). They should be the same.

Key concepts

Understanding Sample Mean

When it comes to statistics, the sample mean (often denoted as \(\bar{x}\)) is a crucial concept. It represents the average of a set of numbers and is calculated by summing up all the observed values and then dividing by the number of observations. For instance, if you have five measurements like 2, 1, 1, 3, and 5, you sum them up to get 12 and divide by 5 (since there are five measurements). This gives us a sample mean of \(\bar{x} = 2.4\).

Understanding the sample mean is vital because it's a measure of the central tendency of a dataset - to put it simply, it gives you an idea of where the 'middle' of the data lies. It's also often used as a starting point for other statistical analyses, like calculating variance and standard deviation.

Grasping Sample Standard Deviation

Sample standard deviation (denoted as \(\text{s}\)) measures how spread out the numbers in a data set are. It's the square root of the sample variance, hinting at how much the individual data points diverge from the sample mean. If the sample standard deviation is small, it means the data points tend to be close to the mean. Conversely, a large standard deviation indicates a wider range of values.

To calculate it, as per our textbook example, you first find the sample variance and then take its square root. If we have already computed the sample variance, \(s^{2} = 3.3\), taking the square root gives us a sample standard deviation of \(s = \(\sqrt{3.3}\) \(\approx\) 1.82\). This step is pivotal as it turns the variance (which is in squared units) back into the original units of the data, making it much more interpretable and useful for comparison purposes.

Computing Formula for Variance

The sample variance (\(s^2\)) quantifies the variability of data points in a sample. There are two formulas to compute this: the definition formula that first requires the sample mean and the computational formula that simplifies the process by not directly requiring it. The computational formula is particularly useful to avoid round-off errors in calculations and when the handling of large datasets is involved.

The formula is: \[ s^{2} = \frac{\sum_{i=1}^{n} x_i^2}{n-1} - \frac{n\bar{x}^2}{(n-1)n} \]
Substituting the numbers from our data, we calculate each term step by step, subtract them, and get the sample variance. Interestingly, while this newer formula might yield the same numerical value as the definition formula, it's generally considered to be more efficient for computational purposes, particularly when using calculators or computers, as it reduces the computational complexity and the chance of making arithmetic errors in the process.