Question

Refer to Exercise \(2.17 .\) The percentage of iron oxide in each of five pottery samples collected at the Island Thorns site was: $$ \begin{array}{lllll} 1.28 & 2.39 & 1.50 & 1.88 & 1.51 \end{array} $$ a. Use the range approximation to find an estimate of s, using an appropriate divisor from Table 2.6 . b. Calculate the standard deviation \(s\). How close did your estimate come to the actual value of \(s\) ?

Short answer

Question: Estimate the standard deviation using the range approximation method for the following pottery samples containing iron oxide (%): 1.28, 2.39, 1.50, 1.88, and 1.51. Compare this estimate to the actual standard deviation value.

Step-by-step solution

Calculate the Range of the Data

First, we need to find the range of the dataset by finding the difference between the largest and smallest values in the dataset. In this case, the values are: $$ \begin{array}{lllll} 1.28 & 2.39 & 1.50 & 1.88 & 1.51 \end{array} $$ So, the range is \(2.39 - 1.28 = 1.11\).

Use the Given Table to Find the Appropriate Divisor

In this step, we need to refer to Table 2.6 (not provided here) to find the appropriate divisor for our range method. Since we have 5 data points, the appropriate divisor according to the table should be used.

Calculate the Estimated Standard Deviation

Now that we have the range and the appropriate divisor, we can calculate the estimated standard deviation. The formula for the range method is: $$ \text{Estimated Standard Deviation (s)} = \frac{\text{Range}}{\text{Divisor}} $$ Plug in the values and calculate the estimated standard deviation.

Calculate the Actual Standard Deviation

Next, we need to calculate the actual standard deviation for comparison. To calculate the standard deviation, use the formula: $$ s = \sqrt{\frac{\sum(x - \bar{x})^2}{n - 1}} $$ Where \(x\) represents each data point, \(\bar{x}\) represents the mean of the dataset, and \(n\) represents the number of data points. 1. Calculate the mean of the dataset. 2. Subtract the mean from each data point and square the result. 3. Sum the squared differences. 4. Divide the sum by \(n - 1 = 4\). 5. Take the square root of the result to get the actual standard deviation.

Compare the Estimated and Actual Standard Deviation

Now that we have the estimated standard deviation and the actual standard deviation, compare these two values to see how close the estimate is to the actual value.

Key concepts

Standard Deviation

Standard deviation is a statistical concept that measures the amount of variability or dispersion in a set of data. It tells us how spread out the data points are from the mean.

To calculate standard deviation, we start by finding the mean (average) of the dataset. We then determine how far each data point is from this mean, by calculating the difference between each value and the mean, and squaring these differences to eliminate negative values.

Next, we sum all these squared differences. The total represents the overall variability in the dataset. We divide this sum by the number of data points minus one (known as degrees of freedom) to get what is known as the variance. Finally, the standard deviation is the square root of this variance.

This process boils down to:

• Calculate the mean of the dataset.
• Subtract the mean from each data point and square the result.
• Sum all the squared results.
• Divide by one less than the number of data points.
• Take the square root of this quotient.

By following these steps, you can find the standard deviation and better understand how your data is distributed.

Range Approximation

Range approximation is a useful technique to estimate the standard deviation of a dataset quickly. While it does not replace the precision of calculating the actual standard deviation, it offers a quick insight into data variability.

The method involves calculating the range, which is the difference between the largest and smallest values in the dataset. With this value calculated, the next step involves using a special divisor to obtain an estimated standard deviation. The specific divisor depends on the number of data points and can often be found in statistical tables.

The general formula for estimating standard deviation through range approximation is: \[ \text{Estimated Standard Deviation} = \frac{\text{Range}}{\text{Divisor}} \] This approach allows for a simple estimate and is particularly useful when a quick analysis is needed without extensive calculations.

Dataset Analysis

Analyzing a dataset involves understanding the properties of the data at hand, such as central tendency, dispersion, and overall distribution. Various statistical measures like the mean, median, mode, and range provide insights into the dataset's characteristics.

When analyzing datasets, standard deviation becomes crucial as it tells us how much the data tends to vary from the average. A low standard deviation means that data points tend to be close to the mean, while a high standard deviation indicates a larger spread in values.

Range gives a simple idea of the spread by indicating the difference between the highest and lowest numbers. For more robust analysis, consider using the range along with standard deviation to provide a better picture of variability.

Good dataset analysis usually involves:

• Calculating basic statistics like mean, median, and mode.
• Understanding dispersion through measures like standard deviation and range.
• Identifying any outliers or unusual data points that might need closer scrutiny.

These foundational steps in dataset analysis are pivotal in interpreting data and making informed decisions based on statistical observations.