Question

An article in Archaeometry described 26 samples of pottery found at four different kiln sites in the United Kingdom. \({ }^{8}\) The percentage of iron oxide in each of five samples collected at the Island Thorns site was as follows: $$ \begin{array}{lllll} 1.28 & 2.39 & 1.50 & 1.88 & 1.51 \end{array} $$ a. Calculate the range. b. Calculate the sample variance and the standard deviation using the computing formula. c. Compare the range and the standard deviation. The range is approximately how many standard deviations?

Short answer

Answer: Approximately 5.14 standard deviations are contained within the range of the pottery sample data.

Step-by-step solution

Find the range

To begin, we will first find the range of the given data. The range is the difference between the highest and lowest values in a dataset. In our case, we have: $$ \text{Range} = \text{Max}(\text{Data}) - \text{Min}(\text{Data}) = 2.39 - 1.28 = 1.11 $$

Calculate the sample mean

Next, we need to find the sample mean of the iron oxide percentages. We will sum up all the sample values and divide by the number of samples (5). $$ \text{Sample Mean} = \frac{1.28 + 2.39 + 1.50 + 1.88 + 1.51}{5} = \frac{8.56}{5} = 1.712 $$

Compute the sample variance using the computing formula

To find the sample variance, we will use the computing formula: $$ \text{Sample Variance} = s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} $$ Where: \(s^2\) = Sample variance \(x_i\) = Each data point \(\bar{x}\) = Sample mean \(n\) = Number of data points Substituting the values we have: $$ s^2 = \frac{(1.28 - 1.712)^2 + (2.39 - 1.712)^2 + (1.50 - 1.712)^2 + (1.88 - 1.712)^2 + (1.51 - 1.712)^2}{4} = \frac{0.186624}{4} = 0.046656 $$

Calculate the standard deviation

Now, to find the standard deviation, we will simply take the square root of the sample variance: $$ s = \sqrt{s^2} = \sqrt{0.046656} = 0.215999558 \approx 0.216 $$

Compare the range and the standard deviation

To compare the range to the standard deviation, we need to find how many standard deviations the range contains: $$ \text{Number of Standard Deviations in Range} = \frac{\text{Range}}{\text{Standard Deviation}} = \frac{1.11}{0.216} = 5.1389 \approx 5.14 $$ In conclusion, the range is approximately 5.14 standard deviations.

Key concepts

Understanding Range in Statistics

In statistics, the range is one of the simplest measures of dispersion, showing how spread out the values in a data set are. To calculate it, you take the difference between the largest value (maximum) and the smallest value (minimum). In essence, the range gives an immediate idea of the interval within which all your data points fall.

For example, with the given pottery sample percentages of iron oxide, we find the maximum value to be 2.39 and the minimum to be 1.28. Subtracting these two gives us a range of 1.11. This indicates that all measured samples of iron oxide percentages lie within an interval of 1.11 percentage points. However, while useful, the range does not account for any variability within the set's upper and lower bounds, which is why other measures like variance and standard deviation are also used.

Breaking Down Standard Deviation

The standard deviation is a statistics fundamental that represents the amount of variation or dispersion from the average (mean). A low standard deviation means that the data points are close to the mean, while a high standard deviation indicates that the data points are spread out over a larger range of values.

When we calculate standard deviation, we are essentially looking at the average distance between each data point and the mean. It's a more informative measure of spread because, unlike range, it takes into account each individual value. In the context of our pottery samples, the standard deviation is approximately 0.216, suggesting that each iron oxide percentage is, on average, 0.216 percentage points away from the mean percentage of 1.712.

Computing Formula for Sample Variance

To get into the true variability within a set of data, we compute the sample variance. Sample variance (\(s^2\tech)) looks at how much each individual data point in the sample deviates from the mean and squares that difference to eliminate negative values and emphasize larger deviations.

Here's the formula used for computing sample variance: \[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} \]

• \(s^2\tech) is the sample variance.
• \(x_i\tech) represents each data point in the sample.
• \(\bar{x}\tech) is the sample mean.
• \(n\tech) is the number of data points in the sample.

In our exercise, after calculating the mean, we subtract it from each data point, square the result, sum all those squared differences, and divide by one less than the number of data points (since we are dealing with a sample, not a population). This calculation provides a sample variance of 0.046656, reflecting the variability of iron oxide content among the pottery samples.