In Exercises 5–20, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as “minutes”) in your results. (The same data were used in Section 3-1, where we found measures of center. Here we find measures of variation.) Then answer the given questions.


Jaws 3 Listed below are the number of unprovoked shark attacks worldwide for the last several years. What extremely important characteristic of the data is not considered when finding the measures of variation?


70 54 68 82 79 83 76 73 98 81

Verified step by step guidance

1. Step 1: Identify the data set provided: {70, 54, 68, 82, 79, 83, 76, 73, 98, 81}. These are the number of unprovoked shark attacks worldwide for several years.
2. Step 2: Calculate the range. The range is the difference between the maximum and minimum values in the data set. Use the formula: $R=\max -\min$. Identify the maximum value (98) and the minimum value (54), then subtract.
3. Step 3: Calculate the variance. First, find the mean of the data set using the formula: $\mathrm{mean}=\frac{\sum x}{n}$, where $\sum x$ is the sum of all data points and $n$ is the number of data points. Then, compute the squared differences from the mean for each data point, sum them, and divide by $n-1$ (since this is a sample). Use the formula: ${\sigma }_2=\frac{\sum (x-\mathrm{mean}{)}^2}{n-1}$.
4. Step 4: Calculate the standard deviation. The standard deviation is the square root of the variance. Use the formula: $\mathrm{SD}=\sqrt{{\sigma }_2}$. Take the square root of the variance obtained in Step 3.
5. Step 5: Reflect on the important characteristic of the data that is not considered. The data does not account for the time period or population size, which could significantly affect the interpretation of the measures of variation. For example, an increase in shark attacks could be due to an increase in the number of people entering the water rather than an increase in shark activity.

Key Concepts

Measures of Variation

Measures of variation, including range, variance, and standard deviation, quantify the spread or dispersion of a dataset. The range indicates the difference between the maximum and minimum values, while variance measures the average squared deviation from the mean. Standard deviation, the square root of variance, provides a more interpretable measure of spread in the same units as the data.

Sample Data

Sample data refers to a subset of a population used to estimate characteristics of the whole group. In statistics, analyzing sample data helps infer properties about the larger population without needing to collect data from every individual. Understanding that the sample may not perfectly represent the population is crucial when interpreting results.

Outliers

Outliers are data points that significantly differ from other observations in a dataset. They can skew the results of measures of variation, leading to misleading conclusions. When calculating range, variance, and standard deviation, it is essential to consider how outliers may affect these statistics, as they can inflate the measures of spread.