For each of the following data sets, decide which has the higher standard deviation (set 1 or set 2), if any, without doing any computation. Explain the rationale behind your choice. Then, verify your choice by computing the standard deviation by hand.

Verified step by step guidance

1. Step 1: Understand the concept of standard deviation. Standard deviation measures the spread or variability of a data set around its mean. A higher standard deviation means the data points are more spread out from the mean, while a lower standard deviation means they are closer to the mean.
2. Step 2: Compare the data sets qualitatively without calculation. Look at the range (difference between the maximum and minimum values) and how the values are distributed: - For (a), Set 1 has values 4, 6, 7, 8, 10 and Set 2 has 4, 7, 7, 7, 10. Set 2 has repeated values clustered around 7, so it likely has less spread. - For (b), Set 1 has values 4, 8, 9, 10, 15 and Set 2 has 40, 80, 90, 100, 150. Set 2 has much larger values and a wider range, so it likely has a higher standard deviation. - For (c), Set 1 has 3, 6, 8, 10, 12, 15 and Set 2 has 93, 96, 98, 100, 102, 105. Both have similar ranges, but Set 1's values are more spread out, while Set 2's values are closer together.
3. Step 3: Calculate the mean for each data set using the formula: \[\text{Mean} = \frac{\sum x_i}{n}\] where \(x_i\) are the data points and \(n\) is the number of points.
4. Step 4: Calculate the variance for each data set using the formula: \[\text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{n}\] This measures the average squared deviation from the mean.
5. Step 5: Calculate the standard deviation by taking the square root of the variance: \[\text{Standard Deviation} = \sqrt{\text{Variance}}\] Compare the computed standard deviations to verify which set has the higher spread.

Key Concepts

Standard Deviation

Standard deviation measures the amount of variation or dispersion in a data set. A higher standard deviation indicates that data points are spread out further from the mean, while a lower standard deviation means they are closer to the mean. It is calculated as the square root of the average squared deviations from the mean.

Range and Data Spread

The range, the difference between the maximum and minimum values, gives a quick sense of data spread. Generally, a larger range suggests a higher standard deviation, but the distribution of values also matters. Clusters or repeated values can reduce variability even if the range is large.

Effect of Repeated Values on Variability

Repeated or identical values in a data set reduce variability because they contribute less to the overall spread around the mean. For example, a set with many repeated middle values will have a lower standard deviation than a set with more evenly spread values, even if their ranges are similar.